1(* ::Package:: *)
2
3(* ::Title:: *)
4(*Integration Problems Involving Two Hyperbolic Functions*)
5
6
7(* ::Subsection::Closed:: *)
8(*Integrands of the form Hyper[a+b x]^m Hyper[a+b x]^n*)
9
10
11(* ::Subsubsection::Closed:: *)
12(*Integrands of the form Cosh[a+b x]^m Sinh[a+b x]^n*)
13
14
15(* Integrands of the form Cosh[a+b*x]^m*Sinh[a+b*x]^n where m and n are positive integers *)
16{Cosh[a + b*x]*Sinh[a + b*x], x, 2, Sinh[a + b*x]^2/(2*b)}
17{Cosh[a + b*x]*Sinh[a + b*x]^n, x, 2, Sinh[a + b*x]^(1 + n)/(b*(1 + n))}
18{Cosh[a + b*x]^3*Sinh[a + b*x]^n, x, 3, Sinh[a + b*x]^(1 + n)/(b*(1 + n)) + Sinh[a + b*x]^(3 + n)/(b*(3 + n))}
19{Cosh[a + b*x]^5*Sinh[a + b*x]^n, x, 3, Sinh[a + b*x]^(1 + n)/(b*(1 + n)) + (2*Sinh[a + b*x]^(3 + n))/(b*(3 + n)) + Sinh[a + b*x]^(5 + n)/(b*(5 + n))}
20
21{Cosh[a + b*x]^m*Sinh[a + b*x], x, 2, Cosh[a + b*x]^(1 + m)/(b*(1 + m))}
22{Cosh[a + b*x]^m*Sinh[a + b*x]^3, x, 3, -(Cosh[a + b*x]^(1 + m)/(b*(1 + m))) + Cosh[a + b*x]^(3 + m)/(b*(3 + m))}
23{Cosh[a + b*x]^m*Sinh[a + b*x]^5, x, 3, Cosh[a + b*x]^(1 + m)/(b*(1 + m)) - (2*Cosh[a + b*x]^(3 + m))/(b*(3 + m)) + Cosh[a + b*x]^(5 + m)/(b*(5 + m))}
24
25{Cosh[a + b*x]^2*Sinh[a + b*x]^2, x, 2, -(x/8) - (Cosh[a + b*x]*Sinh[a + b*x])/(8*b) + (Cosh[a + b*x]^3*Sinh[a + b*x])/(4*b)}
26{Cosh[a + b*x]^2*Sinh[a + b*x]^4, x, 3, x/16 + (Cosh[a + b*x]*Sinh[a + b*x])/(16*b) - (Cosh[a + b*x]^3*Sinh[a + b*x])/(8*b) + (Cosh[a + b*x]^3*Sinh[a + b*x]^3)/(6*b)}
27{Cosh[a + b*x]^2*Sinh[a + b*x]^6, x, 4, -((5*x)/128) - (5*Cosh[a + b*x]*Sinh[a + b*x])/(128*b) + (5*Cosh[a + b*x]^3*Sinh[a + b*x])/(64*b) - (5*Cosh[a + b*x]^3*Sinh[a + b*x]^3)/(48*b) + (Cosh[a + b*x]^3*Sinh[a + b*x]^5)/(8*b)}
28
29{Cosh[a + b*x]^4*Sinh[a + b*x]^2, x, 3, -(x/16) - (Cosh[a + b*x]*Sinh[a + b*x])/(16*b) - (Cosh[a + b*x]^3*Sinh[a + b*x])/(24*b) + (Cosh[a + b*x]^5*Sinh[a + b*x])/(6*b)}
30{Cosh[a + b*x]^4*Sinh[a + b*x]^4, x, 4, (3*x)/128 + (3*Cosh[a + b*x]*Sinh[a + b*x])/(128*b) + (Cosh[a + b*x]^3*Sinh[a + b*x])/(64*b) - (Cosh[a + b*x]^5*Sinh[a + b*x])/(16*b) + (Cosh[a + b*x]^5*Sinh[a + b*x]^3)/(8*b)}
31{Cosh[a + b*x]^4*Sinh[a + b*x]^6, x, 5, -((3*x)/256) - (3*Cosh[a + b*x]*Sinh[a + b*x])/(256*b) - (Cosh[a + b*x]^3*Sinh[a + b*x])/(128*b) + (Cosh[a + b*x]^5*Sinh[a + b*x])/(32*b) - (Cosh[a + b*x]^5*Sinh[a + b*x]^3)/(16*b) + (Cosh[a + b*x]^5*Sinh[a + b*x]^5)/(10*b)}
32
33{Cosh[a + b*x]^6*Sinh[a + b*x]^2, x, 4, -((5*x)/128) - (5*Cosh[a + b*x]*Sinh[a + b*x])/(128*b) - (5*Cosh[a + b*x]^3*Sinh[a + b*x])/(192*b) - (Cosh[a + b*x]^5*Sinh[a + b*x])/(48*b) + (Cosh[a + b*x]^7*Sinh[a + b*x])/(8*b)}
34{Cosh[a + b*x]^6*Sinh[a + b*x]^4, x, 5, (3*x)/256 + (3*Cosh[a + b*x]*Sinh[a + b*x])/(256*b) + (Cosh[a + b*x]^3*Sinh[a + b*x])/(128*b) + (Cosh[a + b*x]^5*Sinh[a + b*x])/(160*b) - (3*Cosh[a + b*x]^7*Sinh[a + b*x])/(80*b) + (Cosh[a + b*x]^7*Sinh[a + b*x]^3)/(10*b)}
35{Cosh[a + b*x]^6*Sinh[a + b*x]^6, x, 6, -((5*x)/1024) - (5*Cosh[a + b*x]*Sinh[a + b*x])/(1024*b) - (5*Cosh[a + b*x]^3*Sinh[a + b*x])/(1536*b) - (Cosh[a + b*x]^5*Sinh[a + b*x])/(384*b) + (Cosh[a + b*x]^7*Sinh[a + b*x])/(64*b) - (Cosh[a + b*x]^7*Sinh[a + b*x]^3)/(24*b) + (Cosh[a + b*x]^7*Sinh[a + b*x]^5)/(12*b)}
36
37
38(* Integrands of the form Csch[a+b*x]^m*Sech[a+b*x]^n where m and n are positive integers *)
39{Csch[a + b*x]*Sech[a + b*x], x, 1, Log[Tanh[a + b*x]]/b}
40{Csch[a + b*x]*Sech[a + b*x]^2, x, 2, -(ArcCoth[Cosh[a + b*x]]/b) + Sech[a + b*x]/b}
41{Csch[a + b*x]*Sech[a + b*x]^3, x, 3, Log[Tanh[a + b*x]]/b - Tanh[a + b*x]^2/(2*b)}
42{Csch[a + b*x]*Sech[a + b*x]^4, x, 3, -(ArcCoth[Cosh[a + b*x]]/b) + Sech[a + b*x]/b + Sech[a + b*x]^3/(3*b)}
43{Csch[a + b*x]*Sech[a + b*x]^5, x, 3, Log[Tanh[a + b*x]]/b - Tanh[a + b*x]^2/b + Tanh[a + b*x]^4/(4*b)}
44
45{Csch[a + b*x]^2*Sech[a + b*x], x, 2, -(ArcTan[Sinh[a + b*x]]/b) - Csch[a + b*x]/b}
46{Csch[a + b*x]^2*Sech[a + b*x]^2, x, 3, -(Coth[a + b*x]/b) - Tanh[a + b*x]/b}
47{Csch[a + b*x]^2*Sech[a + b*x]^3, x, 3, -((3*ArcTan[Sinh[a + b*x]])/(2*b)) - (3*Csch[a + b*x])/(2*b) + (Csch[a + b*x]*Sech[a + b*x]^2)/(2*b)}
48{Csch[a + b*x]^2*Sech[a + b*x]^4, x, 3, -(Coth[a + b*x]/b) - (2*Tanh[a + b*x])/b + Tanh[a + b*x]^3/(3*b)}
49{Csch[a + b*x]^2*Sech[a + b*x]^5, x, 4, -((15*ArcTan[Sinh[a + b*x]])/(8*b)) - (15*Csch[a + b*x])/(8*b) + (5*Csch[a + b*x]*Sech[a + b*x]^2)/(8*b) + (Csch[a + b*x]*Sech[a + b*x]^4)/(4*b)}
50
51{Csch[a + b*x]^3*Sech[a + b*x], x, 3, -(Coth[a + b*x]^2/(2*b)) + Log[Coth[a + b*x]]/b}
52{Csch[a + b*x]^3*Sech[a + b*x]^2, x, 3, (3*ArcCoth[Cosh[a + b*x]])/(2*b) - (3*Sech[a + b*x])/(2*b) - (Csch[a + b*x]^2*Sech[a + b*x])/(2*b)}
53{Csch[a + b*x]^3*Sech[a + b*x]^3, x, 3, -(Coth[a + b*x]^2/(2*b)) - (2*Log[Tanh[a + b*x]])/b + Tanh[a + b*x]^2/(2*b)}
54{Csch[a + b*x]^3*Sech[a + b*x]^4, x, 4, (5*ArcCoth[Cosh[a + b*x]])/(2*b) - (5*Sech[a + b*x])/(2*b) - (5*Sech[a + b*x]^3)/(6*b) - (Csch[a + b*x]^2*Sech[a + b*x]^3)/(2*b)}
55{Csch[a + b*x]^3*Sech[a + b*x]^5, x, 3, -(Coth[a + b*x]^2/(2*b)) - (3*Log[Tanh[a + b*x]])/b + (3*Tanh[a + b*x]^2)/(2*b) - Tanh[a + b*x]^4/(4*b)}
56
57{Csch[a + b*x]^4*Sech[a + b*x], x, 3, ArcTan[Sinh[a + b*x]]/b + Csch[a + b*x]/b - Csch[a + b*x]^3/(3*b)}
58{Csch[a + b*x]^4*Sech[a + b*x]^2, x, 3, (2*Coth[a + b*x])/b - Coth[a + b*x]^3/(3*b) + Tanh[a + b*x]/b}
59{Csch[a + b*x]^4*Sech[a + b*x]^3, x, 4, (5*ArcTan[Sinh[a + b*x]])/(2*b) + (5*Csch[a + b*x])/(2*b) - (5*Csch[a + b*x]^3)/(6*b) + (Csch[a + b*x]^3*Sech[a + b*x]^2)/(2*b)}
60{Csch[a + b*x]^4*Sech[a + b*x]^4, x, 3, (3*Coth[a + b*x])/b - Coth[a + b*x]^3/(3*b) + (3*Tanh[a + b*x])/b - Tanh[a + b*x]^3/(3*b)}
61{Csch[a + b*x]^4*Sech[a + b*x]^5, x, 5, (35*ArcTan[Sinh[a + b*x]])/(8*b) + (35*Csch[a + b*x])/(8*b) - (35*Csch[a + b*x]^3)/(24*b) + (7*Csch[a + b*x]^3*Sech[a + b*x]^2)/(8*b) + (Csch[a + b*x]^3*Sech[a + b*x]^4)/(4*b)}
62
63{Csch[a + b*x]^5*Sech[a + b*x], x, 3, Coth[a + b*x]^2/b - Coth[a + b*x]^4/(4*b) - Log[Coth[a + b*x]]/b}
64{Csch[a + b*x]^5*Sech[a + b*x]^2, x, 4, -((15*ArcCoth[Cosh[a + b*x]])/(8*b)) + (15*Sech[a + b*x])/(8*b) + (5*Csch[a + b*x]^2*Sech[a + b*x])/(8*b) - (Csch[a + b*x]^4*Sech[a + b*x])/(4*b)}
65{Csch[a + b*x]^5*Sech[a + b*x]^3, x, 3, (3*Coth[a + b*x]^2)/(2*b) - Coth[a + b*x]^4/(4*b) - (3*Log[Coth[a + b*x]])/b - Tanh[a + b*x]^2/(2*b)}
66{Csch[a + b*x]^5*Sech[a + b*x]^4, x, 5, -((35*ArcCoth[Cosh[a + b*x]])/(8*b)) + (35*Sech[a + b*x])/(8*b) + (35*Sech[a + b*x]^3)/(24*b) + (7*Csch[a + b*x]^2*Sech[a + b*x]^3)/(8*b) - (Csch[a + b*x]^4*Sech[a + b*x]^3)/(4*b)}
67{Csch[a + b*x]^5*Sech[a + b*x]^5, x, 3, (2*Coth[a + b*x]^2)/b - Coth[a + b*x]^4/(4*b) + (6*Log[Tanh[a + b*x]])/b - (2*Tanh[a + b*x]^2)/b + Tanh[a + b*x]^4/(4*b)}
68
69
70{Sqrt[Sinh[a + b*x]]/Sqrt[Cosh[a + b*x]], x, 4, -(ArcTan[Sqrt[Sinh[a + b*x]]/Sqrt[Cosh[a + b*x]]]/b) + ArcTanh[Sqrt[Sinh[a + b*x]]/Sqrt[Cosh[a + b*x]]]/b}
71{Sqrt[Cosh[a + b*x]]/Sqrt[Sinh[a + b*x]], x, 4, -(ArcTan[Sqrt[Cosh[a + b*x]]/Sqrt[Sinh[a + b*x]]]/b) + ArcTanh[Sqrt[Cosh[a + b*x]]/Sqrt[Sinh[a + b*x]]]/b}
72
73{Sinh[a + b*x]^(1/3)/Cosh[a + b*x]^(1/3), x, 6, -((Sqrt[3]*ArcTan[(1 + (2*Sinh[a + b*x]^(2/3))/Cosh[a + b*x]^(2/3))/Sqrt[3]])/(2*b)) - Log[1 - Sinh[a + b*x]^(2/3)/Cosh[a + b*x]^(2/3)]/(2*b) + Log[1 + Sinh[a + b*x]^(2/3)/Cosh[a + b*x]^(2/3) + Sinh[a + b*x]^(4/3)/Cosh[a + b*x]^(4/3)]/(4*b)}
74{Cosh[a + b*x]^(1/3)/Sinh[a + b*x]^(1/3), x, 6, -((Sqrt[3]*ArcTan[(1 + (2*Cosh[a + b*x]^(2/3))/Sinh[a + b*x]^(2/3))/Sqrt[3]])/(2*b)) - Log[1 - Cosh[a + b*x]^(2/3)/Sinh[a + b*x]^(2/3)]/(2*b) + Log[1 + Cosh[a + b*x]^(4/3)/Sinh[a + b*x]^(4/3) + Cosh[a + b*x]^(2/3)/Sinh[a + b*x]^(2/3)]/(4*b)}
75
76{Cosh[x]^(2/3)/Sinh[x]^(8/3), x, 1, -((3*Cosh[x]^(5/3))/(5*Sinh[x]^(5/3)))}
77{Sinh[x]^(2/3)/Cosh[x]^(8/3), x, 1, (3*Sinh[x]^(5/3))/(5*Cosh[x]^(5/3))}
78
79{Cosh[x]*Csch[x]^(7/3), x, 2, (-3*Csch[x]^(4/3))/4}
80
81
82(* ::Subsubsection::Closed:: *)
83(*Integrands of the form Hyper[a+b x]^m Tanh[a+b x]^n*)
84
85
86(* Integrands of the form Sinh[a+b*x]^m*Tanh[a+b*x]^n where m and n are positive integers *)
87{Sinh[a + b*x]*Tanh[a + b*x], x, 2, -(ArcTan[Sinh[a + b*x]]/b) + Sinh[a + b*x]/b}
88{Sinh[a + b*x]*Tanh[a + b*x]^2, x, 3, Cosh[a + b*x]/b + Sech[a + b*x]/b}
89{Sinh[a + b*x]*Tanh[a + b*x]^3, x, 3, -((3*ArcTan[Sinh[a + b*x]])/(2*b)) + (3*Sinh[a + b*x])/(2*b) - (Sinh[a + b*x]*Tanh[a + b*x]^2)/(2*b)}
90{Sinh[a + b*x]*Tanh[a + b*x]^4, x, 3, Cosh[a + b*x]/b + (2*Sech[a + b*x])/b - Sech[a + b*x]^3/(3*b)}
91
92{Sinh[a + b*x]^2*Tanh[a + b*x], x, 3, Cosh[a + b*x]^2/(2*b) - Log[Cosh[a + b*x]]/b}
93{Sinh[a + b*x]^2*Tanh[a + b*x]^2, x, 2, -((3*x)/2) + (3*Tanh[a + b*x])/(2*b) + (Sinh[a + b*x]^2*Tanh[a + b*x])/(2*b)}
94{Sinh[a + b*x]^2*Tanh[a + b*x]^3, x, 3, Cosh[a + b*x]^2/(2*b) - (2*Log[Cosh[a + b*x]])/b - Sech[a + b*x]^2/(2*b)}
95
96{Sinh[a + b*x]^3*Tanh[a + b*x], x, 3, ArcTan[Sinh[a + b*x]]/b - Sinh[a + b*x]/b + Sinh[a + b*x]^3/(3*b)}
97{Sinh[a + b*x]^3*Tanh[a + b*x]^2, x, 3, -((2*Cosh[a + b*x])/b) + Cosh[a + b*x]^3/(3*b) - Sech[a + b*x]/b}
98{Sinh[a + b*x]^3*Tanh[a + b*x]^3, x, 4, (5*ArcTan[Sinh[a + b*x]])/(2*b) - (5*Sinh[a + b*x])/(2*b) + (5*Sinh[a + b*x]*Tanh[a + b*x]^2)/(6*b) + (Sinh[a + b*x]^3*Tanh[a + b*x]^2)/(3*b)}
99
100{Sinh[a + b*x]^4*Tanh[a + b*x], x, 3, -(Cosh[a + b*x]^2/b) + Cosh[a + b*x]^4/(4*b) + Log[Cosh[a + b*x]]/b}
101
102
103(* Integrands of the form Sech[a+b*x]^m*Tanh[a+b*x]^n where m and n are positive integers *)
104{Sech[a + b*x]*Tanh[a + b*x], x, 1, -Sech[a + b*x]/b}
105{Sech[a + b*x]^2*Tanh[a + b*x], x, 1, -Sech[a + b*x]^2/(2*b)}
106{Sech[a + b*x]^n*Tanh[a + b*x], x, 2, -Sech[a + b*x]^n/(b*n)}
107
108{Sech[a + b*x]^2*Tanh[a + b*x]^2, x, 2, Tanh[a + b*x]^3/(3*b)}
109{Sech[a + b*x]^2*Tanh[a + b*x]^3, x, 2, Tanh[a + b*x]^4/(4*b)}
110{Sech[a + b*x]^2*Tanh[a + b*x]^n, x, 2, Tanh[a + b*x]^(1 + n)/(b*(1 + n))}
111
112{Sech[a + b*x]*Tanh[a + b*x]^3, x, 2, -(Sech[a + b*x]/b) + Sech[a + b*x]^3/(3*b)}
113{Sech[a + b*x]^3*Tanh[a + b*x]^3, x, 3, -(Sech[a + b*x]^3/(3*b)) + Sech[a + b*x]^5/(5*b)}
114{Sech[a + b*x]^n*Tanh[a + b*x]^3, x, 5, -(Sech[a + b*x]^n/(b*n)) + Sech[a + b*x]^(2 + n)/(b*(2 + n))}
115
116{Sech[a + b*x]^4*Tanh[a + b*x]^2, x, 3, Tanh[a + b*x]^3/(3*b) - Tanh[a + b*x]^5/(5*b)}
117{Sech[a + b*x]^4*Sqrt[Tanh[a + b*x]], x, 3, (2*Tanh[a + b*x]^(3/2))/(3*b) - (2*Tanh[a + b*x]^(7/2))/(7*b)}
118{Sech[a + b*x]^4*Tanh[a + b*x]^n, x, 3, Tanh[a + b*x]^(1 + n)/(b*(1 + n)) - Tanh[a + b*x]^(3 + n)/(b*(3 + n))}
119
120{Sech[a + b*x]*Tanh[a + b*x]^2, x, 2, ArcTan[Sinh[a + b*x]]/(2*b) - (Sech[a + b*x]*Tanh[a + b*x])/(2*b)}
121{Sech[a + b*x]*Tanh[a + b*x]^4, x, 3, (3*ArcTan[Sinh[a + b*x]])/(8*b) - (3*Sech[a + b*x]*Tanh[a + b*x])/(8*b) - (Sech[a + b*x]*Tanh[a + b*x]^3)/(4*b)}
122
123{Sech[a + b*x]^3*Tanh[a + b*x]^2, x, 3, ArcTan[Sinh[a + b*x]]/(8*b) - (Sech[a + b*x]*Tanh[a + b*x])/(8*b) + (Sech[a + b*x]*Tanh[a + b*x]^3)/(4*b)}
124
125{Sech[x]*Tanh[x]^5, x, 3, -Sech[x] + (2*Sech[x]^3)/3 - Sech[x]^5/5}
126{Sech[x]^7*Tanh[x]^5, x, 3, (-(1/7))*Sech[x]^7 + (2*Sech[x]^9)/9 - Sech[x]^11/11}
127{Sech[x]^3*Tanh[x]^4, x, 4, (1/16)*ArcTan[Sinh[x]] - (1/16)*Sech[x]*Tanh[x] - (1/24)*Sech[x]*Tanh[x]^3 + (1/6)*Sech[x]*Tanh[x]^5}
128{Sech[x]^5*Tanh[x]^2, x, 4, (1/16)*ArcTan[Sinh[x]] - (1/16)*Sech[x]*Tanh[x] + (1/8)*Sech[x]*Tanh[x]^3 + (1/6)*Sech[x]^3*Tanh[x]^3}
129{Sech[x]^8*Tanh[x]^6, x, 3, Tanh[x]^7/7 - Tanh[x]^9/3 + (3*Tanh[x]^11)/11 - Tanh[x]^13/13}
130
131
132(* ::Subsubsection::Closed:: *)
133(*Integrands of the form Hyper[a+b x]^m Coth[a+b x]^n*)
134
135
136(* Integrands of the form Cosh[a+b*x]^m*Coth[a+b*x]^n where m and n are positive integers *)
137{Cosh[a + b*x]*Coth[a + b*x], x, 2, -(ArcCoth[Cosh[a + b*x]]/b) + Cosh[a + b*x]/b}
138{Cosh[a + b*x]*Coth[a + b*x]^2, x, 3, -(Csch[a + b*x]/b) + Sinh[a + b*x]/b}
139{Cosh[a + b*x]*Coth[a + b*x]^3, x, 3, -((3*ArcCoth[Cosh[a + b*x]])/(2*b)) + (3*Cosh[a + b*x])/(2*b) - (Cosh[a + b*x]*Coth[a + b*x]^2)/(2*b)}
140{Cosh[a + b*x]*Coth[a + b*x]^4, x, 3, -((2*Csch[a + b*x])/b) - Csch[a + b*x]^3/(3*b) + Sinh[a + b*x]/b}
141
142{Cosh[a + b*x]^2*Coth[a + b*x], x, 3, Log[Sinh[a + b*x]]/b + Sinh[a + b*x]^2/(2*b)}
143{Cosh[a + b*x]^2*Coth[a + b*x]^2, x, 2, (3*x)/2 - (3*Coth[a + b*x])/(2*b) + (Cosh[a + b*x]^2*Coth[a + b*x])/(2*b)}
144{Cosh[a + b*x]^2*Coth[a + b*x]^3, x, 3, -(Csch[a + b*x]^2/(2*b)) + (2*Log[Sinh[a + b*x]])/b + Sinh[a + b*x]^2/(2*b)}
145
146{Cosh[a + b*x]^3*Coth[a + b*x], x, 3, -(ArcCoth[Cosh[a + b*x]]/b) + Cosh[a + b*x]/b + Cosh[a + b*x]^3/(3*b)}
147{Cosh[a + b*x]^3*Coth[a + b*x]^2, x, 3, -(Csch[a + b*x]/b) + (2*Sinh[a + b*x])/b + Sinh[a + b*x]^3/(3*b)}
148{Cosh[a + b*x]^3*Coth[a + b*x]^3, x, 4, -((5*ArcCoth[Cosh[a + b*x]])/(2*b)) + (5*Cosh[a + b*x])/(2*b) - (5*Cosh[a + b*x]*Coth[a + b*x]^2)/(6*b) + (Cosh[a + b*x]^3*Coth[a + b*x]^2)/(3*b)}
149
150{Cosh[a + b*x]^4*Coth[a + b*x], x, 3, Log[Sinh[a + b*x]]/b + Sinh[a + b*x]^2/b + Sinh[a + b*x]^4/(4*b)}
151
152
153(* Integrands of the form Coth[a+b*x]^m*Csch[a+b*x]^n where m and n are positive integers *)
154{Coth[a + b*x]*Csch[a + b*x], x, 1, -(Csch[a + b*x]/b)}
155{Coth[a + b*x]*Csch[a + b*x]^2, x, 1, -Csch[a + b*x]^2/(2*b)}
156{Coth[a + b*x]*Csch[a + b*x]^n, x, 2, -Csch[a + b*x]^n/(b*n)}
157
158{Coth[a + b*x]^2*Csch[a + b*x]^2, x, 2, -Coth[a + b*x]^3/(3*b)}
159{Coth[a + b*x]^3*Csch[a + b*x]^2, x, 2, -Coth[a + b*x]^4/(4*b)}
160{Coth[a + b*x]^n*Csch[a + b*x]^2, x, 2, -Coth[a + b*x]^(1 + n)/(b*(1 + n))}
161
162{Coth[a + b*x]^3*Csch[a + b*x], x, 2, -(Csch[a + b*x]/b) - Csch[a + b*x]^3/(3*b)}
163{Coth[a + b*x]^3*Csch[a + b*x]^3, x, 3, -(Csch[a + b*x]^3/(3*b)) - Csch[a + b*x]^5/(5*b)}
164{Coth[a + b*x]^3*Csch[a + b*x]^n, x, 5, -(Csch[a + b*x]^n/(b*n)) - Csch[a + b*x]^(2 + n)/(b*(2 + n))}
165
166{Coth[a + b*x]^2*Csch[a + b*x], x, 2, -(ArcCoth[Cosh[a + b*x]]/(2*b)) - (Coth[a + b*x]*Csch[a + b*x])/(2*b)}
167{Coth[a + b*x]^2*Csch[a + b*x]^3, x, 3, ArcCoth[Cosh[a + b*x]]/(8*b) + (Coth[a + b*x]*Csch[a + b*x])/(8*b) - (Coth[a + b*x]^3*Csch[a + b*x])/(4*b)}
168
169{Coth[a + b*x]^4*Csch[a + b*x], x, 3, -((3*ArcCoth[Cosh[a + b*x]])/(8*b)) - (3*Coth[a + b*x]*Csch[a + b*x])/(8*b) - (Coth[a + b*x]^3*Csch[a + b*x])/(4*b)}
170
171{Coth[x]^2*Csch[x]^4, x, 3, Coth[x]^3/3 - Coth[x]^5/5}
172{Coth[x]^3*Csch[x]^4, x, 3, Coth[x]^4/4 - Coth[x]^6/6}
173{Coth[x]^n*Csch[x]^4, x, 3, Coth[x]^(1 + n)/(1 + n) - Coth[x]^(3 + n)/(3 + n)}
174
175{Coth[x]^4*Csch[x]^3, x, 4, (1/16)*ArcCoth[Cosh[x]] + (1/16)*Coth[x]*Csch[x] + (1/24)*Coth[x]^3*Csch[x] - (1/6)*Coth[x]^5*Csch[x]}
176{Coth[x]^4*Csch[x]^6, x, 3, (-(1/5))*Coth[x]^5 + (2*Coth[x]^7)/7 - Coth[x]^9/9}
177{Coth[6*x]^5*Csch[6*x], x, 3, (-(1/6))*Csch[6*x] - (1/9)*Csch[6*x]^3 - (1/30)*Csch[6*x]^5}
178{Coth[x]^7*Csch[x]^3, x, 3, (-(1/3))*Csch[x]^3 - (3*Csch[x]^5)/5 - (3*Csch[x]^7)/7 - Csch[x]^9/9}
179
180
181(* ::Subsection::Closed:: *)
182(*Integrands of the form Hyper[a+b x]^m Hyper[c+d x]^n*)
183
184
185(* Integrands of the form Sinh[a+b*x]^m*Sinh[c+d*x]^n where m and n are positive integers *)
186{Sinh[a + b*x]*Sinh[c + d*x], x, 3, -(Sinh[a - c + (b - d)*x]/(2*(b - d))) + Sinh[a + c + (b + d)*x]/(2*(b + d))}
187{Sinh[a + b*x]*Sinh[c + d*x]^2, x, 5, -(Cosh[a + b*x]/(2*b)) + Cosh[a - 2*c + (b - 2*d)*x]/(4*(b - 2*d)) + Cosh[a + 2*c + (b + 2*d)*x]/(4*(b + 2*d))}
188{Sinh[a + b*x]*Sinh[c + d*x]^3, x, 6, -(Sinh[a - 3*c + (b - 3*d)*x]/(8*(b - 3*d))) + (3*Sinh[a - c + (b - d)*x])/(8*(b - d)) - (3*Sinh[a + c + (b + d)*x])/(8*(b + d)) + Sinh[a + 3*c + (b + 3*d)*x]/(8*(b + 3*d))}
189
190{Sinh[a + b*x]^2*Sinh[c + d*x]^2, x, 6, x/4 - Sinh[2*a + 2*b*x]/(8*b) + Sinh[2*a - 2*c + 2*(b - d)*x]/(16*(b - d)) - Sinh[2*c + 2*d*x]/(8*d) + Sinh[2*a + 2*c + 2*(b + d)*x]/(16*(b + d))}
191{Sinh[a + b*x]^2*Sinh[c + d*x]^3, x, 8, -(Cosh[2*a - 3*c + (2*b - 3*d)*x]/(16*(2*b - 3*d))) + (3*Cosh[2*a - c + (2*b - d)*x])/(16*(2*b - d)) + (3*Cosh[c + d*x])/(8*d) - Cosh[3*c + 3*d*x]/(24*d) - (3*Cosh[2*a + c + (2*b + d)*x])/(16*(2*b + d)) + Cosh[2*a + 3*c + (2*b + 3*d)*x]/(16*(2*b + 3*d))}
192
193{Sinh[a + b*x]^3*Sinh[c + d*x]^3, x, 10, (3*Sinh[a - 3*c + (b - 3*d)*x])/(32*(b - 3*d)) - (9*Sinh[a - c + (b - d)*x])/(32*(b - d)) - Sinh[3*a - 3*c + 3*(b - d)*x]/(96*(b - d)) + (3*Sinh[3*a - c + (3*b - d)*x])/(32*(3*b - d)) + (9*Sinh[a + c + (b + d)*x])/(32*(b + d)) + Sinh[3*a + 3*c + 3*(b + d)*x]/(96*(b + d)) - (3*Sinh[3*a + c + (3*b + d)*x])/(32*(3*b + d)) - (3*Sinh[a + 3*c + (b + 3*d)*x])/(32*(b + 3*d))}
194
195
196(* Integrands of the form Cosh[a+b*x]^m*Cosh[c+d*x]^n where m and n are positive integers *)
197{Cosh[a + b*x]*Cosh[c + d*x], x, 3, Sinh[a - c + (b - d)*x]/(2*(b - d)) + Sinh[a + c + (b + d)*x]/(2*(b + d))}
198{Cosh[a + b*x]*Cosh[c + d*x]^2, x, 5, Sinh[a + b*x]/(2*b) + Sinh[a - 2*c + (b - 2*d)*x]/(4*(b - 2*d)) + Sinh[a + 2*c + (b + 2*d)*x]/(4*(b + 2*d))}
199{Cosh[a + b*x]*Cosh[c + d*x]^3, x, 6, Sinh[a - 3*c + (b - 3*d)*x]/(8*(b - 3*d)) + (3*Sinh[a - c + (b - d)*x])/(8*(b - d)) + (3*Sinh[a + c + (b + d)*x])/(8*(b + d)) + Sinh[a + 3*c + (b + 3*d)*x]/(8*(b + 3*d))}
200
201{Cosh[a + b*x]^2*Cosh[c + d*x]^2, x, 6, x/4 + Sinh[2*a + 2*b*x]/(8*b) + Sinh[2*a - 2*c + 2*(b - d)*x]/(16*(b - d)) + Sinh[2*c + 2*d*x]/(8*d) + Sinh[2*a + 2*c + 2*(b + d)*x]/(16*(b + d))}
202{Cosh[a + b*x]^2*Cosh[c + d*x]^3, x, 8, Sinh[2*a - 3*c + (2*b - 3*d)*x]/(16*(2*b - 3*d)) + (3*Sinh[2*a - c + (2*b - d)*x])/(16*(2*b - d)) + (3*Sinh[c + d*x])/(8*d) + Sinh[3*c + 3*d*x]/(24*d) + (3*Sinh[2*a + c + (2*b + d)*x])/(16*(2*b + d)) + Sinh[2*a + 3*c + (2*b + 3*d)*x]/(16*(2*b + 3*d))}
203
204{Cosh[a + b*x]^3*Cosh[c + d*x]^3, x, 10, (3*Sinh[a - 3*c + (b - 3*d)*x])/(32*(b - 3*d)) + (9*Sinh[a - c + (b - d)*x])/(32*(b - d)) + Sinh[3*a - 3*c + 3*(b - d)*x]/(96*(b - d)) + (3*Sinh[3*a - c + (3*b - d)*x])/(32*(3*b - d)) + (9*Sinh[a + c + (b + d)*x])/(32*(b + d)) + Sinh[3*a + 3*c + 3*(b + d)*x]/(96*(b + d)) + (3*Sinh[3*a + c + (3*b + d)*x])/(32*(3*b + d)) + (3*Sinh[a + 3*c + (b + 3*d)*x])/(32*(b + 3*d))}
205
206
207(* Integrands of the form Sinh[a+b*x]^m*Cosh[c+d*x]^n where m and n are positive integers *)
208{Sinh[a + b*x]*Cosh[c + d*x], x, 3, Cosh[a - c + (b - d)*x]/(2*(b - d)) + Cosh[a + c + (b + d)*x]/(2*(b + d))}
209{Sinh[a + b*x]*Cosh[c + d*x]^2, x, 5, Cosh[a + b*x]/(2*b) + Cosh[a - 2*c + (b - 2*d)*x]/(4*(b - 2*d)) + Cosh[a + 2*c + (b + 2*d)*x]/(4*(b + 2*d))}
210{Sinh[a + b*x]*Cosh[c + d*x]^3, x, 6, Cosh[a - 3*c + (b - 3*d)*x]/(8*(b - 3*d)) + (3*Cosh[a - c + (b - d)*x])/(8*(b - d)) + (3*Cosh[a + c + (b + d)*x])/(8*(b + d)) + Cosh[a + 3*c + (b + 3*d)*x]/(8*(b + 3*d))}
211
212{Sinh[a + b*x]^2*Cosh[c + d*x], x, 5, Sinh[2*a - c + (2*b - d)*x]/(4*(2*b - d)) - Sinh[c + d*x]/(2*d) + Sinh[2*a + c + (2*b + d)*x]/(4*(2*b + d))}
213{Sinh[a + b*x]^2*Cosh[c + d*x]^2, x, 6, -(x/4) + Sinh[2*a + 2*b*x]/(8*b) + Sinh[2*a - 2*c + 2*(b - d)*x]/(16*(b - d)) - Sinh[2*c + 2*d*x]/(8*d) + Sinh[2*a + 2*c + 2*(b + d)*x]/(16*(b + d))}
214{Sinh[a + b*x]^2*Cosh[c + d*x]^3, x, 8, Sinh[2*a - 3*c + (2*b - 3*d)*x]/(16*(2*b - 3*d)) + (3*Sinh[2*a - c + (2*b - d)*x])/(16*(2*b - d)) - (3*Sinh[c + d*x])/(8*d) - Sinh[3*c + 3*d*x]/(24*d) + (3*Sinh[2*a + c + (2*b + d)*x])/(16*(2*b + d)) + Sinh[2*a + 3*c + (2*b + 3*d)*x]/(16*(2*b + 3*d))}
215
216{Sinh[a + b*x]^3*Cosh[c + d*x], x, 6, -((3*Cosh[a - c + (b - d)*x])/(8*(b - d))) + Cosh[3*a - c + (3*b - d)*x]/(8*(3*b - d)) - (3*Cosh[a + c + (b + d)*x])/(8*(b + d)) + Cosh[3*a + c + (3*b + d)*x]/(8*(3*b + d))}
217{Sinh[a + b*x]^3*Cosh[c + d*x]^2, x, 8, -((3*Cosh[a + b*x])/(8*b)) + Cosh[3*a + 3*b*x]/(24*b) - (3*Cosh[a - 2*c + (b - 2*d)*x])/(16*(b - 2*d)) + Cosh[3*a - 2*c + (3*b - 2*d)*x]/(16*(3*b - 2*d)) - (3*Cosh[a + 2*c + (b + 2*d)*x])/(16*(b + 2*d)) + Cosh[3*a + 2*c + (3*b + 2*d)*x]/(16*(3*b + 2*d))}
218{Sinh[a + b*x]^3*Cosh[c + d*x]^3, x, 10, -((3*Cosh[a - 3*c + (b - 3*d)*x])/(32*(b - 3*d))) - (9*Cosh[a - c + (b - d)*x])/(32*(b - d)) + Cosh[3*a - 3*c + 3*(b - d)*x]/(96*(b - d)) + (3*Cosh[3*a - c + (3*b - d)*x])/(32*(3*b - d)) - (9*Cosh[a + c + (b + d)*x])/(32*(b + d)) + Cosh[3*a + 3*c + 3*(b + d)*x]/(96*(b + d)) + (3*Cosh[3*a + c + (3*b + d)*x])/(32*(3*b + d)) - (3*Cosh[a + 3*c + (b + 3*d)*x])/(32*(b + 3*d))}
219
220
221(* Integrands of the form Sinh[a+b*x]*Hyper[c+b*x]^n where n is a positive integer *)
222{Sinh[a + b*x]*Tanh[c + b*x], x, 3, -((ArcTan[Sinh[c + b*x]]*Cosh[a - c])/b) + Sinh[a + b*x]/b}
223{Sinh[a + b*x]*Tanh[c + b*x]^2, x, 5, Cosh[a + b*x]/b + (Cosh[a - c]*Sech[c + b*x])/b - (ArcTan[Sinh[c + b*x]]*Sinh[a - c])/b}
224{Sinh[a + b*x]*Tanh[c + b*x]^3, x, 8, -((3*ArcTan[Sinh[c + b*x]]*Cosh[a - c])/(2*b)) + (Sech[c + b*x]*Sinh[a - c])/b + Sinh[a + b*x]/b + (Cosh[a - c]*Sech[c + b*x]*Tanh[c + b*x])/(2*b)}
225
226{Sinh[a + b*x]*Coth[c + b*x], x, 3, -((ArcCoth[Cosh[c + b*x]]*Sinh[a - c])/b) + Sinh[a + b*x]/b}
227{Sinh[a + b*x]*Coth[c + b*x]^2, x, 5, -((ArcCoth[Cosh[c + b*x]]*Cosh[a - c])/b) + Cosh[a + b*x]/b - (Csch[c + b*x]*Sinh[a - c])/b}
228{Sinh[a + b*x]*Coth[c + b*x]^3, x, 8, -((Cosh[a - c]*Csch[c + b*x])/b) - (3*ArcCoth[Cosh[c + b*x]]*Sinh[a - c])/(2*b) - (Coth[c + b*x]*Csch[c + b*x]*Sinh[a - c])/(2*b) + Sinh[a + b*x]/b}
229
230{Sinh[a + b*x]*Sech[c + b*x], x, 3, (Cosh[a - c]*Log[Cosh[c + b*x]])/b + x*Sinh[a - c]}
231{Sinh[a + b*x]*Sech[c + b*x]^2, x, 3, -((Cosh[a - c]*Sech[c + b*x])/b) + (ArcTan[Sinh[c + b*x]]*Sinh[a - c])/b}
232{Sinh[a + b*x]*Sech[c + b*x]^3, x, 3, -((Cosh[a - c]*Sech[c + b*x]^2)/(2*b)) + (Sinh[a - c]*Tanh[c + b*x])/b}
233
234{Sinh[a + b*x]*Csch[c + b*x], x, 3, x*Cosh[a - c] + (Log[Sinh[c + b*x]]*Sinh[a - c])/b}
235{Sinh[a + b*x]*Csch[c + b*x]^2, x, 3, -((ArcCoth[Cosh[c + b*x]]*Cosh[a - c])/b) - (Csch[c + b*x]*Sinh[a - c])/b}
236{Sinh[a + b*x]*Csch[c + b*x]^3, x, 3, -((Cosh[a - c]*Coth[c + b*x])/b) - (Csch[c + b*x]^2*Sinh[a - c])/(2*b)}
237
238
239(* Integrands of the form Cosh[a+b*x]*Hyper[c+b*x]^n where n is a positive integer *)
240{Cosh[a + b*x]*Tanh[c + b*x], x, 3, Cosh[a + b*x]/b - (ArcTan[Sinh[c + b*x]]*Sinh[a - c])/b}
241{Cosh[a + b*x]*Tanh[c + b*x]^2, x, 5, -((ArcTan[Sinh[c + b*x]]*Cosh[a - c])/b) + (Sech[c + b*x]*Sinh[a - c])/b + Sinh[a + b*x]/b}
242{Cosh[a + b*x]*Tanh[c + b*x]^3, x, 8, Cosh[a + b*x]/b + (Cosh[a - c]*Sech[c + b*x])/b - (3*ArcTan[Sinh[c + b*x]]*Sinh[a - c])/(2*b) + (Sech[c + b*x]*Sinh[a - c]*Tanh[c + b*x])/(2*b)}
243
244{Cosh[a + b*x]*Coth[c + b*x], x, 3, -((ArcCoth[Cosh[c + b*x]]*Cosh[a - c])/b) + Cosh[a + b*x]/b}
245{Cosh[a + b*x]*Coth[c + b*x]^2, x, 5, -((Cosh[a - c]*Csch[c + b*x])/b) - (ArcCoth[Cosh[c + b*x]]*Sinh[a - c])/b + Sinh[a + b*x]/b}
246{Cosh[a + b*x]*Coth[c + b*x]^3, x, 8, -((3*ArcCoth[Cosh[c + b*x]]*Cosh[a - c])/(2*b)) + Cosh[a + b*x]/b - (Cosh[a - c]*Coth[c + b*x]*Csch[c + b*x])/(2*b) - (Csch[c + b*x]*Sinh[a - c])/b}
247
248{Cosh[a + b*x]*Sech[c + b*x], x, 3, x*Cosh[a - c] + (Log[Cosh[c + b*x]]*Sinh[a - c])/b}
249{Cosh[a + b*x]*Sech[c + b*x]^2, x, 3, (ArcTan[Sinh[c + b*x]]*Cosh[a - c])/b - (Sech[c + b*x]*Sinh[a - c])/b}
250{Cosh[a + b*x]*Sech[c + b*x]^3, x, 3, -((Sech[c + b*x]^2*Sinh[a - c])/(2*b)) + (Cosh[a - c]*Tanh[c + b*x])/b}
251
252{Cosh[a + b*x]*Csch[c + b*x], x, 3, (Cosh[a - c]*Log[Sinh[c + b*x]])/b + x*Sinh[a - c]}
253{Cosh[a + b*x]*Csch[c + b*x]^2, x, 3, -((Cosh[a - c]*Csch[c + b*x])/b) - (ArcCoth[Cosh[c + b*x]]*Sinh[a - c])/b}
254{Cosh[a + b*x]*Csch[c + b*x]^3, x, 3, -((Cosh[a - c]*Csch[c + b*x]^2)/(2*b)) - (Coth[c + b*x]*Sinh[a - c])/b}
255
256
257(* ::Subsection::Closed:: *)
258(*Integrands of the form x^m Hyper[a+b x]^n Hyper[a+ b x]^p*)
259
260
261(* ::Subsubsection::Closed:: *)
262(*Integrands of the form x^m Cosh[a+b x]^n Sinh[a+b x]^p*)
263
264
265{Sinh[x]*Cosh[x]/x, x, 3, (1/2)*SinhIntegral[2*x]}
266{Sinh[x]*Cosh[x]/x^2, x, 4, CoshIntegral[2*x] - Sinh[2*x]/(2*x)}
267{Sinh[x]*Cosh[x]/x^3, x, 5, -(Cosh[2*x]/(2*x)) - Sinh[2*x]/(4*x^2) + SinhIntegral[2*x]}
268
269
270(* Integrands of the form x^m*Sinh[a+b*x]^n*Cosh[a+b*x]^p where m, n and p are integers *)
271{x*Sinh[a + b*x]*Cosh[a + b*x], x, 2, x/(4*b) - (Cosh[a + b*x]*Sinh[a + b*x])/(4*b^2) + (x*Sinh[a + b*x]^2)/(2*b)}
272{x^2*Sinh[a + b*x]*Cosh[a + b*x], x, 3, x^2/(4*b) - (x*Cosh[a + b*x]*Sinh[a + b*x])/(2*b^2) + Sinh[a + b*x]^2/(4*b^3) + (x^2*Sinh[a + b*x]^2)/(2*b)}
273{x^3*Sinh[a + b*x]*Cosh[a + b*x], x, 4, (3*x)/(8*b^3) + x^3/(4*b) - (3*Cosh[a + b*x]*Sinh[a + b*x])/(8*b^4) - (3*x^2*Cosh[a + b*x]*Sinh[a + b*x])/(4*b^2) + (3*x*Sinh[a + b*x]^2)/(4*b^3) + (x^3*Sinh[a + b*x]^2)/(2*b)}
274{Sinh[a + b*x]*Cosh[a + b*x]/x, x, 5, (1/2)*CoshIntegral[2*b*x]*Sinh[2*a] + (1/2)*Cosh[2*a]*SinhIntegral[2*b*x]}
275{Sinh[a + b*x]*Cosh[a + b*x]/x^2, x, 6, b*Cosh[2*a]*CoshIntegral[2*b*x] - Sinh[2*a + 2*b*x]/(2*x) + b*Sinh[2*a]*SinhIntegral[2*b*x]}
276{Sinh[a + b*x]*Cosh[a + b*x]/x^3, x, 7, -((b*Cosh[2*a + 2*b*x])/(2*x)) + b^2*CoshIntegral[2*b*x]*Sinh[2*a] - Sinh[2*a + 2*b*x]/(4*x^2) + b^2*Cosh[2*a]*SinhIntegral[2*b*x]}
277
278{x*Sinh[a + b*x]^2*Cosh[a + b*x], x, 3, Cosh[a + b*x]/(3*b^2) - Cosh[a + b*x]^3/(9*b^2) + (x*Sinh[a + b*x]^3)/(3*b)}
279{x^2*Sinh[a + b*x]^2*Cosh[a + b*x], x, 4, (4*x*Cosh[a + b*x])/(9*b^2) - (4*Sinh[a + b*x])/(9*b^3) - (2*x*Cosh[a + b*x]*Sinh[a + b*x]^2)/(9*b^2) + (2*Sinh[a + b*x]^3)/(27*b^3) + (x^2*Sinh[a + b*x]^3)/(3*b)}
280{x^3*Sinh[a + b*x]^2*Cosh[a + b*x], x, 7, (14*Cosh[a + b*x])/(9*b^4) + (2*x^2*Cosh[a + b*x])/(3*b^2) - (2*Cosh[a + b*x]^3)/(27*b^4) - (4*x*Sinh[a + b*x])/(3*b^3) - (x^2*Cosh[a + b*x]*Sinh[a + b*x]^2)/(3*b^2) + (2*x*Sinh[a + b*x]^3)/(9*b^3) + (x^3*Sinh[a + b*x]^3)/(3*b)}
281{Sinh[a + b*x]^2*Cosh[a + b*x]/x, x, 8, (-(1/4))*Cosh[a]*CoshIntegral[b*x] + (1/4)*Cosh[3*a]*CoshIntegral[3*b*x] - (1/4)*Sinh[a]*SinhIntegral[b*x] + (1/4)*Sinh[3*a]*SinhIntegral[3*b*x]}
282{Sinh[a + b*x]^2*Cosh[a + b*x]/x^2, x, 10, Cosh[a + b*x]/(4*x) - Cosh[3*a + 3*b*x]/(4*x) - (1/4)*b*CoshIntegral[b*x]*Sinh[a] + (3/4)*b*CoshIntegral[3*b*x]*Sinh[3*a] - (1/4)*b*Cosh[a]*SinhIntegral[b*x] + (3/4)*b*Cosh[3*a]*SinhIntegral[3*b*x]}
283{Sinh[a + b*x]^2*Cosh[a + b*x]/x^3, x, 12, Cosh[a + b*x]/(8*x^2) - Cosh[3*a + 3*b*x]/(8*x^2) - (1/8)*b^2*Cosh[a]*CoshIntegral[b*x] + (9/8)*b^2*Cosh[3*a]*CoshIntegral[3*b*x] + (b*Sinh[a + b*x])/(8*x) - (3*b*Sinh[3*a + 3*b*x])/(8*x) - (1/8)*b^2*Sinh[a]*SinhIntegral[b*x] + (9/8)*b^2*Sinh[3*a]*SinhIntegral[3*b*x]}
284
285{x*Sinh[a + b*x]*Cosh[a + b*x]^2, x, 3, (x*Cosh[a + b*x]^3)/(3*b) - Sinh[a + b*x]/(3*b^2) - Sinh[a + b*x]^3/(9*b^2)}
286{x^2*Sinh[a + b*x]*Cosh[a + b*x]^2, x, 4, (4*Cosh[a + b*x])/(9*b^3) + (2*Cosh[a + b*x]^3)/(27*b^3) + (x^2*Cosh[a + b*x]^3)/(3*b) - (4*x*Sinh[a + b*x])/(9*b^2) - (2*x*Cosh[a + b*x]^2*Sinh[a + b*x])/(9*b^2)}
287{x^3*Sinh[a + b*x]*Cosh[a + b*x]^2, x, 7, (4*x*Cosh[a + b*x])/(3*b^3) + (2*x*Cosh[a + b*x]^3)/(9*b^3) + (x^3*Cosh[a + b*x]^3)/(3*b) - (14*Sinh[a + b*x])/(9*b^4) - (2*x^2*Sinh[a + b*x])/(3*b^2) - (x^2*Cosh[a + b*x]^2*Sinh[a + b*x])/(3*b^2) - (2*Sinh[a + b*x]^3)/(27*b^4)}
288{Sinh[a + b*x]*Cosh[a + b*x]^2/x, x, 8, (1/4)*CoshIntegral[b*x]*Sinh[a] + (1/4)*CoshIntegral[3*b*x]*Sinh[3*a] + (1/4)*Cosh[a]*SinhIntegral[b*x] + (1/4)*Cosh[3*a]*SinhIntegral[3*b*x]}
289{Sinh[a + b*x]*Cosh[a + b*x]^2/x^2, x, 10, (1/4)*b*Cosh[a]*CoshIntegral[b*x] + (3/4)*b*Cosh[3*a]*CoshIntegral[3*b*x] - Sinh[a + b*x]/(4*x) - Sinh[3*a + 3*b*x]/(4*x) + (1/4)*b*Sinh[a]*SinhIntegral[b*x] + (3/4)*b*Sinh[3*a]*SinhIntegral[3*b*x]}
290{Sinh[a + b*x]*Cosh[a + b*x]^2/x^3, x, 12, -((b*Cosh[a + b*x])/(8*x)) - (3*b*Cosh[3*a + 3*b*x])/(8*x) + (1/8)*b^2*CoshIntegral[b*x]*Sinh[a] + (9/8)*b^2*CoshIntegral[3*b*x]*Sinh[3*a] - Sinh[a + b*x]/(8*x^2) - Sinh[3*a + 3*b*x]/(8*x^2) + (1/8)*b^2*Cosh[a]*SinhIntegral[b*x] + (9/8)*b^2*Cosh[3*a]*SinhIntegral[3*b*x]}
291
292{x*Sinh[a + b*x]^2*Cosh[a + b*x]^2, x, 5, -(x^2/16) - Cosh[4*a + 4*b*x]/(128*b^2) + (x*Sinh[4*a + 4*b*x])/(32*b)}
293{x^2*Sinh[a + b*x]^2*Cosh[a + b*x]^2, x, 6, -(x^3/24) - (x*Cosh[4*a + 4*b*x])/(64*b^2) + Sinh[4*a + 4*b*x]/(256*b^3) + (x^2*Sinh[4*a + 4*b*x])/(32*b)}
294{x^3*Sinh[a + b*x]^2*Cosh[a + b*x]^2, x, 7, -(x^4/32) - (3*Cosh[4*a + 4*b*x])/(1024*b^4) - (3*x^2*Cosh[4*a + 4*b*x])/(128*b^2) + (3*x*Sinh[4*a + 4*b*x])/(256*b^3) + (x^3*Sinh[4*a + 4*b*x])/(32*b)}
295{Sinh[a + b*x]^2*Cosh[a + b*x]^2/x, x, 6, (1/8)*Cosh[4*a]*CoshIntegral[4*b*x] - Log[x]/8 + (1/8)*Sinh[4*a]*SinhIntegral[4*b*x]}
296{Sinh[a + b*x]^2*Cosh[a + b*x]^2/x^2, x, 7, 1/(8*x) - Cosh[4*a + 4*b*x]/(8*x) + (1/2)*b*CoshIntegral[4*b*x]*Sinh[4*a] + (1/2)*b*Cosh[4*a]*SinhIntegral[4*b*x]}
297{Sinh[a + b*x]^2*Cosh[a + b*x]^2/x^3, x, 8, 1/(16*x^2) - Cosh[4*a + 4*b*x]/(16*x^2) + b^2*Cosh[4*a]*CoshIntegral[4*b*x] - (b*Sinh[4*a + 4*b*x])/(4*x) + b^2*Sinh[4*a]*SinhIntegral[4*b*x]}
298
299
300(* Integrands of the form x*Sinh[a+b*x]*Cosh[a+b*x]^n where n is a half-integer *)
301{x*Sinh[a + b*x]*Cosh[a + b*x]^(3/2), x, 3, (2*x*Cosh[a + b*x]^(5/2))/(5*b) + (12*I*EllipticE[(1/2)*I*(a + b*x), 2])/(25*b^2) - (4*Cosh[a + b*x]^(3/2)*Sinh[a + b*x])/(25*b^2)}
302{x*Sinh[a + b*x]*Sqrt[Cosh[a + b*x]], x, 3, (2*x*Cosh[a + b*x]^(3/2))/(3*b) + (4*I*EllipticF[(1/2)*I*(a + b*x), 2])/(9*b^2) - (4*Sqrt[Cosh[a + b*x]]*Sinh[a + b*x])/(9*b^2)}
303{x*Sinh[a + b*x]/Sqrt[Cosh[a + b*x]], x, 2, (2*x*Sqrt[Cosh[a + b*x]])/b + (4*I*EllipticE[(1/2)*I*(a + b*x), 2])/b^2}
304{x*Sinh[a + b*x]/Cosh[a + b*x]^(3/2), x, 2, -((2*x)/(b*Sqrt[Cosh[a + b*x]])) - (4*I*EllipticF[(1/2)*I*(a + b*x), 2])/b^2}
305{x*Sinh[a + b*x]/Cosh[a + b*x]^(5/2), x, 3, -((2*x)/(3*b*Cosh[a + b*x]^(3/2))) + (4*I*EllipticE[(1/2)*I*(a + b*x), 2])/(3*b^2) + (4*Sinh[a + b*x])/(3*b^2*Sqrt[Cosh[a + b*x]])}
306
307
308(* Integrands of the form x*Cosh[a+b*x]*Sinh[a+b*x]^n where n is a half-integer *)
309{x*Cosh[a + b*x]*Sinh[a + b*x]^(3/2), x, 4, (12*I*EllipticE[Pi/4 - (1/2)*I*(a + b*x), 2]*Sqrt[Sinh[a + b*x]])/(25*b^2*Sqrt[I*Sinh[a + b*x]]) - (4*Cosh[a + b*x]*Sinh[a + b*x]^(3/2))/(25*b^2) + (2*x*Sinh[a + b*x]^(5/2))/(5*b)}
310{x*Cosh[a + b*x]*Sqrt[Sinh[a + b*x]], x, 4, (4*I*EllipticF[Pi/4 - (1/2)*I*(a + b*x), 2]*Sqrt[I*Sinh[a + b*x]])/(9*b^2*Sqrt[Sinh[a + b*x]]) - (4*Cosh[a + b*x]*Sqrt[Sinh[a + b*x]])/(9*b^2) + (2*x*Sinh[a + b*x]^(3/2))/(3*b)}
311{x*Cosh[a + b*x]/Sqrt[Sinh[a + b*x]], x, 3, (2*x*Sqrt[Sinh[a + b*x]])/b - (4*I*EllipticE[Pi/4 - (1/2)*I*(a + b*x), 2]*Sqrt[Sinh[a + b*x]])/(b^2*Sqrt[I*Sinh[a + b*x]])}
312{x*Cosh[a + b*x]/Sinh[a + b*x]^(3/2), x, 3, -((2*x)/(b*Sqrt[Sinh[a + b*x]])) + (4*I*EllipticF[Pi/4 - (1/2)*I*(a + b*x), 2]*Sqrt[I*Sinh[a + b*x]])/(b^2*Sqrt[Sinh[a + b*x]])}
313{x*Cosh[a + b*x]/Sinh[a + b*x]^(5/2), x, 4, -((2*x)/(3*b*Sinh[a + b*x]^(3/2))) - (4*Cosh[a + b*x])/(3*b^2*Sqrt[Sinh[a + b*x]]) + (4*I*EllipticE[Pi/4 - (1/2)*I*(a + b*x), 2]*Sqrt[Sinh[a + b*x]])/(3*b^2*Sqrt[I*Sinh[a + b*x]])}
314
315
316(* ::Subsubsection::Closed:: *)
317(*Integrands of the form x^m Sech[a+b x]^n Sinh[a+b x]^p*)
318
319
320{x*Sinh[a + b*x]*Sech[a + b*x]^(9/2), x, 5, (12*I*Sqrt[Cosh[a + b*x]]*EllipticE[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/(35*b^2) - (2*x*Sech[a + b*x]^(7/2))/(7*b) + (12*Sqrt[Sech[a + b*x]]*Sinh[a + b*x])/(35*b^2) + (4*Sech[a + b*x]^(5/2)*Sinh[a + b*x])/(35*b^2)}
321{x*Sinh[a + b*x]*Sech[a + b*x]^(7/2), x, 4, -((4*I*Sqrt[Cosh[a + b*x]]*EllipticF[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/(15*b^2)) - (2*x*Sech[a + b*x]^(5/2))/(5*b) + (4*Sech[a + b*x]^(3/2)*Sinh[a + b*x])/(15*b^2)}
322{x*Sinh[a + b*x]*Sech[a + b*x]^(5/2), x, 4, (4*I*Sqrt[Cosh[a + b*x]]*EllipticE[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/(3*b^2) - (2*x*Sech[a + b*x]^(3/2))/(3*b) + (4*Sqrt[Sech[a + b*x]]*Sinh[a + b*x])/(3*b^2)}
323{x*Sinh[a + b*x]*Sech[a + b*x]^(3/2), x, 3, -((2*x*Sqrt[Sech[a + b*x]])/b) - (4*I*Sqrt[Cosh[a + b*x]]*EllipticF[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/b^2}
324{x*Sinh[a + b*x]*Sech[a + b*x]^(1/2), x, 3, (2*x)/(b*Sqrt[Sech[a + b*x]]) + (4*I*Sqrt[Cosh[a + b*x]]*EllipticE[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/b^2}
325{x*Sinh[a + b*x]/Sech[a + b*x]^(1/2), x, 4, (2*x)/(3*b*Sech[a + b*x]^(3/2)) + (4*I*Sqrt[Cosh[a + b*x]]*EllipticF[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/(9*b^2) - (4*Sinh[a + b*x])/(9*b^2*Sqrt[Sech[a + b*x]])}
326{x*Sinh[a + b*x]/Sech[a + b*x]^(3/2), x, 4, (2*x)/(5*b*Sech[a + b*x]^(5/2)) + (12*I*Sqrt[Cosh[a + b*x]]*EllipticE[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/(25*b^2) - (4*Sinh[a + b*x])/(25*b^2*Sech[a + b*x]^(3/2))}
327{x*Sinh[a + b*x]/Sech[a + b*x]^(5/2), x, 5, (2*x)/(7*b*Sech[a + b*x]^(7/2)) + (20*I*Sqrt[Cosh[a + b*x]]*EllipticF[(1/2)*I*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/(147*b^2) - (4*Sinh[a + b*x])/(49*b^2*Sech[a + b*x]^(5/2)) - (20*Sinh[a + b*x])/(147*b^2*Sqrt[Sech[a + b*x]])}
328
329
330(* ::Subsubsection::Closed:: *)
331(*Integrands of the form x^m Cosh[a+b x]^n Csch[a+b x]^p*)
332
333
334{x*Cosh[a + b*x]*Csch[a + b*x]^(9/2), x, 6, (12*Cosh[a + b*x]*Sqrt[Csch[a + b*x]])/(35*b^2) - (4*Cosh[a + b*x]*Csch[a + b*x]^(5/2))/(35*b^2) - (2*x*Csch[a + b*x]^(7/2))/(7*b) - (12*I*EllipticE[Pi/4 - (1/2)*I*(a + b*x), 2])/(35*b^2*Sqrt[Csch[a + b*x]]*Sqrt[I*Sinh[a + b*x]])}
335{x*Cosh[a + b*x]*Csch[a + b*x]^(7/2), x, 5, -((4*Cosh[a + b*x]*Csch[a + b*x]^(3/2))/(15*b^2)) - (2*x*Csch[a + b*x]^(5/2))/(5*b) - (4*I*Sqrt[Csch[a + b*x]]*EllipticF[Pi/4 - (1/2)*I*(a + b*x), 2]*Sqrt[I*Sinh[a + b*x]])/(15*b^2)}
336{x*Cosh[a + b*x]*Csch[a + b*x]^(5/2), x, 5, -((4*Cosh[a + b*x]*Sqrt[Csch[a + b*x]])/(3*b^2)) - (2*x*Csch[a + b*x]^(3/2))/(3*b) + (4*I*EllipticE[Pi/4 - (1/2)*I*(a + b*x), 2])/(3*b^2*Sqrt[Csch[a + b*x]]*Sqrt[I*Sinh[a + b*x]])}
337{x*Cosh[a + b*x]*Csch[a + b*x]^(3/2), x, 4, -((2*x*Sqrt[Csch[a + b*x]])/b) + (4*I*Sqrt[Csch[a + b*x]]*EllipticF[Pi/4 - (1/2)*I*(a + b*x), 2]*Sqrt[I*Sinh[a + b*x]])/b^2}
338{x*Cosh[a + b*x]*Csch[a + b*x]^(1/2), x, 4, (2*x)/(b*Sqrt[Csch[a + b*x]]) - (4*I*EllipticE[Pi/4 - (1/2)*I*(a + b*x), 2])/(b^2*Sqrt[Csch[a + b*x]]*Sqrt[I*Sinh[a + b*x]])}
339{x*Cosh[a + b*x]/Csch[a + b*x]^(1/2), x, 5, (2*x)/(3*b*Csch[a + b*x]^(3/2)) - (4*Cosh[a + b*x])/(9*b^2*Sqrt[Csch[a + b*x]]) + (4*I*Sqrt[Csch[a + b*x]]*EllipticF[Pi/4 - (1/2)*I*(a + b*x), 2]*Sqrt[I*Sinh[a + b*x]])/(9*b^2)}
340{x*Cosh[a + b*x]/Csch[a + b*x]^(3/2), x, 5, (2*x)/(5*b*Csch[a + b*x]^(5/2)) - (4*Cosh[a + b*x])/(25*b^2*Csch[a + b*x]^(3/2)) + (12*I*EllipticE[Pi/4 - (1/2)*I*(a + b*x), 2])/(25*b^2*Sqrt[Csch[a + b*x]]*Sqrt[I*Sinh[a + b*x]])}
341{x*Cosh[a + b*x]/Csch[a + b*x]^(5/2), x, 6, (2*x)/(7*b*Csch[a + b*x]^(7/2)) - (4*Cosh[a + b*x])/(49*b^2*Csch[a + b*x]^(5/2)) + (20*Cosh[a + b*x])/(147*b^2*Sqrt[Csch[a + b*x]]) - (20*I*Sqrt[Csch[a + b*x]]*EllipticF[Pi/4 - (1/2)*I*(a + b*x), 2]*Sqrt[I*Sinh[a + b*x]])/(147*b^2)}
342
343
344(* ::Subsubsection::Closed:: *)
345(*Integrands of the form x^m Sech[a+b x]^n Tanh[a+b x]^p*)
346
347
348(* Integrands of the form x^m*Sech[a+b*x]^n*Tanh[a+b*x] where m and n are integers *)
349{x*Sech[a + b*x]*Tanh[a + b*x], x, 2, ArcTan[Sinh[a + b*x]]/b^2 - (x*Sech[a + b*x])/b}
350{x^2*Sech[a + b*x]*Tanh[a + b*x], x, 5, (4*x*ArcTan[E^(a + b*x)])/b^2 - (2*I*PolyLog[2, (-I)*E^(a + b*x)])/b^3 + (2*I*PolyLog[2, I*E^(a + b*x)])/b^3 - (x^2*Sech[a + b*x])/b}
351{x^3*Sech[a + b*x]*Tanh[a + b*x], x, 7, (6*x^2*ArcTan[E^(a + b*x)])/b^2 - (6*I*x*PolyLog[2, (-I)*E^(a + b*x)])/b^3 + (6*I*x*PolyLog[2, I*E^(a + b*x)])/b^3 + (6*I*PolyLog[3, (-I)*E^(a + b*x)])/b^4 - (6*I*PolyLog[3, I*E^(a + b*x)])/b^4 - (x^3*Sech[a + b*x])/b}
352{Sech[a + b*x]*Tanh[a + b*x]/x, x, 0, Int[(Sech[a + b*x]*Tanh[a + b*x])/x, x]}
353
354{x*Sech[a + b*x]^2*Tanh[a + b*x], x, 2, -((x*Sech[a + b*x]^2)/(2*b)) + Tanh[a + b*x]/(2*b^2)}
355{x^2*Sech[a + b*x]^2*Tanh[a + b*x], x, 3, -(Log[Cosh[a + b*x]]/b^3) - (x^2*Sech[a + b*x]^2)/(2*b) + (x*Tanh[a + b*x])/b^2}
356{x^3*Sech[a + b*x]^2*Tanh[a + b*x], x, 6, (3*x^2)/(2*b^2) - (3*x*Log[1 + E^(2*a + 2*b*x)])/b^3 - (3*PolyLog[2, -E^(2*a + 2*b*x)])/(2*b^4) - (x^3*Sech[a + b*x]^2)/(2*b) + (3*x^2*Tanh[a + b*x])/(2*b^2)}
357{Sech[a + b*x]^2*Tanh[a + b*x]/x, x, 0, Int[(Sech[a + b*x]^2*Tanh[a + b*x])/x, x]}
358
359
360(* ::Subsubsection::Closed:: *)
361(*Integrands of the form x^m Csch[a+b x]^n Coth[a+b x]^p*)
362
363
364(* Integrands of the form x^m*Csch[a+b*x]^n*Coth[a+b*x] where m and n are integers *)
365{x^3*Csch[a + b*x]*Coth[a + b*x], x, 7, -((6*x^2*ArcTanh[E^(a + b*x)])/b^2) - (x^3*Csch[a + b*x])/b - (6*x*PolyLog[2, -E^(a + b*x)])/b^3 + (6*x*PolyLog[2, E^(a + b*x)])/b^3 + (6*PolyLog[3, -E^(a + b*x)])/b^4 - (6*PolyLog[3, E^(a + b*x)])/b^4}
366{x^2*Csch[a + b*x]*Coth[a + b*x], x, 5, -((4*x*ArcTanh[E^(a + b*x)])/b^2) - (x^2*Csch[a + b*x])/b - (2*PolyLog[2, -E^(a + b*x)])/b^3 + (2*PolyLog[2, E^(a + b*x)])/b^3}
367{x^1*Csch[a + b*x]*Coth[a + b*x], x, 2, -(ArcCoth[Cosh[a + b*x]]/b^2) - (x*Csch[a + b*x])/b}
368{Csch[a + b*x]*Coth[a + b*x]/x^1, x, 0, Int[(Coth[a + b*x]*Csch[a + b*x])/x, x]}
369
370
371{x^3*Csch[a + b*x]^2*Coth[a + b*x], x, 6, -((3*x^2)/(2*b^2)) - (3*x^2*Coth[a + b*x])/(2*b^2) - (x^3*Csch[a + b*x]^2)/(2*b) + (3*x*Log[1 - E^(2*a + 2*b*x)])/b^3 + (3*PolyLog[2, E^(2*a + 2*b*x)])/(2*b^4)}
372{x^2*Csch[a + b*x]^2*Coth[a + b*x], x, 3, -((x*Coth[a + b*x])/b^2) - (x^2*Csch[a + b*x]^2)/(2*b) + Log[Sinh[a + b*x]]/b^3}
373{x^1*Csch[a + b*x]^2*Coth[a + b*x], x, 2, -(Coth[a + b*x]/(2*b^2)) - (x*Csch[a + b*x]^2)/(2*b)}
374{Csch[a + b*x]^2*Coth[a + b*x]/x^1, x, 0, Int[(Coth[a + b*x]*Csch[a + b*x]^2)/x, x]}
375
376
377(* ::Subsubsection::Closed:: *)
378(*Integrands of the form x^m Cosh[a+b x]^n Coth[a+b x]^p*)
379
380
381(* Integrands of the form x^m*Cosh[x]^2*Coth[x]^m where m and n are integers *)
382{x^1*Cosh[x]^2*Coth[x], x, 14, -(x^2/2) + (1/4)*x*Cosh[2*x] + x*Log[1 - E^(2*x)] + (1/2)*PolyLog[2, E^(2*x)] - (1/8)*Sinh[2*x]}
383{x^2*Cosh[x]^2*Coth[x], x, 17, -(x^3/3) + (1/8)*Cosh[2*x] + (1/4)*x^2*Cosh[2*x] + x^2*Log[1 - E^(2*x)] + x*PolyLog[2, E^(2*x)] - (1/2)*PolyLog[3, E^(2*x)] - (1/4)*x*Sinh[2*x]}
384{x^3*Cosh[x]^2*Coth[x], x, 20, -(x^4/4) + (3/8)*x*Cosh[2*x] + (1/4)*x^3*Cosh[2*x] + x^3*Log[1 - E^(2*x)] + (3/2)*x^2*PolyLog[2, E^(2*x)] - (3/2)*x*PolyLog[3, E^(2*x)] + (3/4)*PolyLog[4, E^(2*x)] - (3/16)*Sinh[2*x] - (3/8)*x^2*Sinh[2*x]}
385
386{x^1*Cosh[x]^2*Coth[x]^2, x, 19, (3*x^2)/4 - (1/8)*Cosh[2*x] - x*Coth[x] + Log[Sinh[x]] + (1/4)*x*Sinh[2*x]}
387{x^2*Cosh[x]^2*Coth[x]^2, x, 29, -x^2 + x^3/2 - (1/4)*x*Cosh[2*x] - x^2*Coth[x] + 2*x*Log[1 - E^(2*x)] + PolyLog[2, E^(2*x)] + (1/8)*Sinh[2*x] + (1/4)*x^2*Sinh[2*x]}
388{x^3*Cosh[x]^2*Coth[x]^2, x, 33, -x^3 + (3*x^4)/8 - (3/16)*Cosh[2*x] - (3/8)*x^2*Cosh[2*x] - x^3*Coth[x] + 3*x^2*Log[1 - E^(2*x)] + 3*x*PolyLog[2, E^(2*x)] - (3/2)*PolyLog[3, E^(2*x)] + (3/8)*x*Sinh[2*x] + (1/4)*x^3*Sinh[2*x]}
389
390{x^1*Cosh[x]^2*Coth[x]^3, x, 22, -x^2 + (1/4)*x*Cosh[2*x] - Coth[x]/2 - (1/2)*x*Csch[x]^2 + 2*x*Log[1 - E^(2*x)] + PolyLog[2, E^(2*x)] - (1/8)*Sinh[2*x]}
391{x^2*Cosh[x]^2*Coth[x]^3, x, 28, -((2*x^3)/3) + (1/8)*Cosh[2*x] + (1/4)*x^2*Cosh[2*x] - x*Coth[x] - (1/2)*x^2*Csch[x]^2 + 2*x^2*Log[1 - E^(2*x)] + Log[Sinh[x]] + 2*x*PolyLog[2, E^(2*x)] - PolyLog[3, E^(2*x)] - (1/4)*x*Sinh[2*x]}
392{x^3*Cosh[x]^2*Coth[x]^3, x, 40, -((3*x^2)/2) - x^4/2 + (3/8)*x*Cosh[2*x] + (1/4)*x^3*Cosh[2*x] - (3/2)*x^2*Coth[x] - (1/2)*x^3*Csch[x]^2 + 3*x*Log[1 - E^(2*x)] + 2*x^3*Log[1 - E^(2*x)] + (3/2)*(1 + 2*x^2)*PolyLog[2, E^(2*x)] - 3*x*PolyLog[3, E^(2*x)] + (3/2)*PolyLog[4, E^(2*x)] - (3/16)*Sinh[2*x] - (3/8)*x^2*Sinh[2*x]}
393
394
395(* ::Subsubsection::Closed:: *)
396(*Integrands of the form x^m Csch[a+b x]^n Sech[a+b x]^p*)
397
398
399{x^3*Csch[a + b*x]*Sech[a + b*x], x, 9, -((2*x^3*ArcTanh[E^(2*a + 2*b*x)])/b) - (3*x^2*PolyLog[2, -E^(2*a + 2*b*x)])/(2*b^2) + (3*x^2*PolyLog[2, E^(2*a + 2*b*x)])/(2*b^2) + (3*x*PolyLog[3, -E^(2*a + 2*b*x)])/(2*b^3) - (3*x*PolyLog[3, E^(2*a + 2*b*x)])/(2*b^3) - (3*PolyLog[4, -E^(2*a + 2*b*x)])/(4*b^4) + (3*PolyLog[4, E^(2*a + 2*b*x)])/(4*b^4)}
400{x^2*Csch[a + b*x]*Sech[a + b*x], x, 7, -((2*x^2*ArcTanh[E^(2*a + 2*b*x)])/b) - (x*PolyLog[2, -E^(2*a + 2*b*x)])/b^2 + (x*PolyLog[2, E^(2*a + 2*b*x)])/b^2 + PolyLog[3, -E^(2*a + 2*b*x)]/(2*b^3) - PolyLog[3, E^(2*a + 2*b*x)]/(2*b^3)}
401{x^1*Csch[a + b*x]*Sech[a + b*x], x, 5, -((2*x*ArcTanh[E^(2*a + 2*b*x)])/b) - PolyLog[2, -E^(2*a + 2*b*x)]/(2*b^2) + PolyLog[2, E^(2*a + 2*b*x)]/(2*b^2)}
402{x^0*Csch[a + b*x]*Sech[a + b*x], x, 1, Log[Tanh[a + b*x]]/b}
403{Csch[a + b*x]*Sech[a + b*x]/x^1, x, 1, 2*Int[Csch[2*a + 2*b*x]/x, x]}
404
405{x^3*Csch[a + b*x]*Sech[a + b*x]^2, x, 19, -((6*x^2*ArcTan[E^(a + b*x)])/b^2) - (2*x^3*ArcTanh[E^(a + b*x)])/b - (3*x^2*PolyLog[2, -E^(a + b*x)])/b^2 + (6*I*x*PolyLog[2, (-I)*E^(a + b*x)])/b^3 - (6*I*x*PolyLog[2, I*E^(a + b*x)])/b^3 + (3*x^2*PolyLog[2, E^(a + b*x)])/b^2 + (6*x*PolyLog[3, -E^(a + b*x)])/b^3 - (6*I*PolyLog[3, (-I)*E^(a + b*x)])/b^4 + (6*I*PolyLog[3, I*E^(a + b*x)])/b^4 - (6*x*PolyLog[3, E^(a + b*x)])/b^3 - (6*PolyLog[4, -E^(a + b*x)])/b^4 + (6*PolyLog[4, E^(a + b*x)])/b^4 + (x^3*Sech[a + b*x])/b}
406{x^2*Csch[a + b*x]*Sech[a + b*x]^2, x, 15, -((4*x*ArcTan[E^(a + b*x)])/b^2) - (2*x^2*ArcTanh[E^(a + b*x)])/b - (2*x*PolyLog[2, -E^(a + b*x)])/b^2 + (2*I*PolyLog[2, (-I)*E^(a + b*x)])/b^3 - (2*I*PolyLog[2, I*E^(a + b*x)])/b^3 + (2*x*PolyLog[2, E^(a + b*x)])/b^2 + (2*PolyLog[3, -E^(a + b*x)])/b^3 - (2*PolyLog[3, E^(a + b*x)])/b^3 + (x^2*Sech[a + b*x])/b}
407{x^1*Csch[a + b*x]*Sech[a + b*x]^2, x, 9, -(ArcTan[Sinh[a + b*x]]/b^2) - (2*x*ArcTanh[E^(a + b*x)])/b - PolyLog[2, -E^(a + b*x)]/b^2 + PolyLog[2, E^(a + b*x)]/b^2 + (x*Sech[a + b*x])/b}
408{x^0*Csch[a + b*x]*Sech[a + b*x]^2, x, 2, -(ArcCoth[Cosh[a + b*x]]/b) + Sech[a + b*x]/b}
409{Csch[a + b*x]*Sech[a + b*x]^2/x^1, x, 0, Int[(Csch[a + b*x]*Sech[a + b*x]^2)/x, x]}
410
411{x^3*Csch[a + b*x]*Sech[a + b*x]^3, x, 19, -((3*x^2)/(2*b^2)) - (2*x^3*ArcTanh[E^(2*a + 2*b*x)])/b + (3*x*Log[1 + E^(2*a + 2*b*x)])/b^3 + (3*(1 - b^2*x^2)*PolyLog[2, -E^(2*a + 2*b*x)])/(2*b^4) + (3*x^2*PolyLog[2, E^(2*a + 2*b*x)])/(2*b^2) + (3*x*PolyLog[3, -E^(2*a + 2*b*x)])/(2*b^3) - (3*x*PolyLog[3, E^(2*a + 2*b*x)])/(2*b^3) - (3*PolyLog[4, -E^(2*a + 2*b*x)])/(4*b^4) + (3*PolyLog[4, E^(2*a + 2*b*x)])/(4*b^4) + (x^3*Sech[a + b*x]^2)/(2*b) - (3*x^2*Tanh[a + b*x])/(2*b^2)}
412{x^2*Csch[a + b*x]*Sech[a + b*x]^3, x, 14, -((2*x^2*ArcTanh[E^(2*a + 2*b*x)])/b) + Log[Cosh[a + b*x]]/b^3 - (x*PolyLog[2, -E^(2*a + 2*b*x)])/b^2 + (x*PolyLog[2, E^(2*a + 2*b*x)])/b^2 + PolyLog[3, -E^(2*a + 2*b*x)]/(2*b^3) - PolyLog[3, E^(2*a + 2*b*x)]/(2*b^3) + (x^2*Sech[a + b*x]^2)/(2*b) - (x*Tanh[a + b*x])/b^2}
413{x^1*Csch[a + b*x]*Sech[a + b*x]^3, x, 9, -((2*x*ArcTanh[E^(2*a + 2*b*x)])/b) - PolyLog[2, -E^(2*a + 2*b*x)]/(2*b^2) + PolyLog[2, E^(2*a + 2*b*x)]/(2*b^2) + (x*Sech[a + b*x]^2)/(2*b) - Tanh[a + b*x]/(2*b^2)}
414{x^0*Csch[a + b*x]*Sech[a + b*x]^3, x, 3, Log[Tanh[a + b*x]]/b - Tanh[a + b*x]^2/(2*b)}
415{Csch[a + b*x]*Sech[a + b*x]^3/x^1, x, 0, Int[(Csch[a + b*x]*Sech[a + b*x]^3)/x, x]}
416
417
418{x^3*Csch[a + b*x]^2*Sech[a + b*x], x, 19, -((2*x^3*ArcTan[E^(a + b*x)])/b) - (6*x^2*ArcTanh[E^(a + b*x)])/b^2 - (x^3*Csch[a + b*x])/b - (6*x*PolyLog[2, -E^(a + b*x)])/b^3 + (3*I*x^2*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - (3*I*x^2*PolyLog[2, I*E^(a + b*x)])/b^2 + (6*x*PolyLog[2, E^(a + b*x)])/b^3 + (6*PolyLog[3, -E^(a + b*x)])/b^4 - (6*I*x*PolyLog[3, (-I)*E^(a + b*x)])/b^3 + (6*I*x*PolyLog[3, I*E^(a + b*x)])/b^3 - (6*PolyLog[3, E^(a + b*x)])/b^4 + (6*I*PolyLog[4, (-I)*E^(a + b*x)])/b^4 - (6*I*PolyLog[4, I*E^(a + b*x)])/b^4}
419{x^2*Csch[a + b*x]^2*Sech[a + b*x], x, 15, -((2*x^2*ArcTan[E^(a + b*x)])/b) - (4*x*ArcTanh[E^(a + b*x)])/b^2 - (x^2*Csch[a + b*x])/b - (2*PolyLog[2, -E^(a + b*x)])/b^3 + (2*I*x*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - (2*I*x*PolyLog[2, I*E^(a + b*x)])/b^2 + (2*PolyLog[2, E^(a + b*x)])/b^3 - (2*I*PolyLog[3, (-I)*E^(a + b*x)])/b^3 + (2*I*PolyLog[3, I*E^(a + b*x)])/b^3}
420{x^1*Csch[a + b*x]^2*Sech[a + b*x], x, 9, -(ArcCoth[Cosh[a + b*x]]/b^2) - (2*x*ArcTan[E^(a + b*x)])/b - (x*Csch[a + b*x])/b + (I*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - (I*PolyLog[2, I*E^(a + b*x)])/b^2}
421{x^0*Csch[a + b*x]^2*Sech[a + b*x], x, 2, -(ArcTan[Sinh[a + b*x]]/b) - Csch[a + b*x]/b}
422{Csch[a + b*x]^2*Sech[a + b*x]/x^1, x, 0, Int[(Csch[a + b*x]^2*Sech[a + b*x])/x, x]}
423
424{x^3*Csch[a + b*x]^2*Sech[a + b*x]^2, x, 7, -((2*x^3)/b) - (2*x^3*Coth[2*a + 2*b*x])/b + (3*x^2*Log[1 - E^(4*a + 4*b*x)])/b^2 + (3*x*PolyLog[2, E^(4*a + 4*b*x)])/(2*b^3) - (3*PolyLog[3, E^(4*a + 4*b*x)])/(8*b^4)}
425{x^2*Csch[a + b*x]^2*Sech[a + b*x]^2, x, 6, -((2*x^2)/b) - (2*x^2*Coth[2*a + 2*b*x])/b + (2*x*Log[1 - E^(4*a + 4*b*x)])/b^2 + PolyLog[2, E^(4*a + 4*b*x)]/(2*b^3)}
426{x^1*Csch[a + b*x]^2*Sech[a + b*x]^2, x, 3, -((2*x*Coth[2*a + 2*b*x])/b) + Log[Sinh[2*a + 2*b*x]]/b^2}
427{x^0*Csch[a + b*x]^2*Sech[a + b*x]^2, x, 3, -(Coth[a + b*x]/b) - Tanh[a + b*x]/b}
428{Csch[a + b*x]^2*Sech[a + b*x]^2/x^1, x, 1, 4*Int[Csch[2*a + 2*b*x]^2/x, x]}
429
430{x^3*Csch[a + b*x]^2*Sech[a + b*x]^3, x, 34, (6*x*ArcTan[E^(a + b*x)])/b^3 - (3*x^3*ArcTan[E^(a + b*x)])/b - (6*x^2*ArcTanh[E^(a + b*x)])/b^2 - (6*x*PolyLog[2, -E^(a + b*x)])/b^3 - (3*I*(2 - 3*b^2*x^2)*PolyLog[2, (-I)*E^(a + b*x)])/(2*b^4) + (3*I*(2 - 3*b^2*x^2)*PolyLog[2, I*E^(a + b*x)])/(2*b^4) + (6*x*PolyLog[2, E^(a + b*x)])/b^3 + (6*PolyLog[3, -E^(a + b*x)])/b^4 - (9*I*x*PolyLog[3, (-I)*E^(a + b*x)])/b^3 + (9*I*x*PolyLog[3, I*E^(a + b*x)])/b^3 - (6*PolyLog[3, E^(a + b*x)])/b^4 + (9*I*PolyLog[4, (-I)*E^(a + b*x)])/b^4 - (9*I*PolyLog[4, I*E^(a + b*x)])/b^4 - (3*x^2*Sech[a + b*x])/(2*b^2) - (x^3*Csch[a + b*x]*(3 - Sech[a + b*x]^2))/(2*b)}
431{x^2*Csch[a + b*x]^2*Sech[a + b*x]^3, x, 24, -((3*x^2*ArcTan[E^(a + b*x)])/b) + ArcTan[Sinh[a + b*x]]/b^3 - (4*x*ArcTanh[E^(a + b*x)])/b^2 - (2*PolyLog[2, -E^(a + b*x)])/b^3 + (3*I*x*PolyLog[2, (-I)*E^(a + b*x)])/b^2 - (3*I*x*PolyLog[2, I*E^(a + b*x)])/b^2 + (2*PolyLog[2, E^(a + b*x)])/b^3 - (3*I*PolyLog[3, (-I)*E^(a + b*x)])/b^3 + (3*I*PolyLog[3, I*E^(a + b*x)])/b^3 - (x*Sech[a + b*x])/b^2 - (x^2*Csch[a + b*x]*(3 - Sech[a + b*x]^2))/(2*b)}
432{x^1*Csch[a + b*x]^2*Sech[a + b*x]^3, x, 11, -(ArcCoth[Cosh[a + b*x]]/b^2) - (3*x*ArcTan[E^(a + b*x)])/b + (3*I*PolyLog[2, (-I)*E^(a + b*x)])/(2*b^2) - (3*I*PolyLog[2, I*E^(a + b*x)])/(2*b^2) - Sech[a + b*x]/(2*b^2) - (x*Csch[a + b*x]*(3 - Sech[a + b*x]^2))/(2*b)}
433{x^0*Csch[a + b*x]^2*Sech[a + b*x]^3, x, 3, -((3*ArcTan[Sinh[a + b*x]])/(2*b)) - (3*Csch[a + b*x])/(2*b) + (Csch[a + b*x]*Sech[a + b*x]^2)/(2*b)}
434{Csch[a + b*x]^2*Sech[a + b*x]^3/x^1, x, 0, Int[(Csch[a + b*x]^2*Sech[a + b*x]^3)/x, x]}
435
436
437{x^3*Csch[a + b*x]^3*Sech[a + b*x], x, 19, -((3*x^2)/(2*b^2)) + (2*x^3*ArcTanh[E^(2*a + 2*b*x)])/b - (3*x^2*Coth[a + b*x])/(2*b^2) - (x^3*Csch[a + b*x]^2)/(2*b) + (3*x*Log[1 - E^(2*a + 2*b*x)])/b^3 + (3*x^2*PolyLog[2, -E^(2*a + 2*b*x)])/(2*b^2) + (3*(1 - b^2*x^2)*PolyLog[2, E^(2*a + 2*b*x)])/(2*b^4) - (3*x*PolyLog[3, -E^(2*a + 2*b*x)])/(2*b^3) + (3*x*PolyLog[3, E^(2*a + 2*b*x)])/(2*b^3) + (3*PolyLog[4, -E^(2*a + 2*b*x)])/(4*b^4) - (3*PolyLog[4, E^(2*a + 2*b*x)])/(4*b^4)}
438{x^2*Csch[a + b*x]^3*Sech[a + b*x], x, 14, (2*x^2*ArcTanh[E^(2*a + 2*b*x)])/b - (x*Coth[a + b*x])/b^2 - (x^2*Csch[a + b*x]^2)/(2*b) + Log[Sinh[a + b*x]]/b^3 + (x*PolyLog[2, -E^(2*a + 2*b*x)])/b^2 - (x*PolyLog[2, E^(2*a + 2*b*x)])/b^2 - PolyLog[3, -E^(2*a + 2*b*x)]/(2*b^3) + PolyLog[3, E^(2*a + 2*b*x)]/(2*b^3)}
439{x^1*Csch[a + b*x]^3*Sech[a + b*x], x, 9, (2*x*ArcTanh[E^(2*a + 2*b*x)])/b - Coth[a + b*x]/(2*b^2) - (x*Csch[a + b*x]^2)/(2*b) + PolyLog[2, -E^(2*a + 2*b*x)]/(2*b^2) - PolyLog[2, E^(2*a + 2*b*x)]/(2*b^2)}
440{x^0*Csch[a + b*x]^3*Sech[a + b*x], x, 3, -(Coth[a + b*x]^2/(2*b)) + Log[Coth[a + b*x]]/b}
441{Csch[a + b*x]^3*Sech[a + b*x]/x^1, x, 0, Int[(Csch[a + b*x]^3*Sech[a + b*x])/x, x]}
442
443{x^3*Csch[a + b*x]^3*Sech[a + b*x]^2, x, 34, (6*x^2*ArcTan[E^(a + b*x)])/b^2 - (6*x*ArcTanh[E^(a + b*x)])/b^3 + (3*x^3*ArcTanh[E^(a + b*x)])/b - (3*x^2*Csch[a + b*x])/(2*b^2) - (3*(2 - 3*b^2*x^2)*PolyLog[2, -E^(a + b*x)])/(2*b^4) - (6*I*x*PolyLog[2, (-I)*E^(a + b*x)])/b^3 + (6*I*x*PolyLog[2, I*E^(a + b*x)])/b^3 + (3*(2 - 3*b^2*x^2)*PolyLog[2, E^(a + b*x)])/(2*b^4) - (9*x*PolyLog[3, -E^(a + b*x)])/b^3 + (6*I*PolyLog[3, (-I)*E^(a + b*x)])/b^4 - (6*I*PolyLog[3, I*E^(a + b*x)])/b^4 + (9*x*PolyLog[3, E^(a + b*x)])/b^3 + (9*PolyLog[4, -E^(a + b*x)])/b^4 - (9*PolyLog[4, E^(a + b*x)])/b^4 - (x^3*(3 + Csch[a + b*x]^2)*Sech[a + b*x])/(2*b)}
444{x^2*Csch[a + b*x]^3*Sech[a + b*x]^2, x, 24, -(ArcCoth[Cosh[a + b*x]]/b^3) + (4*x*ArcTan[E^(a + b*x)])/b^2 + (3*x^2*ArcTanh[E^(a + b*x)])/b - (x*Csch[a + b*x])/b^2 + (3*x*PolyLog[2, -E^(a + b*x)])/b^2 - (2*I*PolyLog[2, (-I)*E^(a + b*x)])/b^3 + (2*I*PolyLog[2, I*E^(a + b*x)])/b^3 - (3*x*PolyLog[2, E^(a + b*x)])/b^2 - (3*PolyLog[3, -E^(a + b*x)])/b^3 + (3*PolyLog[3, E^(a + b*x)])/b^3 - (x^2*(3 + Csch[a + b*x]^2)*Sech[a + b*x])/(2*b)}
445{x^1*Csch[a + b*x]^3*Sech[a + b*x]^2, x, 11, ArcTan[Sinh[a + b*x]]/b^2 + (3*x*ArcTanh[E^(a + b*x)])/b - Csch[a + b*x]/(2*b^2) + (3*PolyLog[2, -E^(a + b*x)])/(2*b^2) - (3*PolyLog[2, E^(a + b*x)])/(2*b^2) - (x*(3 + Csch[a + b*x]^2)*Sech[a + b*x])/(2*b)}
446{x^0*Csch[a + b*x]^3*Sech[a + b*x]^2, x, 3, (3*ArcCoth[Cosh[a + b*x]])/(2*b) - (3*Sech[a + b*x])/(2*b) - (Csch[a + b*x]^2*Sech[a + b*x])/(2*b)}
447{Csch[a + b*x]^3*Sech[a + b*x]^2/x^1, x, 0, Int[(Csch[a + b*x]^3*Sech[a + b*x]^2)/x, x]}
448
449{x^3*Sech[a + b*x]^3*Csch[a + b*x]^3, x, 14, -((6*x*ArcTanh[E^(2*a + 2*b*x)])/b^3) + (4*x^3*ArcTanh[E^(2*a + 2*b*x)])/b - (3*x^2*Csch[2*a + 2*b*x])/b^2 - (2*x^3*Coth[2*a + 2*b*x]*Csch[2*a + 2*b*x])/b - (3*(1 - 2*b^2*x^2)*PolyLog[2, -E^(2*a + 2*b*x)])/(2*b^4) + (3*(1 - 2*b^2*x^2)*PolyLog[2, E^(2*a + 2*b*x)])/(2*b^4) - (3*x*PolyLog[3, -E^(2*a + 2*b*x)])/b^3 + (3*x*PolyLog[3, E^(2*a + 2*b*x)])/b^3 + (3*PolyLog[4, -E^(2*a + 2*b*x)])/(2*b^4) - (3*PolyLog[4, E^(2*a + 2*b*x)])/(2*b^4)}
450{x^2*Sech[a + b*x]^3*Csch[a + b*x]^3, x, 9, -(ArcCoth[Cosh[2*a + 2*b*x]]/b^3) + (4*x^2*ArcTanh[E^(2*a + 2*b*x)])/b - (2*x*Csch[2*a + 2*b*x])/b^2 - (2*x^2*Coth[2*a + 2*b*x]*Csch[2*a + 2*b*x])/b + (2*x*PolyLog[2, -E^(2*a + 2*b*x)])/b^2 - (2*x*PolyLog[2, E^(2*a + 2*b*x)])/b^2 - PolyLog[3, -E^(2*a + 2*b*x)]/b^3 + PolyLog[3, E^(2*a + 2*b*x)]/b^3}
451{x^1*Sech[a + b*x]^3*Csch[a + b*x]^3, x, 6, (4*x*ArcTanh[E^(2*a + 2*b*x)])/b - Csch[2*a + 2*b*x]/b^2 - (2*x*Coth[2*a + 2*b*x]*Csch[2*a + 2*b*x])/b + PolyLog[2, -E^(2*a + 2*b*x)]/b^2 - PolyLog[2, E^(2*a + 2*b*x)]/b^2}
452{x^0*Sech[a + b*x]^3*Csch[a + b*x]^3, x, 3, -(Coth[a + b*x]^2/(2*b)) - (2*Log[Tanh[a + b*x]])/b + Tanh[a + b*x]^2/(2*b)}
453{Sech[a + b*x]^3*Csch[a + b*x]^3/x^1, x, 1, 8*Int[Csch[2*a + 2*b*x]^3/x, x]}
454
455
456(* ::Subsection::Closed:: *)
457(*Integrands of the form Hyper[m x] Hyper[n x]*)
458
459
460(* ::Subsubsection::Closed:: *)
461(*Integrands of the form Hyper[m x] Sinh[n x]*)
462
463
464(* Integrands of the form Sinh[m*x]*Sinh[x] where n is an integer *)
465{Sinh[2*x]*Sinh[x], x, 3, (2*Sinh[x]^3)/3}
466{Sinh[3*x]*Sinh[x], x, 3, (-(1/4))*Sinh[2*x] + (1/8)*Sinh[4*x]}
467{Sinh[4*x]*Sinh[x], x, 3, (-(1/6))*Sinh[3*x] + (1/10)*Sinh[5*x]}
468{Sinh[m*x]*Sinh[x], x, 3, -(Sinh[(1 - m)*x]/(2*(1 - m))) + Sinh[(1 + m)*x]/(2*(1 + m))}
469
470
471(* Integrands of the form Cosh[m*x]*Sinh[x] where n is an integer *)
472{Cosh[2*x]*Sinh[x], x, 4, -(Cosh[x]/2) + (1/6)*Cosh[3*x]}
473{Cosh[3*x]*Sinh[x], x, 4, (-(1/4))*Cosh[2*x] + (1/8)*Cosh[4*x]}
474{Cosh[4*x]*Sinh[x], x, 4, (-(1/6))*Cosh[3*x] + (1/10)*Cosh[5*x]}
475{Cosh[m*x]*Sinh[x], x, 3, Cosh[(1 - m)*x]/(2*(1 - m)) + Cosh[(1 + m)*x]/(2*(1 + m))}
476
477
478(* Integrands of the form Tanh[m*x]*Sinh[x] where n is an integer *)
479{Tanh[2*x]*Sinh[x], x, 5, -(ArcTan[Sqrt[2]*Sinh[x]]/Sqrt[2]) + Sinh[x]}
480{Tanh[3*x]*Sinh[x], x, 3, (-(1/3))*ArcTan[(3*Sinh[x])/(1 - 2*Sinh[x]^2)] + Sinh[x]}
481{Tanh[4*x]*Sinh[x], x, 6, (-(1/4))*Sqrt[2 - Sqrt[2]]*ArcTan[(2*Sinh[x])/Sqrt[2 - Sqrt[2]]] - (1/4)*Sqrt[2 + Sqrt[2]]*ArcTan[(2*Sinh[x])/Sqrt[2 + Sqrt[2]]] + Sinh[x]}
482{Tanh[5*x]*Sinh[x], x, 7, (-(1/5))*ArcTan[Sinh[x]] + (1/10)*(1 + Sqrt[5])*ArcTan[(1 - Sqrt[5])*Sinh[x]] + (1/10)*(1 - Sqrt[5])*ArcTan[(1 + Sqrt[5])*Sinh[x]] + Sinh[x]}
483{Tanh[6*x]*Sinh[x], x, 7, -(ArcTan[Sqrt[2]*Sinh[x]]/(3*Sqrt[2])) + (1/12)*(Sqrt[2] + Sqrt[6])*ArcTan[(Sqrt[2] - Sqrt[6])*Sinh[x]] + (1/12)*(Sqrt[2] - Sqrt[6])*ArcTan[(Sqrt[2] + Sqrt[6])*Sinh[x]] + Sinh[x]}
484(* Before use of TryTrigReduceQ in ExpandExpression, TrigReduce expansion resulted in infinite recursion. *)
485{Tanh[n*x]*Sinh[x], x, 0, Int[Sinh[x]*Tanh[n*x], x]}
486
487
488(* Integrands of the form Coth[m*x]*Sinh[x] where n is an integer *)
489{Coth[2*x]*Sinh[x], x, 4, (-(1/2))*ArcTan[Sinh[x]] + Sinh[x]}
490{Coth[3*x]*Sinh[x], x, 4, -(ArcTan[(2*Sinh[x])/Sqrt[3]]/Sqrt[3]) + Sinh[x]}
491{Coth[4*x]*Sinh[x], x, 5, (-(1/4))*ArcTan[Sinh[x]] - ArcTan[Sqrt[2]*Sinh[x]]/(2*Sqrt[2]) + Sinh[x]}
492{Coth[5*x]*Sinh[x], x, 6, (-(1/10))*Sqrt[10 - 2*Sqrt[5]]*ArcTan[(4*Sinh[x])/Sqrt[10 - 2*Sqrt[5]]] - (1/10)*Sqrt[10 + 2*Sqrt[5]]*ArcTan[(4*Sinh[x])/Sqrt[10 + 2*Sqrt[5]]] + Sinh[x]}
493{Coth[6*x]*Sinh[x], x, 6, (-(1/6))*ArcTan[Sinh[x]] - (1/6)*ArcTan[2*Sinh[x]] - ArcTan[(2*Sinh[x])/Sqrt[3]]/(2*Sqrt[3]) + Sinh[x]}
494
495
496(* Integrands of the form Sech[m*x]*Sinh[x] where n is an integer *)
497{Sech[2*x]*Sinh[x], x, 2, -(ArcTanh[Sqrt[2]*Cosh[x]]/Sqrt[2])}
498{Sech[3*x]*Sinh[x], x, 3, (1/3)*ArcTanh[1 - (8*Cosh[x]^2)/3]}
499{Sech[4*x]*Sinh[x], x, 4, (1/4)*Sqrt[2 + Sqrt[2]]*ArcTanh[(2*Cosh[x])/Sqrt[2 - Sqrt[2]]] - (1/4)*Sqrt[2 - Sqrt[2]]*ArcTanh[(2*Cosh[x])/Sqrt[2 + Sqrt[2]]]}
500{Sech[5*x]*Sinh[x], x, 4, ArcTanh[(5 - 8*Cosh[x]^2)/Sqrt[5]]/(2*Sqrt[5]) + (1/5)*Log[Cosh[x]] - (1/20)*Log[5 - 20*Cosh[x]^2 + 16*Cosh[x]^4]}
501{Sech[6*x]*Sinh[x], x, 7, ArcTanh[Sqrt[2]*Cosh[x]]/(3*Sqrt[2]) - (1/12)*(Sqrt[2] - Sqrt[6])*ArcTanh[(Sqrt[2] - Sqrt[6])*Cosh[x]] - (1/12)*(Sqrt[2] + Sqrt[6])*ArcTanh[(Sqrt[2] + Sqrt[6])*Cosh[x]]}
502
503
504(* Integrands of the form Csch[m*x]*Sinh[x] where n is an integer *)
505{Csch[2*x]*Sinh[x], x, 3, (1/2)*ArcTan[Sinh[x]]}
506{Csch[3*x]*Sinh[x], x, 3, -(ArcTan[Sqrt[3]*Coth[x]]/Sqrt[3])}
507{Csch[4*x]*Sinh[x], x, 4, (-(1/4))*ArcTan[Sinh[x]] + ArcTan[Sqrt[2]*Sinh[x]]/(2*Sqrt[2])}
508{Csch[5*x]*Sinh[x], x, 8, (-(1/10))*Sqrt[10 - 2*Sqrt[5]]*ArcTan[Sqrt[5 - 2*Sqrt[5]]*Coth[x]] + (1/10)*Sqrt[10 + 2*Sqrt[5]]*ArcTan[Sqrt[5 + 2*Sqrt[5]]*Coth[x]]}
509{Csch[6*x]*Sinh[x], x, 6, (1/6)*ArcTan[Sinh[x]] + (1/6)*ArcTan[2*Sinh[x]] - ArcTan[(2*Sinh[x])/Sqrt[3]]/(2*Sqrt[3])}
510
511
512(* ::Subsubsection::Closed:: *)
513(*Integrands of the form Hyper[m x] Cosh[n x]*)
514
515
516(* Integrands of the form Sinh[m*x]*Cosh[x] where n is an integer *)
517{Sinh[2*x]*Cosh[x], x, 3, (2*Cosh[x]^3)/3}
518{Sinh[3*x]*Cosh[x], x, 3, (1/4)*Cosh[2*x] + (1/8)*Cosh[4*x]}
519{Sinh[4*x]*Cosh[x], x, 3, (1/6)*Cosh[3*x] + (1/10)*Cosh[5*x]}
520{Sinh[m*x]*Cosh[x], x, 4, -(Cosh[(1 - m)*x]/(2*(1 - m))) + Cosh[(1 + m)*x]/(2*(1 + m))}
521
522
523(* Integrands of the form Cosh[m*x]*Cosh[x] where n is an integer *)
524{Cosh[2*x]*Cosh[x], x, 3, Sinh[x]/2 + (1/6)*Sinh[3*x]}
525{Cosh[3*x]*Cosh[x], x, 3, (1/4)*Sinh[2*x] + (1/8)*Sinh[4*x]}
526{Cosh[4*x]*Cosh[x], x, 3, (1/6)*Sinh[3*x] + (1/10)*Sinh[5*x]}
527{Cosh[m*x]*Cosh[x], x, 3, Sinh[(1 - m)*x]/(2*(1 - m)) + Sinh[(1 + m)*x]/(2*(1 + m))}
528
529
530(* Integrands of the form Tanh[m*x]*Cosh[x] where n is an integer *)
531{Tanh[2*x]*Cosh[x], x, 5, -(ArcTanh[Sqrt[2]*Cosh[x]]/Sqrt[2]) + Cosh[x]}
532{Tanh[3*x]*Cosh[x], x, 4, -(ArcTanh[(2*Cosh[x])/Sqrt[3]]/Sqrt[3]) + Cosh[x]}
533{Tanh[4*x]*Cosh[x], x, 6, (-(1/4))*Sqrt[2 - Sqrt[2]]*ArcTanh[(2*Cosh[x])/Sqrt[2 - Sqrt[2]]] - (1/4)*Sqrt[2 + Sqrt[2]]*ArcTanh[(2*Cosh[x])/Sqrt[2 + Sqrt[2]]] + Cosh[x]}
534{Tanh[5*x]*Cosh[x], x, 6, (-(1/10))*Sqrt[10 - 2*Sqrt[5]]*ArcTanh[(4*Cosh[x])/Sqrt[10 - 2*Sqrt[5]]] - (1/10)*Sqrt[10 + 2*Sqrt[5]]*ArcTanh[(4*Cosh[x])/Sqrt[10 + 2*Sqrt[5]]] + Cosh[x]}
535{Tanh[6*x]*Cosh[x], x, 8, -(ArcTanh[Sqrt[2]*Cosh[x]]/(3*Sqrt[2])) + (1/12)*(Sqrt[2] + Sqrt[6])*ArcTanh[(Sqrt[2] - Sqrt[6])*Cosh[x]] + (1/12)*(Sqrt[2] - Sqrt[6])*ArcTanh[(Sqrt[2] + Sqrt[6])*Cosh[x]] + Cosh[x]}
536
537
538(* Integrands of the form Coth[m*x]*Cosh[x] where n is an integer *)
539{Coth[2*x]*Cosh[x], x, 4, (-(1/2))*ArcTanh[Cosh[x]] + Cosh[x]}
540{Coth[3*x]*Cosh[x], x, 3, (-(1/3))*ArcTanh[(3*Cosh[x])/(1 + 2*Cosh[x]^2)] + Cosh[x]}
541{Coth[4*x]*Cosh[x], x, 5, (-(1/4))*ArcTanh[Cosh[x]] - ArcTanh[Sqrt[2]*Cosh[x]]/(2*Sqrt[2]) + Cosh[x]}
542{Coth[5*x]*Cosh[x], x, 8, ArcTanh[(1 - 4*Cosh[x])/Sqrt[5]]/(2*Sqrt[5]) - (1/5)*ArcTanh[Cosh[x]] - ArcTanh[(1 + 4*Cosh[x])/Sqrt[5]]/(2*Sqrt[5]) + Cosh[x] - (1/20)*Log[1 - 2*Cosh[x] - 4*Cosh[x]^2] + (1/20)*Log[1 + 2*Cosh[x] - 4*Cosh[x]^2]}
543{Coth[6*x]*Cosh[x], x, 6, (-(1/6))*ArcTanh[Cosh[x]] - (1/6)*ArcTanh[2*Cosh[x]] - ArcTanh[(2*Cosh[x])/Sqrt[3]]/(2*Sqrt[3]) + Cosh[x]}
544
545
546(* Integrands of the form Sech[m*x]*Cosh[x] where n is an integer *)
547{Sech[2*x]*Cosh[x], x, 2, ArcTan[Sqrt[2]*Sinh[x]]/Sqrt[2]}
548{Sech[3*x]*Cosh[x], x, 3, ArcTan[Sqrt[3]*Tanh[x]]/Sqrt[3]}
549{Sech[4*x]*Cosh[x], x, 4, (1/4)*Sqrt[2 + Sqrt[2]]*ArcTan[(2*Sinh[x])/Sqrt[2 - Sqrt[2]]] - (1/4)*Sqrt[2 - Sqrt[2]]*ArcTan[(2*Sinh[x])/Sqrt[2 + Sqrt[2]]]}
550{Sech[5*x]*Cosh[x], x, 8, (-(1/10))*Sqrt[10 - 2*Sqrt[5]]*ArcTan[Sqrt[5 - 2*Sqrt[5]]*Tanh[x]] + (1/10)*Sqrt[10 + 2*Sqrt[5]]*ArcTan[Sqrt[5 + 2*Sqrt[5]]*Tanh[x]]}
551{Sech[6*x]*Cosh[x], x, 7, -(ArcTan[Sqrt[2]*Sinh[x]]/(3*Sqrt[2])) + (1/12)*(Sqrt[2] - Sqrt[6])*ArcTan[(Sqrt[2] - Sqrt[6])*Sinh[x]] + (1/12)*(Sqrt[2] + Sqrt[6])*ArcTan[(Sqrt[2] + Sqrt[6])*Sinh[x]]}
552
553
554(* Integrands of the form Csch[m*x]*Cosh[x] where n is an integer *)
555{Csch[2*x]*Cosh[x], x, 3, (-(1/2))*ArcTanh[Cosh[x]]}
556{Csch[3*x]*Cosh[x], x, 2, (-(1/3))*ArcTanh[1 + (8*Sinh[x]^2)/3]}
557{Csch[4*x]*Cosh[x], x, 4, (-(1/4))*ArcTanh[Cosh[x]] + ArcTanh[Sqrt[2]*Cosh[x]]/(2*Sqrt[2])}
558{Csch[5*x]*Cosh[x], x, 4, ArcTanh[(5 + 8*Sinh[x]^2)/Sqrt[5]]/(2*Sqrt[5]) + (1/5)*Log[Sinh[x]] - (1/20)*Log[5 + 20*Sinh[x]^2 + 16*Sinh[x]^4]}
559{Csch[6*x]*Cosh[x], x, 6, (-(1/6))*ArcTanh[Cosh[x]] - (1/6)*ArcTanh[2*Cosh[x]] + ArcTanh[(2*Cosh[x])/Sqrt[3]]/(2*Sqrt[3])}
560(* Before use of TryTrigReduceQ in ExpandExpression, TrigReduce expansion resulted in infinite recursion. *)
561{Coth[n*x]*Cosh[x], x, 0, Int[Cosh[x]*Coth[n*x], x]}
562
563
564(* ::Subsection::Closed:: *)
565(*Integrands of the form (Hyper[a+b x] Hyper[a+b x])^m*)
566(**)
567
568
569(* Integrands of the form (Sinh[x]*Tanh[x])^m where m is a half-integer *)
570{(Sinh[x]*Tanh[x])^(1/2), x, 2, 2*Coth[x]*Sqrt[Sinh[x]*Tanh[x]]}
571{(Sinh[x]*Tanh[x])^(3/2), x, 3, (2/3)*Csch[x]*(4 + Sinh[x]^2)*Sqrt[Sinh[x]*Tanh[x]]}
572{(Sinh[x]*Tanh[x])^(5/2), x, 4, (-(2/15))*Coth[x]*Sqrt[Sinh[x]*Tanh[x]]*(32 - (8 + 3*Sinh[x]^2)*Tanh[x]^2)}
573
574
575(* Integrands of the form (Cosh[x]*Coth[x])^m where m is a half-integer *)
576{(Cosh[x]*Coth[x])^(1/2), x, 2, 2*Sqrt[Cosh[x]*Coth[x]]*Tanh[x]}
577{(Cosh[x]*Coth[x])^(3/2), x, 3, (-(2/3))*(4 - Cosh[x]^2)*Sqrt[Cosh[x]*Coth[x]]*Sech[x]}
578{(Cosh[x]*Coth[x])^(5/2), x, 4, (2/15)*Sqrt[Cosh[x]*Coth[x]]*(32 - (8 - 3*Cosh[x]^2)*Coth[x]^2)*Tanh[x]}
579
580
581(* ::Subsection::Closed:: *)
582(*Integrands of the form Hyper[x]^m / (a+b Hyper[x])*)
583
584
585(* ::Subsubsection::Closed:: *)
586(*Integrands of the form Hyper[x]^m / (a+b Sinh[x])*)
587
588
589(* Integrands of the form Sinh[x]^m/(a+b*Sinh[x]) where m is a positive integer *)
590{Sinh[x]/(a + b*Sinh[x]), x, 2, x/b + (2*a*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b*Sqrt[a^2 + b^2])}
591{Sinh[x]^2/(a + b*Sinh[x]), x, 4, -((a*x)/b^2) - (2*a^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^2*Sqrt[a^2 + b^2]) + Cosh[x]/b}
592{Sinh[x]^3/(a + b*Sinh[x]), x, 5, (a^2*x)/b^3 - x/(2*b) + (2*a^3*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^3*Sqrt[a^2 + b^2]) - (a*Cosh[x])/b^2 + (Cosh[x]*Sinh[x])/(2*b)}
593{Sinh[x]^4/(a + b*Sinh[x]), x, 7, -((a^3*x)/b^4) + (a*x)/(2*b^2) - (2*a^4*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^4*Sqrt[a^2 + b^2]) + (a^2*Cosh[x])/b^3 - Cosh[x]/b + Cosh[x]^3/(3*b) - (a*Cosh[x]*Sinh[x])/(2*b^2)}
594
595{Sinh[x]/(I + Sinh[x]), x, 2, x + (I*Cosh[x])/(1 - I*Sinh[x])}
596{Sinh[x]^2/(I + Sinh[x]), x, 4, (-I)*x + Cosh[x] + Cosh[x]/(1 - I*Sinh[x])}
597{Sinh[x]^3/(I + Sinh[x]), x, 5, -((3*x)/2) - I*Cosh[x] - (I*Cosh[x])/(1 - I*Sinh[x]) + (1/2)*Cosh[x]*Sinh[x]}
598{Sinh[x]^4/(I + Sinh[x]), x, 7, (3*I*x)/2 - 2*Cosh[x] + Cosh[x]^3/3 - Cosh[x]/(1 - I*Sinh[x]) - (1/2)*I*Cosh[x]*Sinh[x]}
599
600
601(* Integrands of the form Cosh[x]^m/(a+b*Sinh[x]) where m is a positive integer *)
602{Cosh[x]/(a + b*Sinh[x]), x, 2, Log[a + b*Sinh[x]]/b}
603{Cosh[x]^2/(a + b*Sinh[x]), x, 4, -((a*x)/b^2) - (2*Sqrt[a^2 + b^2]*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/b^2 + Cosh[x]/b}
604{Cosh[x]^3/(a + b*Sinh[x]), x, 5, ((a^2 + b^2)*Log[a + b*Sinh[x]])/b^3 - (a*Sinh[x])/b^2 + Sinh[x]^2/(2*b)}
605{Cosh[x]^4/(a + b*Sinh[x]), x, 7, -((a^3*x)/b^4) - (3*a*x)/(2*b^2) - (2*(a^2 + b^2)^(3/2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/b^4 + (a^2*Cosh[x])/b^3 + Cosh[x]/b + Cosh[x]^3/(3*b) - (a*Cosh[x]*Sinh[x])/(2*b^2)}
606
607{Cosh[x]/(I + Sinh[x]), x, 2, Log[I + Sinh[x]]}
608{Cosh[x]^2/(I + Sinh[x]), x, 3, (-I)*x + Cosh[x]}
609{Cosh[x]^3/(I + Sinh[x]), x, 2, (-I)*Sinh[x] + Sinh[x]^2/2}
610{Cosh[x]^4/(I + Sinh[x]), x, 6, -((I*x)/2) + Cosh[x]^3/3 - (1/2)*I*Cosh[x]*Sinh[x]}
611
612
613(* Integrands of the form Tanh[x]^m/(a+b*Sinh[x]) where m is a positive integer *)
614{Tanh[x]/(a + b*Sinh[x]), x, 7, (b*ArcTan[Sinh[x]])/(a^2 + b^2) + (a*Log[Cosh[x]^2])/(2*(a^2 + b^2)) - (a*Log[a + b*Sinh[x]])/(a^2 + b^2)}
615{Tanh[x]^2/(a + b*Sinh[x]), x, 5, -((2*a^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2)) - (b*Sech[x])/(a^2 + b^2) - (a*Tanh[x])/(a^2 + b^2)}
616{Tanh[x]^3/(a + b*Sinh[x]), x, 10, -((b*ArcTan[Sinh[x]])/(2*(a^2 + b^2))) + (b*(2*a^2 + b^2)*ArcTan[Sinh[x]])/(a^2 + b^2)^2 + (a^3*Log[Cosh[x]^2])/(2*(a^2 + b^2)^2) - (a^3*Log[a + b*Sinh[x]])/(a^2 + b^2)^2 + (a*Sech[x]^2)/(2*(a^2 + b^2)) - (b*Sech[x]*Tanh[x])/(2*(a^2 + b^2))}
617{Tanh[x]^4/(a + b*Sinh[x]), x, 8, -((2*a^4*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) - (b*(2*a^2 + b^2)*Sech[x])/(a^2 + b^2)^2 + (b*Sech[x]^3)/(3*(a^2 + b^2)) + (a*Tanh[x])/(a^2 + b^2) - (a*(2*a^2 + b^2)*Tanh[x])/(a^2 + b^2)^2 - (a*Tanh[x]^3)/(3*(a^2 + b^2))}
618
619{Tanh[x]/(I + Sinh[x]), x, 5, (1/2)*ArcTan[Sinh[x]] - 1/(2*(I + Sinh[x]))}
620{Tanh[x]^2/(I + Sinh[x]), x, 6, -((7*Cosh[x])/(12*(1 - I*Sinh[x]))) - Cosh[x]/(4*(1 + I*Sinh[x])) - Cosh[x]/(6*(I + Sinh[x])^2)}
621{Tanh[x]^3/(I + Sinh[x]), x, 7, (3/8)*ArcTan[Sinh[x]] + 1/(8*(I - Sinh[x])) + I/(8*(I + Sinh[x])^2) - 1/(2*(I + Sinh[x]))}
622{Tanh[x]^4/(I + Sinh[x]), x, 11, -(Cosh[x]/(24*(I - Sinh[x])^2)) - (113*Cosh[x])/(240*(1 - I*Sinh[x])) - (13*Cosh[x])/(48*(1 + I*Sinh[x])) + (I*Cosh[x])/(20*(I + Sinh[x])^3) - (13*Cosh[x])/(60*(I + Sinh[x])^2)}
623
624
625(* Integrands of the form Coth[x]^m/(a+b*Sinh[x]) where m is a positive integer *)
626{Coth[x]/(a + b*Sinh[x]), x, 2, Log[Sinh[x]]/a - Log[a + b*Sinh[x]]/a}
627{Coth[x]^2/(a + b*Sinh[x]), x, 5, (b*ArcCoth[Cosh[x]])/a^2 - (2*Sqrt[a^2 + b^2]*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/a^2 - Coth[x]/a}
628{Coth[x]^3/(a + b*Sinh[x]), x, 5, (b*Csch[x])/a^2 - Csch[x]^2/(2*a) + ((a^2 + b^2)*Log[Sinh[x]])/a^3 - ((a^2 + b^2)*Log[a + b*Sinh[x]])/a^3}
629{Coth[x]^4/(a + b*Sinh[x]), x, 9, (3*b*ArcCoth[Cosh[x]])/(2*a^2) + (b^3*ArcCoth[Cosh[x]])/a^4 - (2*(a^2 + b^2)^(3/2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/a^4 - Coth[x]/a - (b^2*Coth[x])/a^3 - Coth[x]^3/(3*a) + (b*Coth[x]*Csch[x])/(2*a^2)}
630
631{Coth[x]/(I + Sinh[x]), x, 2, (-I)*Log[Sinh[x]] + I*Log[I + Sinh[x]]}
632{Coth[x]^2/(I + Sinh[x]), x, 4, -ArcCoth[Cosh[x]] + I*Coth[x]}
633{Coth[x]^3/(I + Sinh[x]), x, 3, -Csch[x] + (1/2)*I*Csch[x]^2}
634{Coth[x]^4/(I + Sinh[x]), x, 8, (-(1/2))*ArcCoth[Cosh[x]] + (1/3)*I*Coth[x]^3 - (1/2)*Coth[x]*Csch[x]}
635
636
637(* Integrands of the form Sech[x]^m/(a+b*Sinh[x]) where m is a positive integer *)
638{Sech[x]/(a + b*Sinh[x]), x, 7, (a*ArcTan[Sinh[x]])/(a^2 + b^2) - (b*Log[Cosh[x]^2])/(2*(a^2 + b^2)) + (b*Log[a + b*Sinh[x]])/(a^2 + b^2)}
639{Sech[x]^2/(a + b*Sinh[x]), x, 5, -((2*b^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2)) + (b*Sech[x])/(a^2 + b^2) + (a*Tanh[x])/(a^2 + b^2)}
640{Sech[x]^3/(a + b*Sinh[x]), x, 10, (a*b^2*ArcTan[Sinh[x]])/(a^2 + b^2)^2 + (a*ArcTan[Sinh[x]])/(2*(a^2 + b^2)) - (b^3*Log[Cosh[x]^2])/(2*(a^2 + b^2)^2) + (b^3*Log[a + b*Sinh[x]])/(a^2 + b^2)^2 + (b*Sech[x]^2)/(2*(a^2 + b^2)) + (a*Sech[x]*Tanh[x])/(2*(a^2 + b^2))}
641{Sech[x]^4/(a + b*Sinh[x]), x, 8, -((2*b^4*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) + (b^3*Sech[x])/(a^2 + b^2)^2 + (b*Sech[x]^3)/(3*(a^2 + b^2)) + (a*b^2*Tanh[x])/(a^2 + b^2)^2 + (a*Tanh[x])/(a^2 + b^2) - (a*Tanh[x]^3)/(3*(a^2 + b^2))}
642
643{Sech[x]/(I + Sinh[x]), x, 6, (-(1/2))*I*ArcTan[Sinh[x]] - I/(2*(I + Sinh[x]))}
644{Sech[x]^2/(I + Sinh[x]), x, 6, -((5*Cosh[x])/(12*(1 - I*Sinh[x]))) + Cosh[x]/(4*(1 + I*Sinh[x])) + Cosh[x]/(6*(I + Sinh[x])^2)}
645{Sech[x]^3/(I + Sinh[x]), x, 8, (-(3/8))*I*ArcTan[Sinh[x]] + I/(8*(I - Sinh[x])) + 1/(8*(I + Sinh[x])^2) - I/(4*(I + Sinh[x]))}
646{Sech[x]^4/(I + Sinh[x]), x, 11, -(Cosh[x]/(24*(I - Sinh[x])^2)) - (73*Cosh[x])/(240*(1 - I*Sinh[x])) + (11*Cosh[x])/(48*(1 + I*Sinh[x])) + (I*Cosh[x])/(20*(I + Sinh[x])^3) + (7*Cosh[x])/(60*(I + Sinh[x])^2)}
647
648
649(* Integrands of the form Csch[x]^m/(a+b*Sinh[x]) where m is a positive integer *)
650{Csch[x]/(a + b*Sinh[x]), x, 4, -(ArcCoth[Cosh[x]]/a) + (2*b*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a*Sqrt[a^2 + b^2])}
651{Csch[x]^2/(a + b*Sinh[x]), x, 5, (b*ArcCoth[Cosh[x]])/a^2 - (2*b^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2*Sqrt[a^2 + b^2]) - Coth[x]/a}
652{Csch[x]^3/(a + b*Sinh[x]), x, 7, ArcCoth[Cosh[x]]/(2*a) - (b^2*ArcCoth[Cosh[x]])/a^3 + (2*b^3*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^3*Sqrt[a^2 + b^2]) + (b*Coth[x])/a^2 - (Coth[x]*Csch[x])/(2*a)}
653{Csch[x]^4/(a + b*Sinh[x]), x, 9, -((b*ArcCoth[Cosh[x]])/(2*a^2)) + (b^3*ArcCoth[Cosh[x]])/a^4 - (2*b^4*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^4*Sqrt[a^2 + b^2]) + Coth[x]/a - (b^2*Coth[x])/a^3 - Coth[x]^3/(3*a) + (b*Coth[x]*Csch[x])/(2*a^2)}
654
655{Csch[x]/(I + Sinh[x]), x, 4, I*ArcCoth[Cosh[x]] - (I*Cosh[x])/(1 - I*Sinh[x])}
656{Csch[x]^2/(I + Sinh[x]), x, 5, -ArcCoth[Cosh[x]] + I*Coth[x] + Cosh[x]/(1 - I*Sinh[x])}
657{Csch[x]^3/(I + Sinh[x]), x, 7, (-(3/2))*I*ArcCoth[Cosh[x]] - Coth[x] + (1/2)*I*Coth[x]*Csch[x] + (I*Cosh[x])/(1 - I*Sinh[x])}
658{Csch[x]^4/(I + Sinh[x]), x, 9, (3/2)*ArcCoth[Cosh[x]] - 2*I*Coth[x] + (1/3)*I*Coth[x]^3 - (1/2)*Coth[x]*Csch[x] - Cosh[x]/(1 - I*Sinh[x])}
659
660
661(* ::Subsubsection::Closed:: *)
662(*Integrands of the form Hyper[x]^m / (a+b Cosh[x])*)
663
664
665(* Integrands of the form Sinh[x]^m/(a+b*Cosh[x]) where m is a positive integer *)
666{Sinh[x]/(a + b*Cosh[x]), x, 2, Log[a + b*Cosh[x]]/b}
667{Sinh[x]^2/(a + b*Cosh[x]), x, 4, -((a*x)/b^2) + (2*Sqrt[a^2 - b^2]*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/b^2 + Sinh[x]/b}
668{Sinh[x]^3/(a + b*Cosh[x]), x, 5, -((a*Cosh[x])/b^2) + Cosh[x]^2/(2*b) + ((a^2 - b^2)*Log[a + b*Cosh[x]])/b^3}
669{Sinh[x]^4/(a + b*Cosh[x]), x, 7, -((a^3*x)/b^4) + (3*a*x)/(2*b^2) + (2*(a^2 - b^2)^(3/2)*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/b^4 + (a^2*Sinh[x])/b^3 - Sinh[x]/b - (a*Cosh[x]*Sinh[x])/(2*b^2) + Sinh[x]^3/(3*b)}
670
671{Sinh[x]/(a + a*Cosh[x]), x, 3, Log[1 + Cosh[x]]/a}
672{Sinh[x]^2/(a + a*Cosh[x]), x, 3, -(x/a) + Sinh[x]/a}
673{Sinh[x]^3/(a + a*Cosh[x]), x, 2, -(Cosh[x]/a) + Cosh[x]^2/(2*a)}
674{Sinh[x]^4/(a + a*Cosh[x]), x, 6, x/(2*a) - (Cosh[x]*Sinh[x])/(2*a) + Sinh[x]^3/(3*a)}
675
676
677(* Integrands of the form Cosh[x]^m/(a+b*Cosh[x]) where m is a positive integer *)
678{Cosh[x]/(a + b*Cosh[x]), x, 2, x/b - (2*a*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(b*Sqrt[a^2 - b^2])}
679{Cosh[x]^2/(a + b*Cosh[x]), x, 4, -((a*x)/b^2) + (2*a^2*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(b^2*Sqrt[a^2 - b^2]) + Sinh[x]/b}
680{Cosh[x]^3/(a + b*Cosh[x]), x, 5, (a^2*x)/b^3 + x/(2*b) - (2*a^3*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(b^3*Sqrt[a^2 - b^2]) - (a*Sinh[x])/b^2 + (Cosh[x]*Sinh[x])/(2*b)}
681{Cosh[x]^4/(a + b*Cosh[x]), x, 7, -((a^3*x)/b^4) - (a*x)/(2*b^2) + (2*a^4*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(b^4*Sqrt[a^2 - b^2]) + (a^2*Sinh[x])/b^3 + Sinh[x]/b - (a*Cosh[x]*Sinh[x])/(2*b^2) + Sinh[x]^3/(3*b)}
682
683{Cosh[x]/(a + a*Cosh[x]), x, 2, x/a - Sinh[x]/(a*(1 + Cosh[x]))}
684{Cosh[x]^2/(a + a*Cosh[x]), x, 4, -(x/a) + Sinh[x]/a + Sinh[x]/(a*(1 + Cosh[x]))}
685{Cosh[x]^3/(a + a*Cosh[x]), x, 5, (3*x)/(2*a) - Sinh[x]/a + (Cosh[x]*Sinh[x])/(2*a) - Sinh[x]/(a*(1 + Cosh[x]))}
686{Cosh[x]^4/(a + a*Cosh[x]), x, 7, -((3*x)/(2*a)) + (2*Sinh[x])/a - (Cosh[x]*Sinh[x])/(2*a) + Sinh[x]/(a*(1 + Cosh[x])) + Sinh[x]^3/(3*a)}
687
688
689(* Integrands of the form Tanh[x]^m/(a+b*Cosh[x]) where m is a positive integer *)
690{Tanh[x]/(a + b*Cosh[x]), x, 2, Log[Cosh[x]]/a - Log[a + b*Cosh[x]]/a}
691{Tanh[x]^2/(a + b*Cosh[x]), x, 5, (b*ArcTan[Sinh[x]])/a^2 + (2*Sqrt[a^2 - b^2]*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/a^2 - Tanh[x]/a}
692{Tanh[x]^3/(a + b*Cosh[x]), x, 5, ((a^2 - b^2)*Log[Cosh[x]])/a^3 - ((a^2 - b^2)*Log[a + b*Cosh[x]])/a^3 - (b*Sech[x])/a^2 + Sech[x]^2/(2*a)}
693{Tanh[x]^4/(a + b*Cosh[x]), x, 9, (3*b*ArcTan[Sinh[x]])/(2*a^2) - (b^3*ArcTan[Sinh[x]])/a^4 + (2*(a^2 - b^2)^(3/2)*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/a^4 - Tanh[x]/a + (b^2*Tanh[x])/a^3 - (b*Sech[x]*Tanh[x])/(2*a^2) - Tanh[x]^3/(3*a)}
694
695{Tanh[x]/(a + a*Cosh[x]), x, 3, -((2*ArcTanh[1 + 2*Cosh[x]])/a)}
696{Tanh[x]^2/(a + a*Cosh[x]), x, 4, ArcTan[Sinh[x]]/a - Tanh[x]/a}
697{Tanh[x]^3/(a + a*Cosh[x]), x, 3, -(Sech[x]/a) + Sech[x]^2/(2*a)}
698{Tanh[x]^4/(a + a*Cosh[x]), x, 8, ArcTan[Sinh[x]]/(2*a) - (Sech[x]*Tanh[x])/(2*a) - Tanh[x]^3/(3*a)}
699
700
701(* Integrands of the form Coth[x]^m/(a+b*Cosh[x]) where m is a positive integer *)
702{Coth[x]/(a + b*Cosh[x]), x, 6, Log[1 - Cosh[x]]/(2*(a + b)) + Log[1 + Cosh[x]]/(2*(a - b)) - (a*Log[a + b*Cosh[x]])/(a^2 - b^2)}
703{Coth[x]^2/(a + b*Cosh[x]), x, 5, (2*a^2*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + Sinh[x]/(2*(a + b)*(1 - Cosh[x])) - Sinh[x]/(2*(a - b)*(1 + Cosh[x]))}
704{Coth[x]^3/(a + b*Cosh[x]), x, 8, 1/(4*(a + b)*(1 - Cosh[x])) + 1/(4*(a - b)*(1 + Cosh[x])) + ((2*a + b)*Log[1 - Cosh[x]])/(4*(a + b)^2) + ((2*a - b)*Log[1 + Cosh[x]])/(4*(a - b)^2) - (a^3*Log[a + b*Cosh[x]])/(a^2 - b^2)^2}
705{Coth[x]^4/(a + b*Cosh[x]), x, 9, (2*a^4*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - Sinh[x]/(12*(a + b)*(1 - Cosh[x])^2) - Sinh[x]/(12*(a + b)*(1 - Cosh[x])) + ((3*a + 2*b)*Sinh[x])/(4*(a + b)^2*(1 - Cosh[x])) + Sinh[x]/(12*(a - b)*(1 + Cosh[x])^2) - ((3*a - 2*b)*Sinh[x])/(4*(a - b)^2*(1 + Cosh[x])) + Sinh[x]/(12*(a - b)*(1 + Cosh[x]))}
706
707{Coth[x]/(a + a*Cosh[x]), x, 6, -(ArcTanh[Cosh[x]]/(2*a)) - 1/(2*a*(1 + Cosh[x]))}
708{Coth[x]^2/(a + a*Cosh[x]), x, 6, Sinh[x]/(4*a*(1 - Cosh[x])) - Sinh[x]/(6*a*(1 + Cosh[x])^2) + (7*Sinh[x])/(12*a*(1 + Cosh[x]))}
709{Coth[x]^3/(a + a*Cosh[x]), x, 8, -((3*ArcTanh[Cosh[x]])/(8*a)) + 1/(8*a*(1 - Cosh[x])) + 1/(8*a*(1 + Cosh[x])^2) - 1/(2*a*(1 + Cosh[x]))}
710{Coth[x]^4/(a + a*Cosh[x]), x, 11, -(Sinh[x]/(24*a*(1 - Cosh[x])^2)) + (13*Sinh[x])/(48*a*(1 - Cosh[x])) + Sinh[x]/(20*a*(1 + Cosh[x])^3) - (13*Sinh[x])/(60*a*(1 + Cosh[x])^2) + (113*Sinh[x])/(240*a*(1 + Cosh[x]))}
711
712
713(* Integrands of the form Sech[x]^m/(a+b*Cosh[x]) where m is a positive integer *)
714{Sech[x]/(a + b*Cosh[x]), x, 4, ArcTan[Sinh[x]]/a - (2*b*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a*Sqrt[a^2 - b^2])}
715{Sech[x]^2/(a + b*Cosh[x]), x, 5, -((b*ArcTan[Sinh[x]])/a^2) + (2*b^2*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2*Sqrt[a^2 - b^2]) + Tanh[x]/a}
716{Sech[x]^3/(a + b*Cosh[x]), x, 7, ArcTan[Sinh[x]]/(2*a) + (b^2*ArcTan[Sinh[x]])/a^3 - (2*b^3*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^3*Sqrt[a^2 - b^2]) - (b*Tanh[x])/a^2 + (Sech[x]*Tanh[x])/(2*a)}
717{Sech[x]^4/(a + b*Cosh[x]), x, 9, -((b*ArcTan[Sinh[x]])/(2*a^2)) - (b^3*ArcTan[Sinh[x]])/a^4 + (2*b^4*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^4*Sqrt[a^2 - b^2]) + Tanh[x]/a + (b^2*Tanh[x])/a^3 - (b*Sech[x]*Tanh[x])/(2*a^2) - Tanh[x]^3/(3*a)}
718
719{Sech[x]/(a + a*Cosh[x]), x, 4, ArcTan[Sinh[x]]/a - Sinh[x]/(a*(1 + Cosh[x]))}
720{Sech[x]^2/(a + a*Cosh[x]), x, 5, -(ArcTan[Sinh[x]]/a) + Sinh[x]/(a*(1 + Cosh[x])) + Tanh[x]/a}
721{Sech[x]^3/(a + a*Cosh[x]), x, 7, (3*ArcTan[Sinh[x]])/(2*a) - Sinh[x]/(a*(1 + Cosh[x])) - Tanh[x]/a + (Sech[x]*Tanh[x])/(2*a)}
722{Sech[x]^4/(a + a*Cosh[x]), x, 9, -((3*ArcTan[Sinh[x]])/(2*a)) + Sinh[x]/(a*(1 + Cosh[x])) + (2*Tanh[x])/a - (Sech[x]*Tanh[x])/(2*a) - Tanh[x]^3/(3*a)}
723
724
725(* Integrands of the form Csch[x]^m/(a+b*Cosh[x]) where m is a positive integer *)
726{Csch[x]/(a + b*Cosh[x]), x, 6, Log[1 - Cosh[x]]/(2*(a + b)) - Log[1 + Cosh[x]]/(2*(a - b)) + (b*Log[a + b*Cosh[x]])/(a^2 - b^2)}
727{Csch[x]^2/(a + b*Cosh[x]), x, 5, (2*b^2*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + Sinh[x]/(2*(a + b)*(1 - Cosh[x])) - Sinh[x]/(2*(a - b)*(1 + Cosh[x]))}
728{Csch[x]^3/(a + b*Cosh[x]), x, 8, 1/(4*(a + b)*(1 - Cosh[x])) - 1/(4*(a - b)*(1 + Cosh[x])) - ((a + 2*b)*Log[1 - Cosh[x]])/(4*(a + b)^2) + ((a - 2*b)*Log[1 + Cosh[x]])/(4*(a - b)^2) + (b^3*Log[a + b*Cosh[x]])/(a^2 - b^2)^2}
729{Csch[x]^4/(a + b*Cosh[x]), x, 9, (2*b^4*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - Sinh[x]/(12*(a + b)*(1 - Cosh[x])^2) - Sinh[x]/(12*(a + b)*(1 - Cosh[x])) - ((a + 2*b)*Sinh[x])/(4*(a + b)^2*(1 - Cosh[x])) + Sinh[x]/(12*(a - b)*(1 + Cosh[x])^2) + ((a - 2*b)*Sinh[x])/(4*(a - b)^2*(1 + Cosh[x])) + Sinh[x]/(12*(a - b)*(1 + Cosh[x]))}
730
731{Csch[x]/(a + a*Cosh[x]), x, 6, -(ArcTanh[Cosh[x]]/(2*a)) + 1/(2*a*(1 + Cosh[x]))}
732{Csch[x]^2/(a + a*Cosh[x]), x, 6, Sinh[x]/(4*a*(1 - Cosh[x])) - Sinh[x]/(6*a*(1 + Cosh[x])^2) - (5*Sinh[x])/(12*a*(1 + Cosh[x]))}
733{Csch[x]^3/(a + a*Cosh[x]), x, 8, (3*ArcTanh[Cosh[x]])/(8*a) + 1/(8*a*(1 - Cosh[x])) - 1/(8*a*(1 + Cosh[x])^2) - 1/(4*a*(1 + Cosh[x]))}
734{Csch[x]^4/(a + a*Cosh[x]), x, 11, -(Sinh[x]/(24*a*(1 - Cosh[x])^2)) - (11*Sinh[x])/(48*a*(1 - Cosh[x])) + Sinh[x]/(20*a*(1 + Cosh[x])^3) + (7*Sinh[x])/(60*a*(1 + Cosh[x])^2) + (73*Sinh[x])/(240*a*(1 + Cosh[x]))}
735
736
737(* ::Subsubsection::Closed:: *)
738(*Integrands of the form Hyper[x]^m / (a+b Tanh[x])*)
739
740
741(* Integrands of the form Sinh[x]^m/(a+b*Tanh[x]) where m is a positive integer *)
742{Sinh[x]/(a + b*Tanh[x]), x, 7, (a*b*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + (a*Cosh[x])/(a^2 - b^2) - (b*Sinh[x])/(a^2 - b^2), (2*a*b*ArcTan[(b + a*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 1/((a + b)*(1 - Tanh[x/2])) + 1/((a - b)*(1 + Tanh[x/2]))}
743{Sinh[x]^2/(a + b*Tanh[x]), x, 5, x/(4*(a - b)) + x/(4*(a + b)) - (a^3*x)/(a^2 - b^2)^2 + (a^2*b*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^2 + 1/(4*(a + b)*(1 - Tanh[x])) - 1/(4*(a - b)*(1 + Tanh[x]))}
744{Sinh[x]^3/(a + b*Tanh[x]), x, 20, -((3*a*b*ArcTan[(b + a*Tanh[x/2])/Sqrt[a^2 - b^2]])/(2*(a^2 - b^2)^(3/2))) - (a*b*(a^2 + 3*b^2)*ArcTan[(b + a*Tanh[x/2])/Sqrt[a^2 - b^2]])/(2*(a^2 - b^2)^(5/2)) + 1/(3*(a + b)*(1 - Tanh[x/2])^3) - 1/(2*(a + b)*(1 - Tanh[x/2])^2) - 3/(4*(a + b)*(1 - Tanh[x/2])) + (a + 3*b)/(4*(a + b)^2*(1 - Tanh[x/2])) + 1/(3*(a - b)*(1 + Tanh[x/2])^3) - 1/(2*(a - b)*(1 + Tanh[x/2])^2) + (a - 3*b)/(4*(a - b)^2*(1 + Tanh[x/2])) - 3/(4*(a - b)*(1 + Tanh[x/2]))}
745{Sinh[x]^4/(a + b*Tanh[x]), x, 9, -(((3*a - 2*b)*x)/(8*(a - b)^2)) + x/(16*(a - b)) + x/(16*(a + b)) - ((3*a + 2*b)*x)/(8*(a + b)^2) + (a^5*x)/(a^2 - b^2)^3 - (a^4*b*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^3 + 1/(16*(a + b)*(1 - Tanh[x])^2) + 1/(16*(a + b)*(1 - Tanh[x])) - (3*a + 2*b)/(8*(a + b)^2*(1 - Tanh[x])) - 1/(16*(a - b)*(1 + Tanh[x])^2) + (3*a - 2*b)/(8*(a - b)^2*(1 + Tanh[x])) - 1/(16*(a - b)*(1 + Tanh[x]))}
746
747{Sinh[x]/(1 + Tanh[x]), x, 6, Cosh[x]^3/3 - Sinh[x]^3/3}
748{Sinh[x]^2/(1 + Tanh[x]), x, 6, -(x/8) - (1/8)*Cosh[x]*Sinh[x] + (1/4)*Cosh[x]^3*Sinh[x] - Sinh[x]^4/4}
749{Sinh[x]^3/(1 + Tanh[x]), x, 7, (-(1/3))*Cosh[x]^3 + Cosh[x]^5/5 - Sinh[x]^5/5}
750{Sinh[x]^4/(1 + Tanh[x]), x, 7, x/16 + (1/16)*Cosh[x]*Sinh[x] - (1/8)*Cosh[x]^3*Sinh[x] + (1/6)*Cosh[x]^3*Sinh[x]^3 - Sinh[x]^6/6}
751
752
753(* Integrands of the form Cosh[x]^m/(a+b*Tanh[x]) where m is a positive integer *)
754{Cosh[x]/(a + b*Tanh[x]), x, 5, -((2*b^2*ArcTan[(b + a*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2)) - (b*Cosh[x])/(a^2 - b^2) + (a*Sinh[x])/(a^2 - b^2)}
755{Cosh[x]^2/(a + b*Tanh[x]), x, 6, -((a*b^2*x)/(a^2 - b^2)^2) + (a*x)/(2*(a^2 - b^2)) + (b^3*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^2 + (a*Cosh[x]*Sinh[x])/(2*(a^2 - b^2)) - (b*Sinh[x]^2)/(2*(a^2 - b^2))}
756{Cosh[x]^3/(a + b*Tanh[x]), x, 10, (2*b^4*ArcTan[(b + a*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) + (b^3*Cosh[x])/(a^2 - b^2)^2 - (b*Cosh[x]^3)/(3*(a^2 - b^2)) - (a*b^2*Sinh[x])/(a^2 - b^2)^2 + (a*Sinh[x])/(a^2 - b^2) + (a*Sinh[x]^3)/(3*(a^2 - b^2))}
757{Cosh[x]^4/(a + b*Tanh[x]), x, 11, (a*b^4*x)/(a^2 - b^2)^3 - (a*b^2*x)/(2*(a^2 - b^2)^2) + (3*a*x)/(8*(a^2 - b^2)) - (b*Cosh[x]^4)/(4*(a^2 - b^2)) - (b^5*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^3 - (a*b^2*Cosh[x]*Sinh[x])/(2*(a^2 - b^2)^2) + (3*a*Cosh[x]*Sinh[x])/(8*(a^2 - b^2)) + (a*Cosh[x]^3*Sinh[x])/(4*(a^2 - b^2)) + (b^3*Sinh[x]^2)/(2*(a^2 - b^2)^2)}
758
759{Cosh[x]/(1 + Tanh[x]), x, 8, (-(1/3))*Cosh[x]^3 + Sinh[x] + Sinh[x]^3/3}
760{Cosh[x]^2/(1 + Tanh[x]), x, 8, (3*x)/8 - Cosh[x]^4/4 + (3/8)*Cosh[x]*Sinh[x] + (1/4)*Cosh[x]^3*Sinh[x]}
761{Cosh[x]^3/(1 + Tanh[x]), x, 8, (-(1/5))*Cosh[x]^5 + Sinh[x] + (2*Sinh[x]^3)/3 + Sinh[x]^5/5}
762{Cosh[x]^4/(1 + Tanh[x]), x, 9, (5*x)/16 - Cosh[x]^6/6 + (5/16)*Cosh[x]*Sinh[x] + (5/24)*Cosh[x]^3*Sinh[x] + (1/6)*Cosh[x]^5*Sinh[x]}
763
764
765(* Integrands of the form Tanh[x]^m/(a+b*Tanh[x]) where m is a positive integer *)
766{Tanh[x]/(a + b*Tanh[x]), x, 2, -((b*x)/(a^2 - b^2)) + (a*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)}
767{Tanh[x]^2/(a + b*Tanh[x]), x, 4, (a*x)/(a^2 - b^2) + Log[Cosh[x]]/b - (a^2*Log[a*Cosh[x] + b*Sinh[x]])/(b*(a^2 - b^2))}
768{Tanh[x]^3/(a + b*Tanh[x]), x, 5, -((b*x)/(a^2 - b^2)) - (a*Log[Cosh[x]])/b^2 + (a^3*Log[a*Cosh[x] + b*Sinh[x]])/(b^2*(a^2 - b^2)) - Tanh[x]/b}
769{Tanh[x]^4/(a + b*Tanh[x]), x, 7, (a*x)/(a^2 - b^2) + (a^2*Log[Cosh[x]])/b^3 + Log[Cosh[x]]/b - (a^4*Log[a*Cosh[x] + b*Sinh[x]])/(b^3*(a^2 - b^2)) + (a*Tanh[x])/b^2 - Tanh[x]^2/(2*b)}
770
771{Tanh[x]/(1 + Tanh[x]), x, 2, x/2 + 1/(2*(1 + Tanh[x]))}
772{Tanh[x]^2/(1 + Tanh[x]), x, 4, -(x/2) + Log[Cosh[x]] - 1/(2*(1 + Tanh[x]))}
773{Tanh[x]^3/(1 + Tanh[x]), x, 5, (3*x)/2 - Log[Cosh[x]] - Tanh[x] + 1/(2*(1 + Tanh[x]))}
774{Tanh[x]^4/(1 + Tanh[x]), x, 7, -((3*x)/2) + 2*Log[Cosh[x]] + Tanh[x] - Tanh[x]^2/2 - 1/(2*(1 + Tanh[x]))}
775
776
777(* Integrands of the form Coth[x]^m/(a+b*Tanh[x]) where m is a positive integer *)
778{Coth[x]/(a + b*Tanh[x]), x, 4, -((b*x)/(a^2 - b^2)) + Log[Sinh[x]]/a + (b^2*Log[a*Cosh[x] + b*Sinh[x]])/(a*(a^2 - b^2))}
779{Coth[x]^2/(a + b*Tanh[x]), x, 5, (a*x)/(a^2 - b^2) - Coth[x]/a - (b*Log[Sinh[x]])/a^2 - (b^3*Log[a*Cosh[x] + b*Sinh[x]])/(a^2*(a^2 - b^2))}
780{Coth[x]^3/(a + b*Tanh[x]), x, 7, -((b*x)/(a^2 - b^2)) + (b*Coth[x])/a^2 - Coth[x]^2/(2*a) + Log[Sinh[x]]/a + (b^2*Log[Sinh[x]])/a^3 + (b^4*Log[a*Cosh[x] + b*Sinh[x]])/(a^3*(a^2 - b^2))}
781{Coth[x]^4/(a + b*Tanh[x]), x, 9, (a*x)/(a^2 - b^2) - Coth[x]/a - (b^2*Coth[x])/a^3 + (b*Coth[x]^2)/(2*a^2) - Coth[x]^3/(3*a) - (b*Log[Sinh[x]])/a^2 - (b^3*Log[Sinh[x]])/a^4 - (b^5*Log[a*Cosh[x] + b*Sinh[x]])/(a^4*(a^2 - b^2))}
782
783{Coth[x]/(1 + Tanh[x]), x, 4, -(x/2) + Log[Sinh[x]] + 1/(2*(1 + Tanh[x]))}
784{Coth[x]^2/(1 + Tanh[x]), x, 5, (3*x)/2 - Coth[x] - Log[Sinh[x]] - 1/(2*(1 + Tanh[x]))}
785{Coth[x]^3/(1 + Tanh[x]), x, 7, -((3*x)/2) + Coth[x] - Coth[x]^2/2 + 2*Log[Sinh[x]] + 1/(2*(1 + Tanh[x]))}
786{Coth[x]^4/(1 + Tanh[x]), x, 9, (5*x)/2 - 2*Coth[x] + Coth[x]^2/2 - Coth[x]^3/3 - 2*Log[Sinh[x]] - 1/(2*(1 + Tanh[x]))}
787
788
789(* Integrands of the form Sech[x]^m/(a+b*Tanh[x]) where m is a positive integer *)
790{Sech[x]/(a + b*Tanh[x]), x, 2, (2*ArcTan[(b + a*Tanh[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2]}
791{Sech[x]^2/(a + b*Tanh[x]), x, 2, Log[a + b*Tanh[x]]/b}
792{Sech[x]^3/(a + b*Tanh[x]), x, 7, (2*a*ArcTan[Tanh[x/2]])/b^2 - (2*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tanh[x/2])/Sqrt[a^2 - b^2]])/b^2 + 2/(b*(1 + Tanh[x/2]^2))}
793{Sech[x]^4/(a + b*Tanh[x]), x, 5, -(((a^2 - b^2)*Log[a + b*Tanh[x]])/b^3) + (a*Tanh[x])/b^2 - Tanh[x]^2/(2*b)}
794
795{Sech[x]/(1 + Tanh[x]), x, 2, -Cosh[x] + Sinh[x]}
796{Sech[x]^2/(1 + Tanh[x]), x, 2, Log[1 + Tanh[x]]}
797{Sech[x]^3/(1 + Tanh[x]), x, 4, ArcTan[Sinh[x]] + Sech[x]}
798{Sech[x]^4/(1 + Tanh[x]), x, 2, Tanh[x] - Tanh[x]^2/2}
799
800
801(* Integrands of the form Csch[x]^m/(a+b*Tanh[x]) where m is a positive integer *)
802{Csch[x]/(a + b*Tanh[x]), x, 4, -(ArcCoth[Cosh[x]]/a) - (2*b*ArcTan[(b + a*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a*Sqrt[a^2 - b^2])}
803{Csch[x]^2/(a + b*Tanh[x]), x, 4, -(Coth[x]/a) + (b*Log[b + a*Coth[x]])/a^2}
804{Csch[x]^3/(a + b*Tanh[x]), x, 6, (2*b*Sqrt[a^2 - b^2]*ArcTan[(b + a*Tanh[x/2])/Sqrt[a^2 - b^2]])/a^3 + (b*Coth[x/2])/(2*a^2) - Coth[x/2]^2/(8*a) - ((a^2 - 2*b^2)*Log[Tanh[x/2]])/(2*a^3) - (b*Tanh[x/2])/(2*a^2) + Tanh[x/2]^2/(8*a)}
805{Csch[x]^4/(a + b*Tanh[x]), x, 5, ((a^2 - b^2)*Coth[x])/a^3 + (b*Coth[x]^2)/(2*a^2) - Coth[x]^3/(3*a) - (b*(a^2 - b^2)*Log[b + a*Coth[x]])/a^4}
806
807{Csch[x]/(1 + Tanh[x]), x, 4, -ArcCoth[Cosh[x]] + Cosh[x] - Sinh[x]}
808{Csch[x]^2/(1 + Tanh[x]), x, 4, -Coth[x] + Log[1 + Coth[x]]}
809{Csch[x]^3/(1 + Tanh[x]), x, 5, (-(1/2))*ArcCoth[Cosh[x]] + Csch[x] - (1/2)*Coth[x]*Csch[x]}
810{Csch[x]^4/(1 + Tanh[x]), x, 2, Coth[x]^2/2 - Coth[x]^3/3}
811
812(* Following hangs Mathematica 6 & 7: *)
813{Csch[x]/(I + Tanh[x]), x, 4, I*ArcCoth[Cosh[x]] - I*Sqrt[2]*ArcTanh[(1 + I*Tanh[x/2])/Sqrt[2]]}
814
815
816(* ::Subsubsection::Closed:: *)
817(*Integrands of the form Hyper[x]^m / (a+b Coth[x])*)
818
819
820(* Integrands of the form Sinh[x]^m/(a+b*Coth[x]) where m is a positive integer *)
821{Sinh[x]/(a + b*Coth[x]), x, 5, -((2*b^2*ArcTanh[(a + b*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2)) + (a*Cosh[x])/(a^2 - b^2) - (b*Sinh[x])/(a^2 - b^2)}
822{Sinh[x]^2/(a + b*Coth[x]), x, 6, (a*b^2*x)/(a^2 - b^2)^2 - (a*x)/(2*(a^2 - b^2)) - (b^3*Log[b*Cosh[x] + a*Sinh[x]])/(a^2 - b^2)^2 + (a*Cosh[x]*Sinh[x])/(2*(a^2 - b^2)) - (b*Sinh[x]^2)/(2*(a^2 - b^2))}
823{Sinh[x]^3/(a + b*Coth[x]), x, 10, -((2*b^4*ArcTanh[(a + b*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2)) + (a*b^2*Cosh[x])/(a^2 - b^2)^2 - (a*Cosh[x])/(a^2 - b^2) + (a*Cosh[x]^3)/(3*(a^2 - b^2)) - (b^3*Sinh[x])/(a^2 - b^2)^2 - (b*Sinh[x]^3)/(3*(a^2 - b^2))}
824{Sinh[x]^4/(a + b*Coth[x]), x, 11, (a*b^4*x)/(a^2 - b^2)^3 - (a*b^2*x)/(2*(a^2 - b^2)^2) + (3*a*x)/(8*(a^2 - b^2)) - (b^5*Log[b*Cosh[x] + a*Sinh[x]])/(a^2 - b^2)^3 + (a*b^2*Cosh[x]*Sinh[x])/(2*(a^2 - b^2)^2) - (3*a*Cosh[x]*Sinh[x])/(8*(a^2 - b^2)) - (b^3*Sinh[x]^2)/(2*(a^2 - b^2)^2) + (a*Cosh[x]*Sinh[x]^3)/(4*(a^2 - b^2)) - (b*Sinh[x]^4)/(4*(a^2 - b^2))}
825
826{Sinh[x]/(1 + Coth[x]), x, 8, Cosh[x] - Cosh[x]^3/3 + Sinh[x]^3/3}
827{Sinh[x]^2/(1 + Coth[x]), x, 8, -((3*x)/8) + (3/8)*Cosh[x]*Sinh[x] - (1/4)*Cosh[x]*Sinh[x]^3 + Sinh[x]^4/4}
828{Sinh[x]^3/(1 + Coth[x]), x, 8, -Cosh[x] + (2*Cosh[x]^3)/3 - Cosh[x]^5/5 + Sinh[x]^5/5}
829{Sinh[x]^4/(1 + Coth[x]), x, 9, (5*x)/16 - (5/16)*Cosh[x]*Sinh[x] + (5/24)*Cosh[x]*Sinh[x]^3 - (1/6)*Cosh[x]*Sinh[x]^5 + Sinh[x]^6/6}
830
831
832(* Integrands of the form Cosh[x]^m/(a+b*Coth[x]) where m is a positive integer *)
833{Cosh[x]/(a + b*Coth[x]), x, 7, (2*a*b*ArcTanh[(a + b*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - (b*Cosh[x])/(a^2 - b^2) + (a*Sinh[x])/(a^2 - b^2), (2*a*b*ArcTanh[(a + b*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + 1/((a + b)*(1 - Tanh[x/2])) - 1/((a - b)*(1 + Tanh[x/2]))}
834{Cosh[x]^2/(a + b*Coth[x]), x, 5, x/(4*(a - b)) + x/(4*(a + b)) + (a*b^2*x)/(a^2 - b^2)^2 - (a^2*b*Log[b*Cosh[x] + a*Sinh[x]])/(a^2 - b^2)^2 + 1/(4*(a + b)*(1 - Tanh[x])) - 1/(4*(a - b)*(1 + Tanh[x]))}
835{Cosh[x]^3/(a + b*Coth[x]), x, 20, (3*a*b*ArcTanh[(a + b*Tanh[x/2])/Sqrt[a^2 - b^2]])/(2*(a^2 - b^2)^(3/2)) + (a*b*(a^2 + 3*b^2)*ArcTanh[(a + b*Tanh[x/2])/Sqrt[a^2 - b^2]])/(2*(a^2 - b^2)^(5/2)) + 1/(3*(a + b)*(1 - Tanh[x/2])^3) - 1/(2*(a + b)*(1 - Tanh[x/2])^2) + (a - b)/(4*(a + b)^2*(1 - Tanh[x/2])) + 3/(4*(a + b)*(1 - Tanh[x/2])) - 1/(3*(a - b)*(1 + Tanh[x/2])^3) + 1/(2*(a - b)*(1 + Tanh[x/2])^2) - 3/(4*(a - b)*(1 + Tanh[x/2])) - (a + b)/(4*(a - b)^2*(1 + Tanh[x/2]))}
836{Cosh[x]^4/(a + b*Coth[x]), x, 9, (a*x)/(8*(a - b)^2) + x/(16*(a - b)) + (a*x)/(8*(a + b)^2) + x/(16*(a + b)) + (a^3*b^2*x)/(a^2 - b^2)^3 - (a^4*b*Log[b*Cosh[x] + a*Sinh[x]])/(a^2 - b^2)^3 + 1/(16*(a + b)*(1 - Tanh[x])^2) + a/(8*(a + b)^2*(1 - Tanh[x])) + 1/(16*(a + b)*(1 - Tanh[x])) - 1/(16*(a - b)*(1 + Tanh[x])^2) - a/(8*(a - b)^2*(1 + Tanh[x])) - 1/(16*(a - b)*(1 + Tanh[x]))}
837
838{Cosh[x]/(1 + Coth[x]), x, 6, Cosh[x]^3/3 - Sinh[x]^3/3}
839{Cosh[x]^2/(1 + Coth[x]), x, 6, x/8 + Cosh[x]^4/4 + (1/8)*Cosh[x]*Sinh[x] - (1/4)*Cosh[x]^3*Sinh[x]}
840{Cosh[x]^3/(1 + Coth[x]), x, 7, Cosh[x]^5/5 - Sinh[x]^3/3 - Sinh[x]^5/5}
841{Cosh[x]^4/(1 + Coth[x]), x, 7, x/16 + Cosh[x]^6/6 + (1/16)*Cosh[x]*Sinh[x] + (1/24)*Cosh[x]^3*Sinh[x] - (1/6)*Cosh[x]^5*Sinh[x]}
842
843
844(* Integrands of the form Tanh[x]^m/(a+b*Coth[x]) where m is a positive integer *)
845{Tanh[x]/(a + b*Coth[x]), x, 4, -((b*x)/(a^2 - b^2)) + Log[Cosh[x]]/a + (b^2*Log[b*Cosh[x] + a*Sinh[x]])/(a*(a^2 - b^2))}
846{Tanh[x]^2/(a + b*Coth[x]), x, 5, (a*x)/(a^2 - b^2) - (b*Log[Cosh[x]])/a^2 - (b^3*Log[b*Cosh[x] + a*Sinh[x]])/(a^2*(a^2 - b^2)) - Tanh[x]/a}
847{Tanh[x]^3/(a + b*Coth[x]), x, 7, -((b*x)/(a^2 - b^2)) + Log[Cosh[x]]/a + (b^2*Log[Cosh[x]])/a^3 + (b^4*Log[b*Cosh[x] + a*Sinh[x]])/(a^3*(a^2 - b^2)) + (b*Tanh[x])/a^2 - Tanh[x]^2/(2*a)}
848{Tanh[x]^4/(a + b*Coth[x]), x, 9, (a*x)/(a^2 - b^2) - (b*Log[Cosh[x]])/a^2 - (b^3*Log[Cosh[x]])/a^4 - (b^5*Log[b*Cosh[x] + a*Sinh[x]])/(a^4*(a^2 - b^2)) - Tanh[x]/a - (b^2*Tanh[x])/a^3 + (b*Tanh[x]^2)/(2*a^2) - Tanh[x]^3/(3*a)}
849
850{Tanh[x]/(1 + Coth[x]), x, 4, -(x/2) + Log[Cosh[x]] - 1/(2*(1 + Tanh[x]))}
851{Tanh[x]^2/(1 + Coth[x]), x, 5, (3*x)/2 - Log[Cosh[x]] - Tanh[x] + 1/(2*(1 + Tanh[x]))}
852{Tanh[x]^3/(1 + Coth[x]), x, 7, -((3*x)/2) + 2*Log[Cosh[x]] + Tanh[x] - Tanh[x]^2/2 - 1/(2*(1 + Tanh[x]))}
853{Tanh[x]^4/(1 + Coth[x]), x, 9, (5*x)/2 - 2*Log[Cosh[x]] - 2*Tanh[x] + Tanh[x]^2/2 - Tanh[x]^3/3 + 1/(2*(1 + Tanh[x]))}
854
855
856(* Integrands of the form Coth[x]^m/(a+b*Coth[x]) where m is a positive integer *)
857{Coth[x]/(a + b*Coth[x]), x, 2, -((b*x)/(a^2 - b^2)) + (a*Log[b*Cosh[x] + a*Sinh[x]])/(a^2 - b^2)}
858{Coth[x]^2/(a + b*Coth[x]), x, 4, (a*x)/(a^2 - b^2) + Log[Sinh[x]]/b - (a^2*Log[b*Cosh[x] + a*Sinh[x]])/(b*(a^2 - b^2))}
859{Coth[x]^3/(a + b*Coth[x]), x, 5, -((b*x)/(a^2 - b^2)) - Coth[x]/b - (a*Log[Sinh[x]])/b^2 + (a^3*Log[b*Cosh[x] + a*Sinh[x]])/(b^2*(a^2 - b^2))}
860{Coth[x]^4/(a + b*Coth[x]), x, 7, (a*x)/(a^2 - b^2) + (a*Coth[x])/b^2 - Coth[x]^2/(2*b) + (a^2*Log[Sinh[x]])/b^3 + Log[Sinh[x]]/b - (a^4*Log[b*Cosh[x] + a*Sinh[x]])/(b^3*(a^2 - b^2))}
861
862{Coth[x]/(1 + Coth[x]), x, 2, x/2 + 1/(2*(1 + Coth[x]))}
863{Coth[x]^2/(1 + Coth[x]), x, 4, -(x/2) + Log[Sinh[x]] + 1/(2*(1 + Tanh[x]))}
864{Coth[x]^3/(1 + Coth[x]), x, 5, (3*x)/2 - Coth[x] - Log[Sinh[x]] - 1/(2*(1 + Tanh[x]))}
865{Coth[x]^4/(1 + Coth[x]), x, 7, -((3*x)/2) + Coth[x] - Coth[x]^2/2 + 2*Log[Sinh[x]] + 1/(2*(1 + Tanh[x]))}
866
867
868(* Integrands of the form Sech[x]^m/(a+b*Coth[x]) where m is a positive integer *)
869{Sech[x]/(a + b*Coth[x]), x, 4, ArcTan[Sinh[x]]/a + (2*b*ArcTanh[(a + b*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a*Sqrt[a^2 - b^2])}
870{Sech[x]^2/(a + b*Coth[x]), x, 4, -((b*Log[b + a*Tanh[x]])/a^2) + Tanh[x]/a}
871{Sech[x]^3/(a + b*Coth[x]), x, 12, ArcTan[Tanh[x/2]]/a - (2*b^2*ArcTan[Tanh[x/2]])/a^3 + (2*b*Sqrt[a^2 - b^2]*ArcTanh[(a + b*Tanh[x/2])/Sqrt[a^2 - b^2]])/a^3 + (2*Tanh[x/2])/(a*(1 + Tanh[x/2]^2)^2) - (2*b)/(a^2*(1 + Tanh[x/2]^2)) - Tanh[x/2]/(a*(1 + Tanh[x/2]^2))}
872{Sech[x]^4/(a + b*Coth[x]), x, 5, -((b*(a^2 - b^2)*Log[b + a*Tanh[x]])/a^4) + ((a^2 - b^2)*Tanh[x])/a^3 + (b*Tanh[x]^2)/(2*a^2) - Tanh[x]^3/(3*a)}
873
874{Sech[x]/(1 + Coth[x]), x, 4, ArcTan[Sinh[x]] + Cosh[x] - Sinh[x]}
875{Sech[x]^2/(1 + Coth[x]), x, 4, -Log[1 + Tanh[x]] + Tanh[x]}
876{Sech[x]^3/(1 + Coth[x]), x, 5, (-(1/2))*ArcTan[Sinh[x]] - Sech[x] + (1/2)*Sech[x]*Tanh[x]}
877{Sech[x]^4/(1 + Coth[x]), x, 2, Tanh[x]^2/2 - Tanh[x]^3/3}
878
879(* Following hangs Mathematica: *)
880{Sech[x]/(I + 2*Coth[x]), x, 4, (-I)*ArcTan[Sinh[x]] + (4*I*ArcTan[(I + 2*Tanh[x/2])/Sqrt[5]])/Sqrt[5]}
881
882
883(* Integrands of the form Csch[x]^m/(a+b*Coth[x]) where m is a positive integer *)
884{Csch[x]/(a + b*Coth[x]), x, 2, -((2*ArcTanh[(a + b*Tanh[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2])}
885{Csch[x]^2/(a + b*Coth[x]), x, 2, -(Log[a + b*Coth[x]]/b)}
886{Csch[x]^3/(a + b*Coth[x]), x, 6, -((2*Sqrt[a^2 - b^2]*ArcTanh[(a + b*Tanh[x/2])/Sqrt[a^2 - b^2]])/b^2) - Coth[x/2]/(2*b) - (a*Log[Tanh[x/2]])/b^2 + Tanh[x/2]/(2*b)}
887{Csch[x]^4/(a + b*Coth[x]), x, 5, (a*Coth[x])/b^2 - Coth[x]^2/(2*b) - ((a^2 - b^2)*Log[a + b*Coth[x]])/b^3}
888
889{Csch[x]/(1 + Coth[x]), x, 2, -Cosh[x] + Sinh[x]}
890{Csch[x]^2/(1 + Coth[x]), x, 2, -Log[1 + Coth[x]]}
891{Csch[x]^3/(1 + Coth[x]), x, 4, ArcCoth[Cosh[x]] - Csch[x]}
892{Csch[x]^4/(1 + Coth[x]), x, 2, Coth[x] - Coth[x]^2/2}
893
894
895(* ::Subsubsection::Closed:: *)
896(*Integrands of the form Hyper[x]^m / (a+b Sech[x])*)
897
898
899(* Integrands of the form Sinh[x]^m/(a+b*Sech[x]) where m is a positive integer *)
900{Sinh[x]/(a + b*Sech[x]), x, 4, Cosh[x]/a - (b*Log[b + a*Cosh[x]])/a^2}
901{Sinh[x]^2/(a + b*Sech[x]), x, 5, -(x/(2*a)) + (b^2*x)/a^3 + (2*b*Sqrt[a^2 - b^2]*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/a^3 - (b*Sinh[x])/a^2 + (Cosh[x]*Sinh[x])/(2*a)}
902{Sinh[x]^3/(a + b*Sech[x]), x, 5, -(((a^2 - b^2)*Cosh[x])/a^3) - (b*Cosh[x]^2)/(2*a^2) + Cosh[x]^3/(3*a) + (b*(a^2 - b^2)*Log[b + a*Cosh[x]])/a^4}
903{Sinh[x]^4/(a + b*Sech[x]), x, 9, -((5*x)/(8*a)) + (b^2*x)/(2*a^3) + ((a^2 - b^2)^2*x)/a^5 - (2*b*(a^2 - b^2)^(3/2)*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/a^5 + (b*Sinh[x])/a^2 - (b^3*Sinh[x])/a^4 - (5*Cosh[x]*Sinh[x])/(8*a) + (b^2*Cosh[x]*Sinh[x])/(2*a^3) + (Cosh[x]^3*Sinh[x])/(4*a) - (b*Sinh[x]^3)/(3*a^2)}
904
905{Sinh[x]/(a + a*Sech[x]), x, 4, Cosh[x]/a - Log[1 + Cosh[x]]/a}
906{Sinh[x]^2/(a + a*Sech[x]), x, 4, x/(2*a) - Sinh[x]/a + (Cosh[x]*Sinh[x])/(2*a)}
907{Sinh[x]^3/(a + a*Sech[x]), x, 2, -(Cosh[x]^2/(2*a)) + Cosh[x]^3/(3*a)}
908{Sinh[x]^4/(a + a*Sech[x]), x, 8, -(x/(8*a)) - (Cosh[x]*Sinh[x])/(8*a) + (Cosh[x]^3*Sinh[x])/(4*a) - Sinh[x]^3/(3*a)}
909
910
911(* Integrands of the form Cosh[x]^m/(a+b*Sech[x]) where m is a positive integer *)
912{Cosh[x]/(a + b*Sech[x]), x, 5, -((b*x)/a^2) + (2*b^2*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2*Sqrt[a^2 - b^2]) + Sinh[x]/a}
913{Cosh[x]^2/(a + b*Sech[x]), x, 6, x/(2*a) + (b^2*x)/a^3 - (2*b^3*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^3*Sqrt[a^2 - b^2]) - (b*Sinh[x])/a^2 + (Cosh[x]*Sinh[x])/(2*a)}
914{Cosh[x]^3/(a + b*Sech[x]), x, 8, -((b*x)/(2*a^2)) - (b^3*x)/a^4 + (2*b^4*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^4*Sqrt[a^2 - b^2]) + Sinh[x]/a + (b^2*Sinh[x])/a^3 - (b*Cosh[x]*Sinh[x])/(2*a^2) + Sinh[x]^3/(3*a)}
915{Cosh[x]^4/(a + b*Sech[x]), x, 10, (3*x)/(8*a) + (b^2*x)/(2*a^3) + (b^4*x)/a^5 - (2*b^5*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^5*Sqrt[a^2 - b^2]) - (b*Sinh[x])/a^2 - (b^3*Sinh[x])/a^4 + (3*Cosh[x]*Sinh[x])/(8*a) + (b^2*Cosh[x]*Sinh[x])/(2*a^3) + (Cosh[x]^3*Sinh[x])/(4*a) - (b*Sinh[x]^3)/(3*a^2)}
916
917{Cosh[x]/(a + a*Sech[x]), x, 5, -(x/a) + Sinh[x]/a + Sinh[x]/(a*(1 + Cosh[x]))}
918{Cosh[x]^2/(a + a*Sech[x]), x, 6, (3*x)/(2*a) - Sinh[x]/a + (Cosh[x]*Sinh[x])/(2*a) - Sinh[x]/(a*(1 + Cosh[x]))}
919{Cosh[x]^3/(a + a*Sech[x]), x, 8, -((3*x)/(2*a)) + (2*Sinh[x])/a - (Cosh[x]*Sinh[x])/(2*a) + Sinh[x]/(a*(1 + Cosh[x])) + Sinh[x]^3/(3*a)}
920{Cosh[x]^4/(a + a*Sech[x]), x, 10, (15*x)/(8*a) - (2*Sinh[x])/a + (7*Cosh[x]*Sinh[x])/(8*a) + (Cosh[x]^3*Sinh[x])/(4*a) - Sinh[x]/(a*(1 + Cosh[x])) - Sinh[x]^3/(3*a)}
921
922
923(* Integrands of the form Tanh[x]^m/(a+b*Sech[x]) where m is a positive integer *)
924{Tanh[x]/(a + b*Sech[x]), x, 2, Log[b + a*Cosh[x]]/a}
925{Tanh[x]^2/(a + b*Sech[x]), x, 4, x/a - ArcTan[Sinh[x]]/b + (2*Sqrt[a^2 - b^2]*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a*b)}
926{Tanh[x]^3/(a + b*Sech[x]), x, 5, (a*Log[Cosh[x]])/b^2 - ((a^2 - b^2)*Log[b + a*Cosh[x]])/(a*b^2) + Sech[x]/b}
927{Tanh[x]^4/(a + b*Sech[x]), x, 7, x/a + (a^2*ArcTan[Sinh[x]])/b^3 - (3*ArcTan[Sinh[x]])/(2*b) - (2*(a^2 - b^2)^(3/2)*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a*b^3) - (a*Tanh[x])/b^2 + (Sech[x]*Tanh[x])/(2*b)}
928
929{Tanh[x]/(a + a*Sech[x]), x, 3, Log[1 + Cosh[x]]/a}
930{Tanh[x]^2/(a + a*Sech[x]), x, 3, x/a - ArcTan[Sinh[x]]/a}
931{Tanh[x]^3/(a + a*Sech[x]), x, 3, Log[Cosh[x]]/a + Sech[x]/a}
932{Tanh[x]^4/(a + a*Sech[x]), x, 6, x/a - ArcTan[Sinh[x]]/(2*a) - Tanh[x]/a + (Sech[x]*Tanh[x])/(2*a)}
933
934
935(* Integrands of the form Coth[x]^m/(a+b*Sech[x]) where m is a positive integer *)
936{Coth[x]/(a + b*Sech[x]), x, 6, Log[1 - Cosh[x]]/(2*(a + b)) + Log[1 + Cosh[x]]/(2*(a - b)) - (b^2*Log[b + a*Cosh[x]])/(a*(a^2 - b^2))}
937{Coth[x]^2/(a + b*Sech[x]), x, 5, x/a + (2*b^3*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(3/2)) + Sinh[x]/(2*(a + b)*(1 - Cosh[x])) - Sinh[x]/(2*(a - b)*(1 + Cosh[x]))}
938{Coth[x]^3/(a + b*Sech[x]), x, 8, 1/(4*(a + b)*(1 - Cosh[x])) + 1/(4*(a - b)*(1 + Cosh[x])) + ((2*a + 3*b)*Log[1 - Cosh[x]])/(4*(a + b)^2) + ((2*a - 3*b)*Log[1 + Cosh[x]])/(4*(a - b)^2) + (b^4*Log[b + a*Cosh[x]])/(a*(a^2 - b^2)^2)}
939{Coth[x]^4/(a + b*Sech[x]), x, 9, x/a - (2*b^5*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a*(a^2 - b^2)^(5/2)) - Sinh[x]/(12*(a + b)*(1 - Cosh[x])^2) - Sinh[x]/(12*(a + b)*(1 - Cosh[x])) + ((3*a + 4*b)*Sinh[x])/(4*(a + b)^2*(1 - Cosh[x])) + Sinh[x]/(12*(a - b)*(1 + Cosh[x])^2) - ((3*a - 4*b)*Sinh[x])/(4*(a - b)^2*(1 + Cosh[x])) + Sinh[x]/(12*(a - b)*(1 + Cosh[x]))}
940
941{Coth[x]/(a + a*Sech[x]), x, 7, 1/(2*a*(1 + Cosh[x])) + Log[1 - Cosh[x]]/(4*a) + (3*Log[1 + Cosh[x]])/(4*a)}
942{Coth[x]^2/(a + a*Sech[x]), x, 6, x/a + Sinh[x]/(4*a*(1 - Cosh[x])) + Sinh[x]/(6*a*(1 + Cosh[x])^2) - (13*Sinh[x])/(12*a*(1 + Cosh[x]))}
943{Coth[x]^3/(a + a*Sech[x]), x, 9, 1/(8*a*(1 - Cosh[x])) - 1/(8*a*(1 + Cosh[x])^2) + 3/(4*a*(1 + Cosh[x])) + (5*Log[1 - Cosh[x]])/(16*a) + (11*Log[1 + Cosh[x]])/(16*a)}
944{Coth[x]^4/(a + a*Sech[x]), x, 11, x/a - Sinh[x]/(24*a*(1 - Cosh[x])^2) + (19*Sinh[x])/(48*a*(1 - Cosh[x])) - Sinh[x]/(20*a*(1 + Cosh[x])^3) + (3*Sinh[x])/(10*a*(1 + Cosh[x])^2) - (91*Sinh[x])/(80*a*(1 + Cosh[x]))}
945
946
947(* Integrands of the form Sech[x]^m/(a+b*Sech[x]) where m is a positive integer *)
948{Sech[x]/(a + b*Sech[x]), x, 2, (2*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2]}
949{Sech[x]^2/(a + b*Sech[x]), x, 6, ArcTan[Sinh[x]]/b - (2*a*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(b*Sqrt[a^2 - b^2])}
950{Sech[x]^3/(a + b*Sech[x]), x, 7, -((a*ArcTan[Sinh[x]])/b^2) + (2*a^2*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(b^2*Sqrt[a^2 - b^2]) + Tanh[x]/b}
951{Sech[x]^4/(a + b*Sech[x]), x, 9, (a^2*ArcTan[Sinh[x]])/b^3 + ArcTan[Sinh[x]]/(2*b) - (2*a^3*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(b^3*Sqrt[a^2 - b^2]) - (a*Tanh[x])/b^2 + (Sech[x]*Tanh[x])/(2*b)}
952
953{Sech[x]/(a + a*Sech[x]), x, 2, Sinh[x]/(a*(1 + Cosh[x]))}
954{Sech[x]^2/(a + a*Sech[x]), x, 6, ArcTan[Sinh[x]]/a - Sinh[x]/(a*(1 + Cosh[x]))}
955{Sech[x]^3/(a + a*Sech[x]), x, 7, -(ArcTan[Sinh[x]]/a) + Sinh[x]/(a*(1 + Cosh[x])) + Tanh[x]/a}
956{Sech[x]^4/(a + a*Sech[x]), x, 9, (3*ArcTan[Sinh[x]])/(2*a) - Sinh[x]/(a*(1 + Cosh[x])) - Tanh[x]/a + (Sech[x]*Tanh[x])/(2*a)}
957
958
959(* Integrands of the form Csch[x]^m/(a+b*Sech[x]) where m is a positive integer *)
960{Csch[x]/(a + b*Sech[x]), x, 6, Log[1 - Cosh[x]]/(2*(a + b)) - Log[1 + Cosh[x]]/(2*(a - b)) + (b*Log[b + a*Cosh[x]])/(a^2 - b^2)}
961{Csch[x]^2/(a + b*Sech[x]), x, 5, (2*a*b*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) + Sinh[x]/(2*(a + b)*(1 - Cosh[x])) - Sinh[x]/(2*(a - b)*(1 + Cosh[x]))}
962{Csch[x]^3/(a + b*Sech[x]), x, 8, 1/(4*(a + b)*(1 - Cosh[x])) - 1/(4*(a - b)*(1 + Cosh[x])) - (a*Log[1 - Cosh[x]])/(4*(a + b)^2) + (a*Log[1 + Cosh[x]])/(4*(a - b)^2) - (a^2*b*Log[b + a*Cosh[x]])/(a^2 - b^2)^2}
963{Csch[x]^4/(a + b*Sech[x]), x, 9, -((2*a^3*b*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2)) - Sinh[x]/(12*(a + b)*(1 - Cosh[x])^2) - (a*Sinh[x])/(4*(a + b)^2*(1 - Cosh[x])) - Sinh[x]/(12*(a + b)*(1 - Cosh[x])) + Sinh[x]/(12*(a - b)*(1 + Cosh[x])^2) + (a*Sinh[x])/(4*(a - b)^2*(1 + Cosh[x])) + Sinh[x]/(12*(a - b)*(1 + Cosh[x]))}
964
965{Csch[x]/(a + a*Sech[x]), x, 6, -(ArcTanh[Cosh[x]]/(2*a)) - 1/(2*a*(1 + Cosh[x]))}
966{Csch[x]^2/(a + a*Sech[x]), x, 6, Sinh[x]/(4*a*(1 - Cosh[x])) + Sinh[x]/(6*a*(1 + Cosh[x])^2) - Sinh[x]/(12*a*(1 + Cosh[x]))}
967{Csch[x]^3/(a + a*Sech[x]), x, 7, ArcTanh[Cosh[x]]/(8*a) + 1/(8*a*(1 - Cosh[x])) + 1/(8*a*(1 + Cosh[x])^2)}
968{Csch[x]^4/(a + a*Sech[x]), x, 9, -(Sinh[x]/(24*a*(1 - Cosh[x])^2)) - (5*Sinh[x])/(48*a*(1 - Cosh[x])) - Sinh[x]/(20*a*(1 + Cosh[x])^3) - Sinh[x]/(30*a*(1 + Cosh[x])^2) + (7*Sinh[x])/(240*a*(1 + Cosh[x]))}
969
970
971(* ::Subsubsection::Closed:: *)
972(*Integrands of the form Hyper[x]^m / (a+b Csch[x])*)
973
974
975(* Integrands of the form Sinh[x]^m/(a+b*Csch[x]) where m is a positive integer *)
976{Sinh[x]/(a + b*Csch[x]), x, 5, -((b*x)/a^2) - (2*b^2*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2*Sqrt[a^2 + b^2]) + Cosh[x]/a}
977{Sinh[x]^2/(a + b*Csch[x]), x, 6, -(x/(2*a)) + (b^2*x)/a^3 + (2*b^3*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^3*Sqrt[a^2 + b^2]) - (b*Cosh[x])/a^2 + (Cosh[x]*Sinh[x])/(2*a)}
978{Sinh[x]^3/(a + b*Csch[x]), x, 8, (b*x)/(2*a^2) - (b^3*x)/a^4 - (2*b^4*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^4*Sqrt[a^2 + b^2]) - Cosh[x]/a + (b^2*Cosh[x])/a^3 + Cosh[x]^3/(3*a) - (b*Cosh[x]*Sinh[x])/(2*a^2)}
979{Sinh[x]^4/(a + b*Csch[x]), x, 10, (3*x)/(8*a) - (b^2*x)/(2*a^3) + (b^4*x)/a^5 + (2*b^5*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^5*Sqrt[a^2 + b^2]) + (b*Cosh[x])/a^2 - (b^3*Cosh[x])/a^4 - (b*Cosh[x]^3)/(3*a^2) - (3*Cosh[x]*Sinh[x])/(8*a) + (b^2*Cosh[x]*Sinh[x])/(2*a^3) + (Cosh[x]*Sinh[x]^3)/(4*a)}
980
981{Sinh[x]/(I + Csch[x]), x, 5, x - I*Cosh[x] - (I*Cosh[x])/(1 + I*Sinh[x])}
982{Sinh[x]^2/(I + Csch[x]), x, 6, (3*I*x)/2 + Cosh[x] + Cosh[x]/(1 + I*Sinh[x]) - (1/2)*I*Cosh[x]*Sinh[x]}
983{Sinh[x]^3/(I + Csch[x]), x, 8, -((3*x)/2) + 2*I*Cosh[x] - (1/3)*I*Cosh[x]^3 + (I*Cosh[x])/(1 + I*Sinh[x]) + (1/2)*Cosh[x]*Sinh[x]}
984{Sinh[x]^4/(I + Csch[x]), x, 10, -((15*I*x)/8) - 2*Cosh[x] + Cosh[x]^3/3 - Cosh[x]/(1 + I*Sinh[x]) + (7/8)*I*Cosh[x]*Sinh[x] - (1/4)*I*Cosh[x]*Sinh[x]^3}
985
986
987(* Integrands of the form Cosh[x]^m/(a+b*Csch[x]) where m is a positive integer *)
988{Cosh[x]/(a + b*Csch[x]), x, 4, -((b*Log[b + a*Sinh[x]])/a^2) + Sinh[x]/a}
989{Cosh[x]^2/(a + b*Csch[x]), x, 5, x/(2*a) + (b^2*x)/a^3 + (2*b*Sqrt[a^2 + b^2]*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/a^3 - (b*Cosh[x])/a^2 + (Cosh[x]*Sinh[x])/(2*a)}
990{Cosh[x]^3/(a + b*Csch[x]), x, 5, -((b*(a^2 + b^2)*Log[b + a*Sinh[x]])/a^4) + ((a^2 + b^2)*Sinh[x])/a^3 - (b*Sinh[x]^2)/(2*a^2) + Sinh[x]^3/(3*a)}
991{Cosh[x]^4/(a + b*Csch[x]), x, 9, -((5*x)/(8*a)) - (b^2*x)/(2*a^3) + ((a^2 + b^2)^2*x)/a^5 + (2*b*(a^2 + b^2)^(3/2)*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/a^5 - (b*Cosh[x])/a^2 - (b^3*Cosh[x])/a^4 - (b*Cosh[x]^3)/(3*a^2) + (5*Cosh[x]*Sinh[x])/(8*a) + (b^2*Cosh[x]*Sinh[x])/(2*a^3) + (Cosh[x]*Sinh[x]^3)/(4*a)}
992
993{Cosh[x]/(I + Csch[x]), x, 4, Log[-I + Sinh[x]] - I*Sinh[x]}
994{Cosh[x]^2/(I + Csch[x]), x, 4, (I*x)/2 + Cosh[x] - (1/2)*I*Cosh[x]*Sinh[x]}
995{Cosh[x]^3/(I + Csch[x]), x, 2, Sinh[x]^2/2 - (1/3)*I*Sinh[x]^3}
996{Cosh[x]^4/(I + Csch[x]), x, 8, (I*x)/8 + Cosh[x]^3/3 - (1/8)*I*Cosh[x]*Sinh[x] - (1/4)*I*Cosh[x]*Sinh[x]^3}
997
998
999(* Integrands of the form Tanh[x]^m/(a+b*Csch[x]) where m is a positive integer *)
1000{Tanh[x]/(a + b*Csch[x]), x, 7, -((b*ArcTan[Sinh[x]])/(a^2 + b^2)) + (a*Log[Cosh[x]^2])/(2*(a^2 + b^2)) + (b^2*Log[b + a*Sinh[x]])/(a*(a^2 + b^2))}
1001{Tanh[x]^2/(a + b*Csch[x]), x, 5, x/a + (2*b^3*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a*(a^2 + b^2)^(3/2)) + (b*Sech[x])/(a^2 + b^2) - (a*Tanh[x])/(a^2 + b^2)}
1002{Tanh[x]^3/(a + b*Csch[x]), x, 10, (b*ArcTan[Sinh[x]])/(2*(a^2 + b^2)) - (b*(a^2 + 2*b^2)*ArcTan[Sinh[x]])/(a^2 + b^2)^2 + (a*(a^2 + 2*b^2)*Log[Cosh[x]^2])/(2*(a^2 + b^2)^2) + (b^4*Log[b + a*Sinh[x]])/(a*(a^2 + b^2)^2) + (a*Sech[x]^2)/(2*(a^2 + b^2)) + (b*Sech[x]*Tanh[x])/(2*(a^2 + b^2))}
1003{Tanh[x]^4/(a + b*Csch[x]), x, 8, x/a + (2*b^5*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a*(a^2 + b^2)^(5/2)) + (b*(a^2 + 2*b^2)*Sech[x])/(a^2 + b^2)^2 - (b*Sech[x]^3)/(3*(a^2 + b^2)) + (a*Tanh[x])/(a^2 + b^2) - (a*(2*a^2 + 3*b^2)*Tanh[x])/(a^2 + b^2)^2 - (a*Tanh[x]^3)/(3*(a^2 + b^2))}
1004
1005{Tanh[x]/(I + Csch[x]), x, 6, (-(3/4))*I*Log[-I + Sinh[x]] - (1/4)*I*Log[I + Sinh[x]] + 1/(2*(I - Sinh[x]))}
1006{Tanh[x]^2/(I + Csch[x]), x, 6, (-I)*x - Cosh[x]/(6*(I - Sinh[x])^2) + Cosh[x]/(4*(1 - I*Sinh[x])) - (13*Cosh[x])/(12*(1 + I*Sinh[x]))}
1007{Tanh[x]^3/(I + Csch[x]), x, 8, (-(11/16))*I*Log[-I + Sinh[x]] - (5/16)*I*Log[I + Sinh[x]] - I/(8*(I - Sinh[x])^2) + 3/(4*(I - Sinh[x])) + 1/(8*(I + Sinh[x]))}
1008{Tanh[x]^4/(I + Csch[x]), x, 11, (-I)*x + (I*Cosh[x])/(20*(I - Sinh[x])^3) - (3*Cosh[x])/(10*(I - Sinh[x])^2) + (19*Cosh[x])/(48*(1 - I*Sinh[x])) - (91*Cosh[x])/(80*(1 + I*Sinh[x])) + Cosh[x]/(24*(I + Sinh[x])^2)}
1009
1010
1011(* Integrands of the form Coth[x]^m/(a+b*Csch[x]) where m is a positive integer *)
1012{Coth[x]/(a + b*Csch[x]), x, 2, Log[b + a*Sinh[x]]/a}
1013{Coth[x]^2/(a + b*Csch[x]), x, 4, x/a - ArcCoth[Cosh[x]]/b + (2*Sqrt[a^2 + b^2]*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a*b)}
1014{Coth[x]^3/(a + b*Csch[x]), x, 5, -(Csch[x]/b) - (a*Log[Sinh[x]])/b^2 + ((a^2 + b^2)*Log[b + a*Sinh[x]])/(a*b^2)}
1015{Coth[x]^4/(a + b*Csch[x]), x, 7, x/a - (a^2*ArcCoth[Cosh[x]])/b^3 - (3*ArcCoth[Cosh[x]])/(2*b) + (2*(a^2 + b^2)^(3/2)*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a*b^3) + (a*Coth[x])/b^2 - (Coth[x]*Csch[x])/(2*b)}
1016
1017{Coth[x]/(I + Csch[x]), x, 3, (-I)*Log[-I + Sinh[x]]}
1018{Coth[x]^2/(I + Csch[x]), x, 3, (-I)*x - ArcCoth[Cosh[x]]}
1019{Coth[x]^3/(I + Csch[x]), x, 3, -Csch[x] - I*Log[Sinh[x]]}
1020{Coth[x]^4/(I + Csch[x]), x, 6, (-I)*x - (1/2)*ArcCoth[Cosh[x]] + I*Coth[x] - (1/2)*Coth[x]*Csch[x]}
1021
1022
1023(* Integrands of the form Sech[x]^m/(a+b*Csch[x]) where m is a positive integer *)
1024{Sech[x]/(a + b*Csch[x]), x, 7, (a*ArcTan[Sinh[x]])/(a^2 + b^2) + (b*Log[Cosh[x]^2])/(2*(a^2 + b^2)) - (b*Log[b + a*Sinh[x]])/(a^2 + b^2)}
1025{Sech[x]^2/(a + b*Csch[x]), x, 5, (2*a*b*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) - (b*Sech[x])/(a^2 + b^2) + (a*Tanh[x])/(a^2 + b^2)}
1026{Sech[x]^3/(a + b*Csch[x]), x, 10, -((a*b^2*ArcTan[Sinh[x]])/(a^2 + b^2)^2) + (a*ArcTan[Sinh[x]])/(2*(a^2 + b^2)) + (a^2*b*Log[Cosh[x]^2])/(2*(a^2 + b^2)^2) - (a^2*b*Log[b + a*Sinh[x]])/(a^2 + b^2)^2 - (b*Sech[x]^2)/(2*(a^2 + b^2)) + (a*Sech[x]*Tanh[x])/(2*(a^2 + b^2))}
1027{Sech[x]^4/(a + b*Csch[x]), x, 8, (2*a^3*b*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (a^2*b*Sech[x])/(a^2 + b^2)^2 - (b*Sech[x]^3)/(3*(a^2 + b^2)) - (a*b^2*Tanh[x])/(a^2 + b^2)^2 + (a*Tanh[x])/(a^2 + b^2) - (a*Tanh[x]^3)/(3*(a^2 + b^2))}
1028
1029{Sech[x]/(I + Csch[x]), x, 5, (-(1/2))*I*ArcTan[Sinh[x]] - I/(2*(I - Sinh[x]))}
1030{Sech[x]^2/(I + Csch[x]), x, 6, Cosh[x]/(6*(I - Sinh[x])^2) - Cosh[x]/(4*(1 - I*Sinh[x])) + Cosh[x]/(12*(1 + I*Sinh[x]))}
1031{Sech[x]^3/(I + Csch[x]), x, 6, (-(1/8))*I*ArcTan[Sinh[x]] + 1/(8*(I - Sinh[x])^2) - I/(8*(I + Sinh[x]))}
1032{Sech[x]^4/(I + Csch[x]), x, 9, (I*Cosh[x])/(20*(I - Sinh[x])^3) + Cosh[x]/(30*(I - Sinh[x])^2) - (5*Cosh[x])/(48*(1 - I*Sinh[x])) + (7*Cosh[x])/(240*(1 + I*Sinh[x])) + Cosh[x]/(24*(I + Sinh[x])^2)}
1033
1034
1035(* Integrands of the form Csch[x]^m/(a+b*Csch[x]) where m is a positive integer *)
1036{Csch[x]/(a + b*Csch[x]), x, 2, -((2*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2])}
1037{Csch[x]^2/(a + b*Csch[x]), x, 6, -(ArcCoth[Cosh[x]]/b) + (2*a*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b*Sqrt[a^2 + b^2])}
1038{Csch[x]^3/(a + b*Csch[x]), x, 7, (a*ArcCoth[Cosh[x]])/b^2 - (2*a^2*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^2*Sqrt[a^2 + b^2]) - Coth[x]/b}
1039{Csch[x]^4/(a + b*Csch[x]), x, 9, -((a^2*ArcCoth[Cosh[x]])/b^3) + ArcCoth[Cosh[x]]/(2*b) + (2*a^3*ArcTanh[(a - b*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^3*Sqrt[a^2 + b^2]) + (a*Coth[x])/b^2 - (Coth[x]*Csch[x])/(2*b)}
1040
1041{Csch[x]/(I + Csch[x]), x, 2, -(Cosh[x]/(I - Sinh[x]))}
1042{Csch[x]^2/(I + Csch[x]), x, 6, -ArcCoth[Cosh[x]] + (I*Cosh[x])/(I - Sinh[x])}
1043{Csch[x]^3/(I + Csch[x]), x, 7, I*ArcCoth[Cosh[x]] - Coth[x] + Cosh[x]/(I - Sinh[x])}
1044{Csch[x]^4/(I + Csch[x]), x, 9, (3/2)*ArcCoth[Cosh[x]] + I*Coth[x] - (1/2)*Coth[x]*Csch[x] - (I*Cosh[x])/(I - Sinh[x])}
1045
1046
1047(* ::Subsection::Closed:: *)
1048(*Integrands of the form Hyper[x]^m / (a+b Hyper[x])^2*)
1049
1050
1051(* ::Subsubsection::Closed:: *)
1052(*Integrands of the form Hyper[x]^m / (a+b Sinh[x])^2*)
1053
1054
1055(* Integrands of the form Sinh[x]^m/(a+b*Sinh[x])^2 where m is a positive integer *)
1056{Sinh[x]/(a + b*Sinh[x])^2, x, 2, -((2*b*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2)) + (a*Cosh[x])/((a^2 + b^2)*(a + b*Sinh[x]))}
1057{Sinh[x]^2/(a + b*Sinh[x])^2, x, 5, x/b^2 - (2*a^3*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^2*(a^2 + b^2)^(3/2)) + (4*a*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^2*Sqrt[a^2 + b^2]) - (a^2*Cosh[x])/(b*(a^2 + b^2)*(a + b*Sinh[x]))}
1058{Sinh[x]^3/(a + b*Sinh[x])^2, x, 6, -((2*a*x)/b^3) + (2*a^4*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^3*(a^2 + b^2)^(3/2)) - (6*a^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^3*Sqrt[a^2 + b^2]) + Cosh[x]/b^2 + (a^3*Cosh[x])/(b^2*(a^2 + b^2)*(a + b*Sinh[x]))}
1059{Sinh[x]^4/(a + b*Sinh[x])^2, x, 7, (3*a^2*x)/b^4 - x/(2*b^2) - (2*a^5*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^4*(a^2 + b^2)^(3/2)) + (8*a^3*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^4*Sqrt[a^2 + b^2]) - (2*a*Cosh[x])/b^3 + (Cosh[x]*Sinh[x])/(2*b^2) - (a^4*Cosh[x])/(b^3*(a^2 + b^2)*(a + b*Sinh[x]))}
1060
1061{Sinh[x]/(I + Sinh[x])^2, x, 4, -((2*Cosh[x])/(3*(1 - I*Sinh[x]))) - Cosh[x]/(3*(I + Sinh[x])^2)}
1062{Sinh[x]^2/(I + Sinh[x])^2, x, 5, x + (5*I*Cosh[x])/(3*(1 - I*Sinh[x])) + (I*Cosh[x])/(3*(I + Sinh[x])^2)}
1063{Sinh[x]^3/(I + Sinh[x])^2, x, 6, -2*I*x + Cosh[x] + (8*Cosh[x])/(3*(1 - I*Sinh[x])) + Cosh[x]/(3*(I + Sinh[x])^2)}
1064{Sinh[x]^4/(I + Sinh[x])^2, x, 7, -((7*x)/2) - 2*I*Cosh[x] - (11*I*Cosh[x])/(3*(1 - I*Sinh[x])) + (1/2)*Cosh[x]*Sinh[x] - (I*Cosh[x])/(3*(I + Sinh[x])^2)}
1065
1066
1067(* Integrands of the form Cosh[x]^m/(a+b*Sinh[x])^2 where m is a positive integer *)
1068{Cosh[x]/(a + b*Sinh[x])^2, x, 2, -(1/(b*(a + b*Sinh[x])))}
1069{Cosh[x]^2/(a + b*Sinh[x])^2, x, 5, x/b^2 + (2*a*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^2*Sqrt[a^2 + b^2]) - Cosh[x]/(b*(a + b*Sinh[x]))}
1070{Cosh[x]^3/(a + b*Sinh[x])^2, x, 5, -((2*a*Log[a + b*Sinh[x]])/b^3) + Sinh[x]/b^2 - (a^2 + b^2)/(b^3*(a + b*Sinh[x]))}
1071{Cosh[x]^4/(a + b*Sinh[x])^2, x, 7, (3*a^2*x)/b^4 + (3*x)/(2*b^2) + (6*a*Sqrt[a^2 + b^2]*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/b^4 - (2*a*Cosh[x])/b^3 + (Cosh[x]*Sinh[x])/(2*b^2) - ((a^2 + b^2)*Cosh[x])/(b^3*(a + b*Sinh[x]))}
1072
1073{Cosh[x]/(I + Sinh[x])^2, x, 2, -(1/(I + Sinh[x]))}
1074{Cosh[x]^2/(I + Sinh[x])^2, x, 3, x + (2*I*Cosh[x])/(1 - I*Sinh[x])}
1075{Cosh[x]^3/(I + Sinh[x])^2, x, 4, -2*I*Log[I + Sinh[x]] + Sinh[x]}
1076{Cosh[x]^4/(I + Sinh[x])^2, x, 4, -((3*x)/2) - 2*I*Cosh[x] + (1/2)*Cosh[x]*Sinh[x]}
1077
1078
1079(* Integrands of the form Tanh[x]^m/(a+b*Sinh[x])^2 where m is a positive integer *)
1080{Tanh[x]/(a + b*Sinh[x])^2, x, 8, (2*a*b*ArcTan[Sinh[x]])/(a^2 + b^2)^2 + ((a^2 - b^2)*Log[Cosh[x]^2])/(2*(a^2 + b^2)^2) - ((a^2 - b^2)*Log[a + b*Sinh[x]])/(a^2 + b^2)^2 + a/((a^2 + b^2)*(a + b*Sinh[x]))}
1081{Tanh[x]^2/(a + b*Sinh[x])^2, x, 7, -((2*a^3*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) + (4*a*b^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2) - (2*a*b*Sech[x])/(a^2 + b^2)^2 - (a^2*b*Cosh[x])/((a^2 + b^2)^2*(a + b*Sinh[x])) - ((a^2 - b^2)*Tanh[x])/(a^2 + b^2)^2}
1082{Tanh[x]^3/(a + b*Sinh[x])^2, x, 11, (4*a^3*b*ArcTan[Sinh[x]])/(a^2 + b^2)^3 - (a*b*ArcTan[Sinh[x]])/(a^2 + b^2)^2 + (a^2*(a^2 - 3*b^2)*Log[Cosh[x]^2])/(2*(a^2 + b^2)^3) - (a^2*(a^2 - 3*b^2)*Log[a + b*Sinh[x]])/(a^2 + b^2)^3 + ((a^2 - b^2)*Sech[x]^2)/(2*(a^2 + b^2)^2) + a^3/((a^2 + b^2)^2*(a + b*Sinh[x])) - (a*b*Sech[x]*Tanh[x])/(a^2 + b^2)^2}
1083(* {Tanh[x]^4/(a + b*Sinh[x])^2, x, 12, -((2*a^5*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2)) + (8*a^3*b^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) - (4*a^3*b*Sech[x])/(a^2 + b^2)^3 + (2*a*b*Sech[x]^3)/(3*(a^2 + b^2)^2) - (a^4*b*Cosh[x])/((a^2 + b^2)^3*(a + b*Sinh[x])) + (a^2*Tanh[x])/(a^2 + b^2)^2 - (b^2*Tanh[x])/(a^2 + b^2)^2 - ((2*a^4 - 3*a^2*b^2 - b^4)*Tanh[x])/(a^2 + b^2)^3 - (a^2*Tanh[x]^3)/(3*(a^2 + b^2)^2) + (b^2*Tanh[x]^3)/(3*(a^2 + b^2)^2)} *)
1084
1085{Tanh[x]/(I + Sinh[x])^2, x, 6, (-(1/4))*I*ArcTan[Sinh[x]] - 1/(4*(I + Sinh[x])^2) - I/(4*(I + Sinh[x]))}
1086{Tanh[x]^2/(I + Sinh[x])^2, x, 9, (7*I*Cosh[x])/(120*(1 - I*Sinh[x])) + (I*Cosh[x])/(8*(1 + I*Sinh[x])) - Cosh[x]/(10*(I + Sinh[x])^3) - (11*I*Cosh[x])/(60*(I + Sinh[x])^2)}
1087{Tanh[x]^3/(I + Sinh[x])^2, x, 8, (-(1/8))*I*ArcTan[Sinh[x]] - I/(16*(I - Sinh[x])) + I/(12*(I + Sinh[x])^3) - 1/(4*(I + Sinh[x])^2) - (3*I)/(16*(I + Sinh[x]))}
1088(* {Tanh[x]^4/(I + Sinh[x])^2, x, 14, (I*Cosh[x])/(48*(I - Sinh[x])^2) + (241*I*Cosh[x])/(1680*(1 - I*Sinh[x])) - (I*Cosh[x])/(48*(1 + I*Sinh[x])) + (I*Cosh[x])/(28*(I + Sinh[x])^4) - (9*Cosh[x])/(70*(I + Sinh[x])^3) - (241*I*Cosh[x])/(1680*(I + Sinh[x])^2) + Tanh[x]/4} *)
1089
1090
1091(* Integrands of the form Coth[x]^m/(a+b*Sinh[x])^2 where m is a positive integer *)
1092{Coth[x]/(a + b*Sinh[x])^2, x, 5, Log[Sinh[x]]/a^2 - Log[a + b*Sinh[x]]/a^2 + 1/(a*(a + b*Sinh[x]))}
1093{Coth[x]^2/(a + b*Sinh[x])^2, x, 7, (2*b*ArcCoth[Cosh[x]])/a^3 - (2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a*Sqrt[a^2 + b^2]) - (4*b^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^3*Sqrt[a^2 + b^2]) - Coth[x]/a^2 - (b*Cosh[x])/(a^2*(a + b*Sinh[x]))}
1094{Coth[x]^3/(a + b*Sinh[x])^2, x, 6, (2*b*Csch[x])/a^3 - Csch[x]^2/(2*a^2) + ((a^2 + 3*b^2)*Log[Sinh[x]])/a^4 - ((a^2 + 3*b^2)*Log[a + b*Sinh[x]])/a^4 + (a^2 + b^2)/(a^3*(a + b*Sinh[x]))}
1095(* {Coth[x]^4/(a + b*Sinh[x])^2, x, 11, (3*b*ArcCoth[Cosh[x]])/a^3 + (4*b^3*ArcCoth[Cosh[x]])/a^5 - (2*Sqrt[a^2 + b^2]*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/a^3 - (8*b^2*Sqrt[a^2 + b^2]*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/a^5 - Coth[x]/a^2 - (3*b^2*Coth[x])/a^4 - Coth[x]^3/(3*a^2) + (b*Coth[x]*Csch[x])/a^3 - (b*(a^2 + b^2)*Cosh[x])/(a^4*(a + b*Sinh[x]))} *)
1096
1097{Coth[x]/(I + Sinh[x])^2, x, 5, -Log[Sinh[x]] + Log[I + Sinh[x]] - I/(I + Sinh[x])}
1098{Coth[x]^2/(I + Sinh[x])^2, x, 5, 2*I*ArcCoth[Cosh[x]] + Coth[x] - (2*I*Cosh[x])/(1 - I*Sinh[x])}
1099{Coth[x]^3/(I + Sinh[x])^2, x, 5, 2*I*Csch[x] + Csch[x]^2/2 + 2*Log[Sinh[x]] - 2*Log[I + Sinh[x]]}
1100(* {Coth[x]^4/(I + Sinh[x])^2, x, 7, (-I)*ArcCoth[Cosh[x]] - 2*Coth[x] + Coth[x]^3/3 + I*Coth[x]*Csch[x]} *)
1101
1102
1103(* Integrands of the form Sech[x]^m/(a+b*Sinh[x])^2 where m is a positive integer *)
1104{Sech[x]/(a + b*Sinh[x])^2, x, 8, ((a^2 - b^2)*ArcTan[Sinh[x]])/(a^2 + b^2)^2 - (a*b*Log[Cosh[x]^2])/(a^2 + b^2)^2 + (2*a*b*Log[a + b*Sinh[x]])/(a^2 + b^2)^2 - b/((a^2 + b^2)*(a + b*Sinh[x]))}
1105{Sech[x]^2/(a + b*Sinh[x])^2, x, 7, -((6*a*b^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) + (2*a*b*Sech[x])/(a^2 + b^2)^2 - (b^3*Cosh[x])/((a^2 + b^2)^2*(a + b*Sinh[x])) + ((a^2 - b^2)*Tanh[x])/(a^2 + b^2)^2}
1106{Sech[x]^3/(a + b*Sinh[x])^2, x, 11, (b^2*(3*a^2 - b^2)*ArcTan[Sinh[x]])/(a^2 + b^2)^3 + ((a^2 - b^2)*ArcTan[Sinh[x]])/(2*(a^2 + b^2)^2) - (2*a*b^3*Log[Cosh[x]^2])/(a^2 + b^2)^3 + (4*a*b^3*Log[a + b*Sinh[x]])/(a^2 + b^2)^3 + (a*b*Sech[x]^2)/(a^2 + b^2)^2 - b^3/((a^2 + b^2)^2*(a + b*Sinh[x])) + ((a^2 - b^2)*Sech[x]*Tanh[x])/(2*(a^2 + b^2)^2)}
1107(* {Sech[x]^4/(a + b*Sinh[x])^2, x, 12, -((10*a*b^4*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2)) + (4*a*b^3*Sech[x])/(a^2 + b^2)^3 + (2*a*b*Sech[x]^3)/(3*(a^2 + b^2)^2) - (b^5*Cosh[x])/((a^2 + b^2)^3*(a + b*Sinh[x])) + (b^2*(3*a^2 - b^2)*Tanh[x])/(a^2 + b^2)^3 + (a^2*Tanh[x])/(a^2 + b^2)^2 - (b^2*Tanh[x])/(a^2 + b^2)^2 - (a^2*Tanh[x]^3)/(3*(a^2 + b^2)^2) + (b^2*Tanh[x]^3)/(3*(a^2 + b^2)^2)} *)
1108
1109{Sech[x]/(I + Sinh[x])^2, x, 7, (-(1/4))*ArcTan[Sinh[x]] - I/(4*(I + Sinh[x])^2) - 1/(4*(I + Sinh[x]))}
1110{Sech[x]^2/(I + Sinh[x])^2, x, 9, (11*I*Cosh[x])/(40*(1 - I*Sinh[x])) - (I*Cosh[x])/(8*(1 + I*Sinh[x])) + Cosh[x]/(10*(I + Sinh[x])^3) - (3*I*Cosh[x])/(20*(I + Sinh[x])^2)}
1111{Sech[x]^3/(I + Sinh[x])^2, x, 9, (-(1/4))*ArcTan[Sinh[x]] + 1/(16*(I - Sinh[x])) + 1/(12*(I + Sinh[x])^3) - I/(8*(I + Sinh[x])^2) - 3/(16*(I + Sinh[x]))}
1112(* {Sech[x]^4/(I + Sinh[x])^2, x, 14, (I*Cosh[x])/(48*(I - Sinh[x])^2) + (37*I*Cosh[x])/(336*(1 - I*Sinh[x])) - (I*Cosh[x])/(48*(1 + I*Sinh[x])) + (I*Cosh[x])/(28*(I + Sinh[x])^4) + Cosh[x]/(14*(I + Sinh[x])^3) - (37*I*Cosh[x])/(336*(I + Sinh[x])^2) - Tanh[x]/4} *)
1113
1114
1115(* Integrands of the form Csch[x]^m/(a+b*Sinh[x])^2 where m is a positive integer *)
1116{Csch[x]/(a + b*Sinh[x])^2, x, 6, -(ArcCoth[Cosh[x]]/a^2) + (2*b*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2) + (2*b*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2*Sqrt[a^2 + b^2]) + (b^2*Cosh[x])/(a*(a^2 + b^2)*(a + b*Sinh[x]))}
1117{Csch[x]^2/(a + b*Sinh[x])^2, x, 7, (2*b*ArcCoth[Cosh[x]])/a^3 - (2*b^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a*(a^2 + b^2)^(3/2)) - (4*b^2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^3*Sqrt[a^2 + b^2]) - Coth[x]/a^2 - (b^3*Cosh[x])/(a^2*(a^2 + b^2)*(a + b*Sinh[x]))}
1118{Csch[x]^3/(a + b*Sinh[x])^2, x, 9, ArcCoth[Cosh[x]]/(2*a^2) - (3*b^2*ArcCoth[Cosh[x]])/a^4 + (2*b^3*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2*(a^2 + b^2)^(3/2)) + (6*b^3*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^4*Sqrt[a^2 + b^2]) + (2*b*Coth[x])/a^3 - (Coth[x]*Csch[x])/(2*a^2) + (b^4*Cosh[x])/(a^3*(a^2 + b^2)*(a + b*Sinh[x]))}
1119(* {Csch[x]^4/(a + b*Sinh[x])^2, x, 11, -((b*ArcCoth[Cosh[x]])/a^3) + (4*b^3*ArcCoth[Cosh[x]])/a^5 - (2*b^4*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^3*(a^2 + b^2)^(3/2)) - (8*b^4*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^5*Sqrt[a^2 + b^2]) + Coth[x]/a^2 - (3*b^2*Coth[x])/a^4 - Coth[x]^3/(3*a^2) + (b*Coth[x]*Csch[x])/a^3 - (b^5*Cosh[x])/(a^4*(a^2 + b^2)*(a + b*Sinh[x]))} *)
1120
1121{Csch[x]/(I + Sinh[x])^2, x, 6, ArcCoth[Cosh[x]] - (4*Cosh[x])/(3*(1 - I*Sinh[x])) + Cosh[x]/(3*(I + Sinh[x])^2)}
1122{Csch[x]^2/(I + Sinh[x])^2, x, 7, 2*I*ArcCoth[Cosh[x]] + Coth[x] - (7*I*Cosh[x])/(3*(1 - I*Sinh[x])) + (I*Cosh[x])/(3*(I + Sinh[x])^2)}
1123{Csch[x]^3/(I + Sinh[x])^2, x, 9, (-(7/2))*ArcCoth[Cosh[x]] + 2*I*Coth[x] + (1/2)*Coth[x]*Csch[x] + (10*Cosh[x])/(3*(1 - I*Sinh[x])) - Cosh[x]/(3*(I + Sinh[x])^2)}
1124(* {Csch[x]^4/(I + Sinh[x])^2, x, 11, -5*I*ArcCoth[Cosh[x]] - 4*Coth[x] + Coth[x]^3/3 + I*Coth[x]*Csch[x] + (13*I*Cosh[x])/(3*(1 - I*Sinh[x])) - (I*Cosh[x])/(3*(I + Sinh[x])^2)} *)
1125
1126
1127(* ::Subsubsection::Closed:: *)
1128(*Integrands of the form Hyper[x]^m / (a+b Cosh[x])^2*)
1129
1130
1131{Sinh[x]^2/(a + b*Cosh[x])^2, x, 5, x/b^2 - (2*a*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(b^2*Sqrt[a^2 - b^2]) - Sinh[x]/(b*(a + b*Cosh[x]))}
1132
1133
1134(* Integrands of the form Sinh[x]^m/(1+/-Cosh[x])^n *)
1135{Sinh[x]/(1 + Cosh[x])^2, x, 2, -(1/(1 + Cosh[x]))}
1136{Sinh[x]/(1 - Cosh[x])^2, x, 2, 1/(1 - Cosh[x])}
1137{Sinh[x]^2/(1 + Cosh[x])^2, x, 2, x - 2*Tanh[x/2]}
1138{Sinh[x]^2/(1 - Cosh[x])^2, x, 2, x - 2*Coth[x/2]}
1139{Sinh[x]^3/(1 + Cosh[x])^2, x, 4, Cosh[x] - 2*Log[1 + Cosh[x]]}
1140{Sinh[x]^3/(1 - Cosh[x])^2, x, 4, Cosh[x] + 2*Log[1 - Cosh[x]]}
1141
1142
1143(* ::Subsection::Closed:: *)
1144(*Integrands of the form Hyper[x]^m / (a+b Hyper[x])^3*)
1145
1146
1147(* ::Subsubsection::Closed:: *)
1148(*Integrands of the form Hyper[x]^m / (a+b Sinh[x])^3*)
1149
1150
1151{Cosh[x]/(1 + I*Sinh[x])^3, x, 2, I/(2*(1 + I*Sinh[x])^2)}
1152{Cosh[x]/(1 - I*Sinh[x])^3, x, 2, -(I/(2*(1 - I*Sinh[x])^2))}
1153{Cosh[x]^2/(1 + I*Sinh[x])^3, x, 5, -((2*I*Cosh[x])/(3*(I - Sinh[x])^2)) - (I*Cosh[x])/(3*(1 + I*Sinh[x]))}
1154{Cosh[x]^2/(1 - I*Sinh[x])^3, x, 5, (I*Cosh[x])/(3*(1 - I*Sinh[x])) + (2*I*Cosh[x])/(3*(I + Sinh[x])^2)}
1155{Cosh[x]^3/(1 + I*Sinh[x])^3, x, 6, I*Log[-I + Sinh[x]] - 2/(I - Sinh[x])}
1156{Cosh[x]^3/(1 - I*Sinh[x])^3, x, 6, (-I)*Log[I + Sinh[x]] + 2/(I + Sinh[x])}
1157
1158
1159(* ::Subsubsection::Closed:: *)
1160(*Integrands of the form Hyper[x]^m / (a+b Cosh[x])^3*)
1161
1162
1163{Sinh[x]/(1 + Cosh[x])^3, x, 2, -(1/(2*(1 + Cosh[x])^2))}
1164{Sinh[x]/(1 - Cosh[x])^3, x, 2, 1/(2*(1 - Cosh[x])^2)}
1165{Sinh[x]^2/(1 + Cosh[x])^3, x, 5, -((2*Sinh[x])/(3*(1 + Cosh[x])^2)) + Sinh[x]/(3*(1 + Cosh[x]))}
1166{Sinh[x]^2/(1 - Cosh[x])^3, x, 5, (2*Sinh[x])/(3*(1 - Cosh[x])^2) - Sinh[x]/(3*(1 - Cosh[x]))}
1167{Sinh[x]^3/(1 + Cosh[x])^3, x, 5, 2/(1 + Cosh[x]) + Log[1 + Cosh[x]]}
1168{Sinh[x]^3/(1 - Cosh[x])^3, x, 6, -(2/(1 - Cosh[x])) - Log[1 - Cosh[x]]}
1169
1170
1171(* ::Subsection::Closed:: *)
1172(*Integrands of the form Hyper[x]^m / (a+b Hyper[x]^2)*)
1173
1174
1175(* ::Subsubsection::Closed:: *)
1176(*Integrands of the form Hyper[x]^m / (a+b Sinh[x]^2)*)
1177
1178
1179(* Integrands of the form Sinh[x]^m/(a+b*Sinh[x]^2) where m is a positive integer *)
1180{Sinh[x]/(a + b*Sinh[x]^2), x, 2, ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a - b]]/(Sqrt[a - b]*Sqrt[b])}
1181{Sinh[x]^2/(a + b*Sinh[x]^2), x, 4, x/b - (Sqrt[a]*ArcTanh[(Sqrt[a]*Coth[x])/Sqrt[a - b]])/(Sqrt[a - b]*b)}
1182{Sinh[x]^3/(a + b*Sinh[x]^2), x, 4, -((a*ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a - b]])/(Sqrt[a - b]*b^(3/2))) + Cosh[x]/b}
1183{Sinh[x]^4/(a + b*Sinh[x]^2), x, 5, -((a*x)/b^2) - x/(2*b) + (a^(3/2)*ArcTanh[(Sqrt[a]*Coth[x])/Sqrt[a - b]])/(Sqrt[a - b]*b^2) + (Cosh[x]*Sinh[x])/(2*b)}
1184{Sinh[x]^5/(a + b*Sinh[x]^2), x, 6, (a^2*ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a - b]])/(Sqrt[a - b]*b^(5/2)) - (a*Cosh[x])/b^2 - Cosh[x]/b + Cosh[x]^3/(3*b)}
1185{Sinh[x]^6/(a + b*Sinh[x]^2), x, 7, (a^2*x)/b^3 + (a*x)/(2*b^2) + (3*x)/(8*b) - (a^(5/2)*ArcTanh[(Sqrt[a]*Coth[x])/Sqrt[a - b]])/(Sqrt[a - b]*b^3) - (a*Cosh[x]*Sinh[x])/(2*b^2) - (3*Cosh[x]*Sinh[x])/(8*b) + (Cosh[x]*Sinh[x]^3)/(4*b)}
1186{Sinh[x]^7/(a + b*Sinh[x]^2), x, 9, -((a^3*ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a - b]])/(Sqrt[a - b]*b^(7/2))) + (a^2*Cosh[x])/b^3 + (a*Cosh[x])/b^2 + Cosh[x]/b - (a*Cosh[x]^3)/(3*b^2) - (2*Cosh[x]^3)/(3*b) + Cosh[x]^5/(5*b)}
1187(* {Sinh[x]^8/(a + b*Sinh[x]^2), x, 9, -((a^3*x)/b^4) - (a^2*x)/(2*b^3) - (3*a*x)/(8*b^2) - (5*x)/(16*b) + (a^(7/2)*ArcTanh[(Sqrt[a - b]*Tanh[x])/Sqrt[a]])/(Sqrt[a - b]*b^4) + (a^2*Cosh[x]*Sinh[x])/(2*b^3) + (3*a*Cosh[x]*Sinh[x])/(8*b^2) + (5*Cosh[x]*Sinh[x])/(16*b) - (a*Cosh[x]*Sinh[x]^3)/(4*b^2) - (5*Cosh[x]*Sinh[x]^3)/(24*b) + (Cosh[x]*Sinh[x]^5)/(6*b)} *)
1188
1189
1190(* Integrands of the form Cosh[x]^m/(a+b*Sinh[x]^2) where m is a positive integer *)
1191{Cosh[x]/(a + b*Sinh[x]^2), x, 2, ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])}
1192{Cosh[x]^2/(a + b*Sinh[x]^2), x, 4, x/b - (Sqrt[a - b]*ArcTanh[(Sqrt[a]*Coth[x])/Sqrt[a - b]])/(Sqrt[a]*b)}
1193{Cosh[x]^3/(a + b*Sinh[x]^2), x, 4, -(((a - b)*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a]])/(Sqrt[a]*b^(3/2))) + Sinh[x]/b}
1194{Cosh[x]^4/(a + b*Sinh[x]^2), x, 5, -((a*x)/b^2) + (3*x)/(2*b) + ((a - b)^(3/2)*ArcTanh[(Sqrt[a]*Coth[x])/Sqrt[a - b]])/(Sqrt[a]*b^2) + (Cosh[x]*Sinh[x])/(2*b)}
1195{Cosh[x]^5/(a + b*Sinh[x]^2), x, 6, ((a - b)^2*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a]])/(Sqrt[a]*b^(5/2)) - (a*Sinh[x])/b^2 + (2*Sinh[x])/b + Sinh[x]^3/(3*b)}
1196{Cosh[x]^6/(a + b*Sinh[x]^2), x, 7, ((a - b)^2*x)/b^3 - (a*x)/(2*b^2) + (7*x)/(8*b) - ((a - b)^(5/2)*ArcTanh[(Sqrt[a]*Coth[x])/Sqrt[a - b]])/(Sqrt[a]*b^3) - (a*Cosh[x]*Sinh[x])/(2*b^2) + (7*Cosh[x]*Sinh[x])/(8*b) + (Cosh[x]^3*Sinh[x])/(4*b)}
1197{Cosh[x]^7/(a + b*Sinh[x]^2), x, 9, -(((a - b)^3*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a]])/(Sqrt[a]*b^(7/2))) + ((a - b)^2*Sinh[x])/b^3 - (a*Sinh[x])/b^2 + (2*Sinh[x])/b - (a*Sinh[x]^3)/(3*b^2) + Sinh[x]^3/b + Sinh[x]^5/(5*b)}
1198(* {Cosh[x]^8/(a + b*Sinh[x]^2), x, 9, -(((a - b)^3*x)/b^4) + ((a - b)^2*x)/(2*b^3) - (3*a*x)/(8*b^2) + (11*x)/(16*b) + ((a - b)^(7/2)*ArcTanh[(Sqrt[a - b]*Tanh[x])/Sqrt[a]])/(Sqrt[a]*b^4) + ((a - b)^2*Cosh[x]*Sinh[x])/(2*b^3) - (3*a*Cosh[x]*Sinh[x])/(8*b^2) + (11*Cosh[x]*Sinh[x])/(16*b) - (a*Cosh[x]^3*Sinh[x])/(4*b^2) + (11*Cosh[x]^3*Sinh[x])/(24*b) + (Cosh[x]^5*Sinh[x])/(6*b)} *)
1199
1200
1201(* Integrands of the form Cosh[x]^m/(a+b*Sinh[x]^n) where m and n are integers *)
1202{Cosh[x]^2/(a + a*Sinh[x]^2), x, 2, x/a}
1203{Cosh[x]^3/(a + a*Sinh[x]^2), x, 3, Sinh[x]/a}
1204{Cosh[x]^4/(a + a*Sinh[x]^2), x, 3, x/(2*a) + (Cosh[x]*Sinh[x])/(2*a)}
1205
1206{Cosh[x]^2/(1 - Sinh[x]^2), x, 4, -x + Sqrt[2]*ArcTanh[Coth[x]/Sqrt[2]]}
1207{Cosh[x]^3/(1 - Sinh[x]^2), x, 4, 2*ArcTanh[Sinh[x]] - Sinh[x]}
1208{Cosh[x]^4/(1 - Sinh[x]^2), x, 5, -((5*x)/2) + 2*Sqrt[2]*ArcTanh[Coth[x]/Sqrt[2]] - (1/2)*Cosh[x]*Sinh[x]}
1209
1210
1211{Coth[x]/(1 - Sinh[x]^2), x, 2, -ArcTanh[1 - 2*Sinh[x]^2]}
1212
1213
1214(* ::Subsubsection::Closed:: *)
1215(*Integrands of the form Hyper[x]^m / (a+b Cosh[x]^2)*)
1216
1217
1218(* Integrands of the form Sinh[x]^m/(a+b*Cosh[x]^2) where m is a positive integer *)
1219{Sinh[x]/(a + b*Cosh[x]^2), x, 2, ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b])}
1220{Sinh[x]^2/(a + b*Cosh[x]^2), x, 4, x/b - (Sqrt[a + b]*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(Sqrt[a]*b)}
1221{Sinh[x]^3/(a + b*Cosh[x]^2), x, 4, -(((a + b)*ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]])/(Sqrt[a]*b^(3/2))) + Cosh[x]/b}
1222{Sinh[x]^4/(a + b*Cosh[x]^2), x, 5, -((a*x)/b^2) - (3*x)/(2*b) + ((a + b)^(3/2)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(Sqrt[a]*b^2) + (Cosh[x]*Sinh[x])/(2*b)}
1223{Sinh[x]^5/(a + b*Cosh[x]^2), x, 6, ((a + b)^2*ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]])/(Sqrt[a]*b^(5/2)) - (a*Cosh[x])/b^2 - (2*Cosh[x])/b + Cosh[x]^3/(3*b)}
1224{Sinh[x]^6/(a + b*Cosh[x]^2), x, 7, (a*x)/(2*b^2) + (7*x)/(8*b) + ((a + b)^2*x)/b^3 - ((a + b)^(5/2)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(Sqrt[a]*b^3) - (a*Cosh[x]*Sinh[x])/(2*b^2) - (7*Cosh[x]*Sinh[x])/(8*b) + (Cosh[x]*Sinh[x]^3)/(4*b)}
1225{Sinh[x]^7/(a + b*Cosh[x]^2), x, 9, -(((a + b)^3*ArcTan[(Sqrt[b]*Cosh[x])/Sqrt[a]])/(Sqrt[a]*b^(7/2))) + (a*Cosh[x])/b^2 + (2*Cosh[x])/b + ((a + b)^2*Cosh[x])/b^3 - (a*Cosh[x]^3)/(3*b^2) - Cosh[x]^3/b + Cosh[x]^5/(5*b)}
1226(* {Sinh[x]^8/(a + b*Cosh[x]^2), x, 9, -((3*a*x)/(8*b^2)) - (11*x)/(16*b) - ((a + b)^2*x)/(2*b^3) - ((a + b)^3*x)/b^4 + ((a + b)^(7/2)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(Sqrt[a]*b^4) + (3*a*Cosh[x]*Sinh[x])/(8*b^2) + (11*Cosh[x]*Sinh[x])/(16*b) + ((a + b)^2*Cosh[x]*Sinh[x])/(2*b^3) - (a*Cosh[x]*Sinh[x]^3)/(4*b^2) - (11*Cosh[x]*Sinh[x]^3)/(24*b) + (Cosh[x]*Sinh[x]^5)/(6*b)} *)
1227
1228
1229(* Integrands of the form Cosh[x]^m/(a+b*Cosh[x]^2) where m is a positive integer *)
1230{Cosh[x]/(a + b*Cosh[x]^2), x, 2, ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]]/(Sqrt[b]*Sqrt[a + b])}
1231{Cosh[x]^2/(a + b*Cosh[x]^2), x, 4, x/b - (Sqrt[a]*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(b*Sqrt[a + b])}
1232{Cosh[x]^3/(a + b*Cosh[x]^2), x, 4, -((a*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]])/(b^(3/2)*Sqrt[a + b])) + Sinh[x]/b}
1233{Cosh[x]^4/(a + b*Cosh[x]^2), x, 5, -((a*x)/b^2) + x/(2*b) + (a^(3/2)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(b^2*Sqrt[a + b]) + (Cosh[x]*Sinh[x])/(2*b)}
1234{Cosh[x]^5/(a + b*Cosh[x]^2), x, 6, (a^2*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]])/(b^(5/2)*Sqrt[a + b]) - (a*Sinh[x])/b^2 + Sinh[x]/b + Sinh[x]^3/(3*b)}
1235{Cosh[x]^6/(a + b*Cosh[x]^2), x, 7, (a^2*x)/b^3 - (a*x)/(2*b^2) + (3*x)/(8*b) - (a^(5/2)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(b^3*Sqrt[a + b]) - (a*Cosh[x]*Sinh[x])/(2*b^2) + (3*Cosh[x]*Sinh[x])/(8*b) + (Cosh[x]^3*Sinh[x])/(4*b)}
1236{Cosh[x]^7/(a + b*Cosh[x]^2), x, 9, -((a^3*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a + b]])/(b^(7/2)*Sqrt[a + b])) + (a^2*Sinh[x])/b^3 - (a*Sinh[x])/b^2 + Sinh[x]/b - (a*Sinh[x]^3)/(3*b^2) + (2*Sinh[x]^3)/(3*b) + Sinh[x]^5/(5*b)}
1237(* {Cosh[x]^8/(a + b*Cosh[x]^2), x, 9, -((a^3*x)/b^4) + (a^2*x)/(2*b^3) - (3*a*x)/(8*b^2) + (5*x)/(16*b) + (a^(7/2)*ArcTanh[(Sqrt[a]*Tanh[x])/Sqrt[a + b]])/(b^4*Sqrt[a + b]) + (a^2*Cosh[x]*Sinh[x])/(2*b^3) - (3*a*Cosh[x]*Sinh[x])/(8*b^2) + (5*Cosh[x]*Sinh[x])/(16*b) - (a*Cosh[x]^3*Sinh[x])/(4*b^2) + (5*Cosh[x]^3*Sinh[x])/(24*b) + (Cosh[x]^5*Sinh[x])/(6*b)} *)
1238
1239
1240(* Integrands of the form Sinh[x]^m/(a+b*Cosh[x]^2) where a^2==b^2 and m is an integer *)
1241{Sinh[x]^2/(a - a*Cosh[x]^2), x, 2, -x/a}
1242{Sinh[x]^3/(a - a*Cosh[x]^2), x, 3, -(Cosh[x]/a)}
1243{Sinh[x]^4/(a - a*Cosh[x]^2), x, 3, x/(2*a) - (Cosh[x]*Sinh[x])/(2*a)}
1244
1245
1246{Tanh[x]/(1 + Cosh[x]^2), x, 2, -ArcTanh[1 + 2*Cosh[x]^2]}
1247
1248
1249(* ::Subsubsection::Closed:: *)
1250(*Integrands of the form x^m Hyper[x]^2 / (a+b Tanh[x]^2)*)
1251
1252
1253{Sech[c + d*x]^2/(a + b*Tanh[c + d*x]^2), x, 2, ArcTan[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a]]/(Sqrt[a]*Sqrt[b]*d)}
1254{x*Sech[c + d*x]^2/(a + b*Tanh[c + d*x]^2), x, 10, (x*Log[1 + ((a + b)*E^(2*c + 2*d*x))/(a - 2*Sqrt[-a]*Sqrt[b] - b)])/(2*Sqrt[-a]*Sqrt[b]*d) - (x*Log[1 + ((a + b)*E^(2*c + 2*d*x))/(a + 2*Sqrt[-a]*Sqrt[b] - b)])/(2*Sqrt[-a]*Sqrt[b]*d) + PolyLog[2, -(((a + b)*E^(2*c + 2*d*x))/(a - 2*Sqrt[-a]*Sqrt[b] - b))]/(4*Sqrt[-a]*Sqrt[b]*d^2) - PolyLog[2, -(((a + b)*E^(2*c + 2*d*x))/(a + 2*Sqrt[-a]*Sqrt[b] - b))]/(4*Sqrt[-a]*Sqrt[b]*d^2)}
1255{x^2*Sech[c + d*x]^2/(a + b*Tanh[c + d*x]^2), x, 12, (x^2*Log[1 + ((a + b)*E^(2*c + 2*d*x))/(a - 2*Sqrt[-a]*Sqrt[b] - b)])/(2*Sqrt[-a]*Sqrt[b]*d) - (x^2*Log[1 + ((a + b)*E^(2*c + 2*d*x))/(a + 2*Sqrt[-a]*Sqrt[b] - b)])/(2*Sqrt[-a]*Sqrt[b]*d) + (x*PolyLog[2, -(((a + b)*E^(2*c + 2*d*x))/(a - 2*Sqrt[-a]*Sqrt[b] - b))])/(2*Sqrt[-a]*Sqrt[b]*d^2) - (x*PolyLog[2, -(((a + b)*E^(2*c + 2*d*x))/(a + 2*Sqrt[-a]*Sqrt[b] - b))])/(2*Sqrt[-a]*Sqrt[b]*d^2) - PolyLog[3, -(((a + b)*E^(2*c + 2*d*x))/(a - 2*Sqrt[-a]*Sqrt[b] - b))]/(4*Sqrt[-a]*Sqrt[b]*d^3) + PolyLog[3, -(((a + b)*E^(2*c + 2*d*x))/(a + 2*Sqrt[-a]*Sqrt[b] - b))]/(4*Sqrt[-a]*Sqrt[b]*d^3)}
1256
1257
1258(* ::Subsection::Closed:: *)
1259(*Integrands of the form Hyper[x]^m / (a+b Hyper[x]^n)*)
1260
1261
1262(* ::Subsubsection::Closed:: *)
1263(*Integrands of the form Hyper[x]^m / (a+b Sinh[x]^3)*)
1264
1265
1266{Coth[x]^3/(a + b*Sinh[x]^3), x, 10, (b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Sinh[x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3)) - Csch[x]^2/(2*a) + Log[Sinh[x]]/a - (b^(2/3)*Log[a^(1/3) + b^(1/3)*Sinh[x]])/(3*a^(5/3)) + (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Sinh[x] + b^(2/3)*Sinh[x]^2])/(6*a^(5/3)) - Log[a + b*Sinh[x]^3]/(3*a)}
1267
1268
1269(* ::Subsubsection::Closed:: *)
1270(*Integrands of the form Hyper[x]^m / (a+b Cosh[x]^3)*)
1271
1272
1273{Sinh[x]/(4 - 3*Cosh[x]^3), x, 6, ArcTan[(6^(2/3) + 6*Cosh[x])/(3*2^(2/3)*3^(1/6))]/(2*2^(1/3)*3^(5/6)) - Log[6^(2/3) - 3*Cosh[x]]/(6*6^(1/3)) + Log[2*6^(1/3) + 6^(2/3)*Cosh[x] + 3*Cosh[x]^2]/(12*6^(1/3))}
1274
1275
1276{Tanh[x]^3/(a + b*Cosh[x]^3), x, 10, -((b^(2/3)*ArcTan[(a^(1/3) - 2*b^(1/3)*Cosh[x])/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3))) + Log[Cosh[x]]/a + (b^(2/3)*Log[a^(1/3) + b^(1/3)*Cosh[x]])/(3*a^(5/3)) - (b^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*Cosh[x] + b^(2/3)*Cosh[x]^2])/(6*a^(5/3)) - Log[a + b*Cosh[x]^3]/(3*a) + Sech[x]^2/(2*a)}
1277
1278
1279(* ::Subsection::Closed:: *)
1280(*Integrands of the form Hyper[x]^m (a+b Hyper[x])^n*)
1281
1282
1283(* ::Subsubsection::Closed:: *)
1284(*Integrands of the form Hyper[x]^m (a+b Sinh[x])^n*)
1285
1286
1287{Sinh[x]/Sqrt[a + a*I*Sinh[x]], x, 3, -((2*ArcTanh[Sin[Pi/4 - (I*x)/2]]*Cos[Pi/4 - (I*x)/2])/Sqrt[a + I*a*Sinh[x]]) + (2*Cosh[x])/Sqrt[a + I*a*Sinh[x]]}
1288{Sinh[x]/Sqrt[a - a*I*Sinh[x]], x, 3, (2*Cosh[x])/Sqrt[a - I*a*Sinh[x]] - (2*ArcTanh[Cos[Pi/4 - (I*x)/2]]*Sin[Pi/4 - (I*x)/2])/Sqrt[a - I*a*Sinh[x]]}
1289{Sinh[x]/Sqrt[a + b*Sinh[x]], x, 5, (2*I*EllipticE[Pi/4 - (I*x)/2, -((2*I*b)/(a - I*b))]*Sqrt[a + b*Sinh[x]])/(b*Sqrt[(a + b*Sinh[x])/(a - I*b)]) - (2*I*a*EllipticF[Pi/4 - (I*x)/2, -((2*I*b)/(a - I*b))]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(b*Sqrt[a + b*Sinh[x]])}
1290
1291
1292(* ::Subsubsection::Closed:: *)
1293(*Integrands of the form Hyper[x]^m (a+b Cosh[x])^n*)
1294
1295
1296{Cosh[x]/Sqrt[a + a*Cosh[x]], x, 3, -((2*ArcTan[Sinh[x/2]]*Cosh[x/2])/Sqrt[a + a*Cosh[x]]) + (2*Sinh[x])/Sqrt[a + a*Cosh[x]]}
1297{Cosh[x]/Sqrt[a - a*Cosh[x]], x, 3, -((2*ArcTanh[Cosh[x/2]]*Sinh[x/2])/Sqrt[a - a*Cosh[x]]) + (2*Sinh[x])/Sqrt[a - a*Cosh[x]]}
1298{Cosh[x]/Sqrt[a + b*Cosh[x]], x, 5, -((2*I*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(b*Sqrt[(a + b*Cosh[x])/(a + b)])) + (2*I*a*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + b)])/(b*Sqrt[a + b*Cosh[x]])}
1299
1300
1301(* ::Subsubsection::Closed:: *)
1302(*Integrands of the form Hyper[x]^m (a+b Tanh[x])^n*)
1303
1304
1305(* Integrands of the form Tanh[x]^m*(1+Tanh[x])^n where m is an integer and n is a half-integer *)
1306{Tanh[x]*(1 + Tanh[x])^(3/2), x, 3, 2*Sqrt[2]*ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]] - 2*Sqrt[1 + Tanh[x]] - (2/3)*(1 + Tanh[x])^(3/2)}
1307{Tanh[x]*Sqrt[1 + Tanh[x]], x, 2, Sqrt[2]*ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]] - 2*Sqrt[1 + Tanh[x]]}
1308{Tanh[x]/Sqrt[1 + Tanh[x]], x, 6, ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]]/Sqrt[2] + 1/Sqrt[1 + Tanh[x]]}
1309{Tanh[x]/(1 + Tanh[x])^(3/2), x, 6, ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]]/(2*Sqrt[2]) + 1/(3*(1 + Tanh[x])^(3/2)) - 1/(2*Sqrt[1 + Tanh[x]])}
1310
1311{Tanh[x]^2*(1 + Tanh[x])^(3/2), x, 6, 2*Sqrt[2]*ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]] - 2*Sqrt[1 + Tanh[x]] - (2/5)*(1 + Tanh[x])^(5/2)}
1312{Tanh[x]^2*Sqrt[1 + Tanh[x]], x, 7, Sqrt[2]*ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]] - (2/3)*(1 + Tanh[x])^(3/2)}
1313{Tanh[x]^2/Sqrt[1 + Tanh[x]], x, 6, ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]]/Sqrt[2] - 1/Sqrt[1 + Tanh[x]] - 2*Sqrt[1 + Tanh[x]]}
1314{Tanh[x]^2/(1 + Tanh[x])^(3/2), x, 6, ArcTanh[Sqrt[1 + Tanh[x]]/Sqrt[2]]/(2*Sqrt[2]) - 1/(3*(1 + Tanh[x])^(3/2)) + 3/(2*Sqrt[1 + Tanh[x]])}
1315
1316
1317(* ::Subsubsection::Closed:: *)
1318(*Integrands of the form Hyper[x]^m (a+b Coth[x])^n*)
1319
1320
1321(* Integrands of the form Coth[x]^m*(1+Coth[x])^n where m is an integer and n is a half-integer *)
1322{Coth[x]*(1 + Coth[x])^(3/2), x, 3, 2*Sqrt[2]*ArcCoth[Sqrt[1 + Coth[x]]/Sqrt[2]] - 2*Sqrt[1 + Coth[x]] - (2/3)*(1 + Coth[x])^(3/2)}
1323{Coth[x]*Sqrt[1 + Coth[x]], x, 2, Sqrt[2]*ArcCoth[Sqrt[1 + Coth[x]]/Sqrt[2]] - 2*Sqrt[1 + Coth[x]]}
1324{Coth[x]/Sqrt[1 + Coth[x]], x, 6, ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]/Sqrt[2] + 1/Sqrt[1 + Coth[x]]}
1325{Coth[x]/(1 + Coth[x])^(3/2), x, 6, ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]/(2*Sqrt[2]) + 1/(3*(1 + Coth[x])^(3/2)) - 1/(2*Sqrt[1 + Coth[x]])}
1326
1327{Coth[x]^2*(1 + Coth[x])^(3/2), x, 6, 2*Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]] - 2*Sqrt[1 + Coth[x]] - (2/5)*(1 + Coth[x])^(5/2)}
1328{Coth[x]^2*Sqrt[1 + Coth[x]], x, 7, Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]] - (2/3)*(1 + Coth[x])^(3/2)}
1329{Coth[x]^2/Sqrt[1 + Coth[x]], x, 6, ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]/Sqrt[2] - 1/Sqrt[1 + Coth[x]] - 2*Sqrt[1 + Coth[x]]}
1330{Coth[x]^2/(1 + Coth[x])^(3/2), x, 6, ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]/(2*Sqrt[2]) - 1/(3*(1 + Coth[x])^(3/2)) + 3/(2*Sqrt[1 + Coth[x]])}
1331
1332
1333{Sqrt[1 + Coth[x]]*Sech[x]^2, x, 4, ArcTanh[Sqrt[1 + Coth[x]]] + Sqrt[1 + Coth[x]]*Tanh[x]}
1334
1335
1336(* ::Subsection::Closed:: *)
1337(*Integrands of the form Hyper[x]^m (a+b Hyper[x]^2)^n*)
1338
1339
1340(* ::Subsubsection::Closed:: *)
1341(*Integrands of the form Hyper[x]^m (a+b Sinh[x]^2)^n*)
1342
1343
1344{Cosh[x]^1/(a + b*Sinh[x]^2)^2, x, 3, ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[b]) + Sinh[x]/(2*a*(a + b*Sinh[x]^2))}
1345{Cosh[x]^2/(a + b*Sinh[x]^2)^2, x, 7, ArcTanh[(Sqrt[a - b]*Tanh[x])/Sqrt[a]]/(2*a^(3/2)*Sqrt[a - b]) + (Cosh[x]*Sinh[x])/(2*a*(a + b*Sinh[x]^2)), ArcTanh[(Sqrt[a]*Coth[x])/Sqrt[a - b]]/(Sqrt[a]*Sqrt[a - b]*b) - ((2*a - b)*ArcTanh[(Sqrt[a - b]*Tanh[x])/Sqrt[a]])/(2*a^(3/2)*Sqrt[a - b]*b) + Sinh[2*x]/(2*a*(2*a - b + b*Cosh[2*x]))}
1346{Cosh[x]^3/(a + b*Sinh[x]^2)^2, x, 5, ((a + b)*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a]])/(2*a^(3/2)*b^(3/2)) - ((a - b)*Sinh[x])/(2*a*b*(a + b*Sinh[x]^2)), ((a + b)*ArcTan[(Sqrt[b]*Sinh[x])/Sqrt[a]])/(2*a^(3/2)*b^(3/2)) + Sinh[x]/(2*a*(a + b*Sinh[x]^2)) - Sinh[x]/(2*b*(a + b*Sinh[x]^2))}
1347{Cosh[x]^4/(a + b*Sinh[x]^2)^2, x, 7, x/b^2 + ((-2*a^2 + a*b + b^2)*ArcTanh[((2*a - 2*b)*Tanh[x])/(2*Sqrt[a]*Sqrt[a - b])])/(2*a^(3/2)*Sqrt[a - b]*b^2) - ((a - b)*Cosh[x]*Sinh[x])/(2*a*b*(a + b*Sinh[x]^2)), x/b^2 - (2*Sqrt[a - b]*ArcTanh[(Sqrt[a]*Coth[x])/Sqrt[a - b]])/(Sqrt[a]*b^2) + (Sqrt[a - b]*(2*a - b)*ArcTanh[(Sqrt[a - b]*Tanh[x])/Sqrt[a]])/(2*a^(3/2)*b^2) - ((a - b)*Sinh[2*x])/(2*a*b*(2*a - b + b*Cosh[2*x]))}
1348
1349
1350{Coth[x]/Sqrt[a + b*Sinh[x]^2], x, 2, -(ArcTanh[Sqrt[a + b*Sinh[x]^2]/Sqrt[a]]/Sqrt[a])}
1351{Coth[x]/Sqrt[1 + Sinh[x]^2], x, 2, -ArcTanh[Sqrt[Cosh[x]^2]]}
1352{Coth[x]/Sqrt[1 - Sinh[x]^2], x, 2, -ArcTanh[Sqrt[1 - Sinh[x]^2]]}
1353
1354
1355(* ::Subsubsection::Closed:: *)
1356(*Integrands of the form Hyper[x]^m (a+b Cosh[x]^2)^n*)
1357
1358
1359{Tanh[x]/Sqrt[a + b*Cosh[x]^2], x, 2, -(ArcTanh[Sqrt[a + b*Cosh[x]^2]/Sqrt[a]]/Sqrt[a])}
1360{Tanh[x]/Sqrt[1 + Cosh[x]^2], x, 2, -ArcTanh[Sqrt[1 + Cosh[x]^2]]}
1361{Tanh[x]/Sqrt[1 - Cosh[x]^2], x, 2, -ArcTanh[Sqrt[-Sinh[x]^2]]}
1362
1363
1364(* ::Subsubsection::Closed:: *)
1365(*Integrands of the form Hyper[x]^m (a+b Tanh[x]^2)^n*)
1366
1367
1368(* Integrands of the form Tanh[x]^m/Sqrt[a+b*Tanh[x]^2] where m is an integer *)
1369{Tanh[x]^3/Sqrt[a + b*Tanh[x]^2], x, 6, ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/Sqrt[a + b] - Sqrt[a + b*Tanh[x]^2]/b}
1370{Tanh[x]^2/Sqrt[a + b*Tanh[x]^2], x, 5, -(ArcTanh[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/Sqrt[b]) + ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/Sqrt[a + b]}
1371{Tanh[x]/Sqrt[a + b*Tanh[x]^2], x, 3, ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/Sqrt[a + b]}
1372{Coth[x]/Sqrt[a + b*Tanh[x]^2], x, 7, -(ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]]/Sqrt[a]) + ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/Sqrt[a + b]}
1373{Coth[x]^2/Sqrt[a + b*Tanh[x]^2], x, 5, ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/Sqrt[a + b] - (Coth[x]*Sqrt[a + b*Tanh[x]^2])/a}
1374
1375
1376(* Integrands of the form Tanh[x]^m*Sqrt[a+b*Tanh[x]^2] where m is an integer *)
1377{Tanh[x]^3*Sqrt[a + b*Tanh[x]^2], x, 7, Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - Sqrt[a + b*Tanh[x]^2] - (a + b*Tanh[x]^2)^(3/2)/(3*b)}
1378{Tanh[x]^2*Sqrt[a + b*Tanh[x]^2], x, 14, (-Sqrt[a + b])*ArcTanh[(Sqrt[b] - Sqrt[b]*Tanh[x] - Sqrt[a + b*Tanh[x]^2])/Sqrt[a + b]] - Sqrt[a + b]*ArcTanh[(Sqrt[b] + Sqrt[b]*Tanh[x] + Sqrt[a + b*Tanh[x]^2])/Sqrt[a + b]] - ((a + 2*b)*Log[Sqrt[b]*Tanh[x] + Sqrt[a + b*Tanh[x]^2]])/(2*Sqrt[b]) + a^2/(8*Sqrt[b]*(Sqrt[b]*Tanh[x] + Sqrt[a + b*Tanh[x]^2])^2) - (Sqrt[b]*Tanh[x] + Sqrt[a + b*Tanh[x]^2])^2/(8*Sqrt[b])}
1379{Tanh[x]*Sqrt[a + b*Tanh[x]^2], x, 4, Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - Sqrt[a + b*Tanh[x]^2]}
1380{Coth[x]*Sqrt[a + b*Tanh[x]^2], x, 7, (-Sqrt[a])*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]] + Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]}
1381{Coth[x]^2*Sqrt[a + b*Tanh[x]^2], x, 18, (-Sqrt[a + b])*ArcTanh[(Sqrt[b] - Sqrt[b]*Tanh[x] - Sqrt[a + b*Tanh[x]^2])/Sqrt[a + b]] - Sqrt[a + b]*ArcTanh[(Sqrt[b] + Sqrt[b]*Tanh[x] + Sqrt[a + b*Tanh[x]^2])/Sqrt[a + b]] + (2*a*Sqrt[b])/(a - (Sqrt[b]*Tanh[x] + Sqrt[a + b*Tanh[x]^2])^2)}
1382
1383
1384(* Integrands of the form Tanh[x]^m*(a+b*Tanh[x]^2)^(3/2) where m is an integer *)
1385{Tanh[x]^3*(a + b*Tanh[x]^2)^(3/2), x, 8, (a + b)^(3/2)*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - (a + b)*Sqrt[a + b*Tanh[x]^2] - (1/3)*(a + b*Tanh[x]^2)^(3/2) - (a + b*Tanh[x]^2)^(5/2)/(5*b)}
1386{Tanh[x]^2*(a + b*Tanh[x]^2)^(3/2), x, 13, -((3*a^2*ArcTanh[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]])/(8*Sqrt[b])) - (3/2)*a*Sqrt[b]*ArcTanh[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]] - b^(3/2)*ArcTanh[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]] + (a + b)^(3/2)*ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]] - (3/8)*a*Tanh[x]*Sqrt[a + b*Tanh[x]^2] - (1/2)*b*Tanh[x]*Sqrt[a + b*Tanh[x]^2] - (1/4)*Tanh[x]*(a + b*Tanh[x]^2)^(3/2)}
1387{Tanh[x]*(a + b*Tanh[x]^2)^(3/2), x, 6, (a + b)^(3/2)*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - (a + b)*Sqrt[a + b*Tanh[x]^2] - (1/3)*(a + b*Tanh[x]^2)^(3/2)}
1388{Coth[x]*(a + b*Tanh[x]^2)^(3/2), x, 8, (-a^(3/2))*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]] + (a + b)^(3/2)*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - b*Sqrt[a + b*Tanh[x]^2]}
1389{Coth[x]^2*(a + b*Tanh[x]^2)^(3/2), x, 13, (-b^(3/2))*ArcTanh[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]] + (a + b)^(3/2)*ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]] + b*Tanh[x]*Sqrt[a + b*Tanh[x]^2] - Coth[x]*(a + b*Tanh[x]^2)^(3/2)}
1390
1391
1392(* Integrands of the form Tanh[x]^m/(a+b*Tanh[x]^2)^(3/2) where m is an integer *)
1393{Tanh[x]^3/(a + b*Tanh[x]^2)^(3/2), x, 7, ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/(a + b)^(3/2) + a/(b*(a + b)*Sqrt[a + b*Tanh[x]^2])}
1394{Tanh[x]^2/(a + b*Tanh[x]^2)^(3/2), x, 7, ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/(a + b)^(3/2) - Tanh[x]/((a + b)*Sqrt[a + b*Tanh[x]^2])}
1395{Tanh[x]/(a + b*Tanh[x]^2)^(3/2), x, 4, ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/(a + b)^(3/2) - 1/((a + b)*Sqrt[a + b*Tanh[x]^2])}
1396{Coth[x]/(a + b*Tanh[x]^2)^(3/2), x, 8, -(ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]]/a^(3/2)) + ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/(a + b)^(3/2) + b/(a*(a + b)*Sqrt[a + b*Tanh[x]^2])}
1397{Coth[x]^2/(a + b*Tanh[x]^2)^(3/2), x, 8, ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/(a + b)^(3/2) + Coth[x]/(a*Sqrt[a + b*Tanh[x]^2]) + (b*Tanh[x])/(a*(a + b)*Sqrt[a + b*Tanh[x]^2]) - (2*Coth[x]*Sqrt[a + b*Tanh[x]^2])/a^2}
1398
1399
1400(* Integrands of the form Tanh[x]^m/(a+b*Tanh[x]^2)^(5/2) where m is an integer *)
1401{Tanh[x]^3/(a + b*Tanh[x]^2)^(5/2), x, 7, ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/(a + b)^(5/2) + a/(3*b*(a + b)*(a + b*Tanh[x]^2)^(3/2)) - 1/((a + b)^2*Sqrt[a + b*Tanh[x]^2])}
1402{Tanh[x]^2/(a + b*Tanh[x]^2)^(5/2), x, 11, ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/(a + b)^(5/2) - Tanh[x]/(3*(a + b)*(a + b*Tanh[x]^2)^(3/2)) + (b*Tanh[x])/(a*(a + b)^2*Sqrt[a + b*Tanh[x]^2]) - (2*Tanh[x])/(3*a*(a + b)*Sqrt[a + b*Tanh[x]^2])}
1403{Tanh[x]/(a + b*Tanh[x]^2)^(5/2), x, 5, ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/(a + b)^(5/2) - 1/(3*(a + b)*(a + b*Tanh[x]^2)^(3/2)) - 1/((a + b)^2*Sqrt[a + b*Tanh[x]^2])}
1404{Coth[x]/(a + b*Tanh[x]^2)^(5/2), x, 8, -(ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]]/a^(5/2)) + ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]]/(a + b)^(5/2) + b/(3*a*(a + b)*(a + b*Tanh[x]^2)^(3/2)) + (b*(2*a + b))/(a^2*(a + b)^2*Sqrt[a + b*Tanh[x]^2])}
1405{Coth[x]^2/(a + b*Tanh[x]^2)^(5/2), x, 12, ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/(a + b)^(5/2) + Coth[x]/(3*a*(a + b*Tanh[x]^2)^(3/2)) + (b*Tanh[x])/(3*a*(a + b)*(a + b*Tanh[x]^2)^(3/2)) + (4*Coth[x])/(3*a^2*Sqrt[a + b*Tanh[x]^2]) + (b*Tanh[x])/(a*(a + b)^2*Sqrt[a + b*Tanh[x]^2]) + (2*b*Tanh[x])/(3*a^2*(a + b)*Sqrt[a + b*Tanh[x]^2]) - (8*Coth[x]*Sqrt[a + b*Tanh[x]^2])/(3*a^3)}
1406
1407
1408(* ::Subsubsection::Closed:: *)
1409(*Integrands of the form Hyper[x]^m (a+b Coth[x]^2)^n*)
1410
1411
1412(* Integrands of the form Coth[x]^m/Sqrt[a+b*Coth[x]^2] where m is an integer *)
1413{Coth[x]^3/Sqrt[a + b*Coth[x]^2], x, 6, ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/Sqrt[a + b] - Sqrt[a + b*Coth[x]^2]/b}
1414{Coth[x]^2/Sqrt[a + b*Coth[x]^2], x, 5, -(ArcTanh[(Sqrt[b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/Sqrt[b]) + ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/Sqrt[a + b]}
1415{Coth[x]/Sqrt[a + b*Coth[x]^2], x, 3, ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/Sqrt[a + b]}
1416{Tanh[x]/Sqrt[a + b*Coth[x]^2], x, 7, -(ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a]]/Sqrt[a]) + ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/Sqrt[a + b]}
1417{Tanh[x]^2/Sqrt[a + b*Coth[x]^2], x, 5, ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/Sqrt[a + b] - (Sqrt[a + b*Coth[x]^2]*Tanh[x])/a}
1418
1419
1420(* Integrands of the form Coth[x]^m*Sqrt[a+b*Coth[x]^2] where m is an integer *)
1421{Coth[x]^3*Sqrt[a + b*Coth[x]^2], x, 7, Sqrt[a + b]*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - Sqrt[a + b*Coth[x]^2] - (a + b*Coth[x]^2)^(3/2)/(3*b)}
1422{Coth[x]^2*Sqrt[a + b*Coth[x]^2], x, 14, (-Sqrt[a + b])*ArcTanh[(Sqrt[b] - Sqrt[b]*Coth[x] - Sqrt[a + b*Coth[x]^2])/Sqrt[a + b]] - Sqrt[a + b]*ArcTanh[(Sqrt[b] + Sqrt[b]*Coth[x] + Sqrt[a + b*Coth[x]^2])/Sqrt[a + b]] + a^2/(8*Sqrt[b]*(Sqrt[b]*Coth[x] + Sqrt[a + b*Coth[x]^2])^2) - (Sqrt[b]*Coth[x] + Sqrt[a + b*Coth[x]^2])^2/(8*Sqrt[b]) - ((a + 2*b)*Log[Sqrt[b]*Coth[x] + Sqrt[a + b*Coth[x]^2]])/(2*Sqrt[b])}
1423{Coth[x]*Sqrt[a + b*Coth[x]^2], x, 4, Sqrt[a + b]*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - Sqrt[a + b*Coth[x]^2]}
1424{Tanh[x]*Sqrt[a + b*Coth[x]^2], x, 7, (-Sqrt[a])*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a]] + Sqrt[a + b]*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]}
1425{Tanh[x]^2*Sqrt[a + b*Coth[x]^2], x, 18, (-Sqrt[a + b])*ArcTanh[(Sqrt[b] - Sqrt[b]*Coth[x] - Sqrt[a + b*Coth[x]^2])/Sqrt[a + b]] - Sqrt[a + b]*ArcTanh[(Sqrt[b] + Sqrt[b]*Coth[x] + Sqrt[a + b*Coth[x]^2])/Sqrt[a + b]] + (2*a*Sqrt[b])/(a - (Sqrt[b]*Coth[x] + Sqrt[a + b*Coth[x]^2])^2)}
1426
1427
1428(* Integrands of the form Coth[x]^m*(a+b*Coth[x]^2)^(3/2) where m is an integer *)
1429{Coth[x]^3*(a + b*Coth[x]^2)^(3/2), x, 8, (a + b)^(3/2)*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - (a + b)*Sqrt[a + b*Coth[x]^2] - (1/3)*(a + b*Coth[x]^2)^(3/2) - (a + b*Coth[x]^2)^(5/2)/(5*b)}
1430{Coth[x]^2*(a + b*Coth[x]^2)^(3/2), x, 13, -((3*a^2*ArcTanh[(Sqrt[b]*Coth[x])/Sqrt[a + b*Coth[x]^2]])/(8*Sqrt[b])) - (3/2)*a*Sqrt[b]*ArcTanh[(Sqrt[b]*Coth[x])/Sqrt[a + b*Coth[x]^2]] - b^(3/2)*ArcTanh[(Sqrt[b]*Coth[x])/Sqrt[a + b*Coth[x]^2]] + (a + b)^(3/2)*ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]] - (3/8)*a*Coth[x]*Sqrt[a + b*Coth[x]^2] - (1/2)*b*Coth[x]*Sqrt[a + b*Coth[x]^2] - (1/4)*Coth[x]*(a + b*Coth[x]^2)^(3/2)}
1431{Coth[x]*(a + b*Coth[x]^2)^(3/2), x, 6, (a + b)^(3/2)*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - (a + b)*Sqrt[a + b*Coth[x]^2] - (1/3)*(a + b*Coth[x]^2)^(3/2)}
1432{Tanh[x]*(a + b*Coth[x]^2)^(3/2), x, 8, (-a^(3/2))*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a]] + (a + b)^(3/2)*ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]] - b*Sqrt[a + b*Coth[x]^2]}
1433{Tanh[x]^2*(a + b*Coth[x]^2)^(3/2), x, 13, (-b^(3/2))*ArcTanh[(Sqrt[b]*Coth[x])/Sqrt[a + b*Coth[x]^2]] + (a + b)^(3/2)*ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]] + b*Coth[x]*Sqrt[a + b*Coth[x]^2] - (a + b*Coth[x]^2)^(3/2)*Tanh[x]}
1434
1435
1436(* Integrands of the form Coth[x]^m/(a+b*Coth[x]^2)^(3/2) where m is an integer *)
1437{Coth[x]^3/(a + b*Coth[x]^2)^(3/2), x, 7, ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/(a + b)^(3/2) + a/(b*(a + b)*Sqrt[a + b*Coth[x]^2])}
1438{Coth[x]^2/(a + b*Coth[x]^2)^(3/2), x, 7, ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/(a + b)^(3/2) - Coth[x]/((a + b)*Sqrt[a + b*Coth[x]^2])}
1439{Coth[x]/(a + b*Coth[x]^2)^(3/2), x, 4, ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/(a + b)^(3/2) - 1/((a + b)*Sqrt[a + b*Coth[x]^2])}
1440{Tanh[x]/(a + b*Coth[x]^2)^(3/2), x, 8, -(ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a]]/a^(3/2)) + ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/(a + b)^(3/2) + b/(a*(a + b)*Sqrt[a + b*Coth[x]^2])}
1441{Tanh[x]^2/(a + b*Coth[x]^2)^(3/2), x, 8, ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/(a + b)^(3/2) + (b*Coth[x])/(a*(a + b)*Sqrt[a + b*Coth[x]^2]) + Tanh[x]/(a*Sqrt[a + b*Coth[x]^2]) - (2*Sqrt[a + b*Coth[x]^2]*Tanh[x])/a^2}
1442
1443
1444(* Integrands of the form Coth[x]^m/(a+b*Coth[x]^2)^(5/2) where m is an integer *)
1445{Coth[x]^3/(a + b*Coth[x]^2)^(5/2), x, 7, ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/(a + b)^(5/2) + a/(3*b*(a + b)*(a + b*Coth[x]^2)^(3/2)) - 1/((a + b)^2*Sqrt[a + b*Coth[x]^2])}
1446{Coth[x]^2/(a + b*Coth[x]^2)^(5/2), x, 11, ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/(a + b)^(5/2) - Coth[x]/(3*(a + b)*(a + b*Coth[x]^2)^(3/2)) + (b*Coth[x])/(a*(a + b)^2*Sqrt[a + b*Coth[x]^2]) - (2*Coth[x])/(3*a*(a + b)*Sqrt[a + b*Coth[x]^2])}
1447{Coth[x]/(a + b*Coth[x]^2)^(5/2), x, 5, ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/(a + b)^(5/2) - 1/(3*(a + b)*(a + b*Coth[x]^2)^(3/2)) - 1/((a + b)^2*Sqrt[a + b*Coth[x]^2])}
1448{Tanh[x]/(a + b*Coth[x]^2)^(5/2), x, 8, -(ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a]]/a^(5/2)) + ArcTanh[Sqrt[a + b*Coth[x]^2]/Sqrt[a + b]]/(a + b)^(5/2) + b/(3*a*(a + b)*(a + b*Coth[x]^2)^(3/2)) + (b*(2*a + b))/(a^2*(a + b)^2*Sqrt[a + b*Coth[x]^2])}
1449{Tanh[x]^2/(a + b*Coth[x]^2)^(5/2), x, 12, ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/(a + b)^(5/2) + (b*Coth[x])/(3*a*(a + b)*(a + b*Coth[x]^2)^(3/2)) + (b*Coth[x])/(a*(a + b)^2*Sqrt[a + b*Coth[x]^2]) + (2*b*Coth[x])/(3*a^2*(a + b)*Sqrt[a + b*Coth[x]^2]) + Tanh[x]/(3*a*(a + b*Coth[x]^2)^(3/2)) + (4*Tanh[x])/(3*a^2*Sqrt[a + b*Coth[x]^2]) - (8*Sqrt[a + b*Coth[x]^2]*Tanh[x])/(3*a^3)}
1450
1451
1452(* ::Subsection::Closed:: *)
1453(*Integrands of the form Hyper[x]^m (a+b Hyper[x]^n)^p*)
1454
1455
1456(* ::Subsubsection::Closed:: *)
1457(*Integrands of the form Coth[x] (a+b Sinh[x]^n)^p*)
1458
1459
1460(* Integrands of the form Coth[x]/Sqrt[a+b*Sinh[x]^n] where n is an integer *)
1461{Coth[x]/Sqrt[a + b*Sinh[x]], x, 2, -((2*ArcTanh[Sqrt[a + b*Sinh[x]]/Sqrt[a]])/Sqrt[a])}
1462{Coth[x]/Sqrt[a + b*Sinh[x]^2], x, 2, -(ArcTanh[Sqrt[a + b*Sinh[x]^2]/Sqrt[a]]/Sqrt[a])}
1463{Coth[x]/Sqrt[a + b*Sinh[x]^3], x, 2, -((2*ArcTanh[Sqrt[a + b*Sinh[x]^3]/Sqrt[a]])/(3*Sqrt[a]))}
1464{Coth[x]/Sqrt[a + b*Sinh[x]^4], x, 2, -((2*ArcTanh[Sqrt[a + b*Sinh[x]^4]/Sqrt[a]])/(4*Sqrt[a]))}
1465{Coth[x]/Sqrt[a + b*Sinh[x]^n], x, 2, -((2*ArcTanh[Sqrt[a + b*Sinh[x]^n]/Sqrt[a]])/(Sqrt[a]*n))}
1466
1467
1468(* Integrands of the form Coth[x]*Sqrt[a+b*Sinh[x]^n] where n is an integer *)
1469{Coth[x]*Sqrt[a + b*Sinh[x]], x, 3, -2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sinh[x]]/Sqrt[a]] + 2*Sqrt[a + b*Sinh[x]]}
1470{Coth[x]*Sqrt[a + b*Sinh[x]^2], x, 3, (-Sqrt[a])*ArcTanh[Sqrt[a + b*Sinh[x]^2]/Sqrt[a]] + Sqrt[a + b*Sinh[x]^2]}
1471{Coth[x]*Sqrt[a + b*Sinh[x]^3], x, 3, (-(2/3))*Sqrt[a]*ArcTanh[Sqrt[a + b*Sinh[x]^3]/Sqrt[a]] + (2/3)*Sqrt[a + b*Sinh[x]^3]}
1472{Coth[x]*Sqrt[a + b*Sinh[x]^4], x, 3, (-(1/2))*Sqrt[a]*ArcTanh[Sqrt[a + b*Sinh[x]^4]/Sqrt[a]] + (1/2)*Sqrt[a + b*Sinh[x]^4]}
1473{Coth[x]*Sqrt[a + b*Sinh[x]^n], x, 3, -((2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sinh[x]^n]/Sqrt[a]])/n) + (2*Sqrt[a + b*Sinh[x]^n])/n}
1474
1475
1476(* ::Subsubsection::Closed:: *)
1477(*Integrands of the form Tanh[x] (a+b Cosh[x]^n)^p*)
1478
1479
1480(* Integrands of the form Tanh[x]/Sqrt[a+b*Cosh[x]^n] where n is an integer *)
1481{Tanh[x]/Sqrt[a + b*Cosh[x]], x, 2, -((2*ArcTanh[Sqrt[a + b*Cosh[x]]/Sqrt[a]])/Sqrt[a])}
1482{Tanh[x]/Sqrt[a + b*Cosh[x]^2], x, 2, -(ArcTanh[Sqrt[a + b*Cosh[x]^2]/Sqrt[a]]/Sqrt[a])}
1483{Tanh[x]/Sqrt[a + b*Cosh[x]^3], x, 2, -((2*ArcTanh[Sqrt[a + b*Cosh[x]^3]/Sqrt[a]])/(3*Sqrt[a]))}
1484{Tanh[x]/Sqrt[a + b*Cosh[x]^4], x, 2, -(ArcTanh[Sqrt[a + b*Cosh[x]^4]/Sqrt[a]]/(2*Sqrt[a]))}
1485{Tanh[x]/Sqrt[a + b*Cosh[x]^n], x, 2, -((2*ArcTanh[Sqrt[a + b*Cosh[x]^n]/Sqrt[a]])/(Sqrt[a]*n))}
1486
1487
1488(* Integrands of the form Tanh[x]*Sqrt[a+b*Cosh[x]^n] where n is an integer *)
1489{Tanh[x]*Sqrt[a + b*Cosh[x]], x, 3, -2*Sqrt[a]*ArcTanh[Sqrt[a + b*Cosh[x]]/Sqrt[a]] + 2*Sqrt[a + b*Cosh[x]]}
1490{Tanh[x]*Sqrt[a + b*Cosh[x]^2], x, 3, (-Sqrt[a])*ArcTanh[Sqrt[a + b*Cosh[x]^2]/Sqrt[a]] + Sqrt[a + b*Cosh[x]^2]}
1491{Tanh[x]*Sqrt[a + b*Cosh[x]^3], x, 3, (-(2/3))*Sqrt[a]*ArcTanh[Sqrt[a + b*Cosh[x]^3]/Sqrt[a]] + (2/3)*Sqrt[a + b*Cosh[x]^3]}
1492{Tanh[x]*Sqrt[a + b*Cosh[x]^4], x, 3, (-(1/2))*Sqrt[a]*ArcTanh[Sqrt[a + b*Cosh[x]^4]/Sqrt[a]] + (1/2)*Sqrt[a + b*Cosh[x]^4]}
1493{Tanh[x]*Sqrt[a + b*Cosh[x]^n], x, 3, -((2*Sqrt[a]*ArcTanh[Sqrt[a + b*Cosh[x]^n]/Sqrt[a]])/n) + (2*Sqrt[a + b*Cosh[x]^n])/n}
1494
1495
1496(* ::Subsubsection::Closed:: *)
1497(*Integrands of the form Tanh[x] (a+b Tanh[x]^n)^p*)
1498
1499
1500(* Integrands of the form Tanh[x]*(a+b*Tanh[x]^4)^m where m is a half-integer *)
1501{Tanh[x]*Sqrt[a + b*Tanh[x]^4], x, 8, (-Sqrt[a + b])*ArcTanh[(Sqrt[b]*Sech[x]^2 - Sqrt[a + b*Tanh[x]^4])/Sqrt[a + b]] - (1/2)*Sqrt[b]*Log[Sqrt[b]*Tanh[x]^2 + Sqrt[a + b*Tanh[x]^4]] + (1/4)*((-Sqrt[b])*Tanh[x]^2 - Sqrt[a + b*Tanh[x]^4]) - a/(4*(Sqrt[b]*Tanh[x]^2 + Sqrt[a + b*Tanh[x]^4]))}
1502{Tanh[x]/Sqrt[a + b*Tanh[x]^4], x, 3, ArcTanh[(a + b*Tanh[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Tanh[x]^4])]/(2*Sqrt[a + b])}
1503{Tanh[x]/(a + b*Tanh[x]^4)^(3/2), x, 11, -(ArcTanh[(Sqrt[b]*Sech[x]^2 - Sqrt[a + b*Tanh[x]^4])/Sqrt[a + b]]/(a + b)^(3/2)) - Sqrt[b]/((a + b)*(a + (Sqrt[b]*Tanh[x]^2 + Sqrt[a + b*Tanh[x]^4])^2)) - (Sqrt[b]*Tanh[x]^2 + Sqrt[a + b*Tanh[x]^4])/((a + b)*(a + (Sqrt[b]*Tanh[x]^2 + Sqrt[a + b*Tanh[x]^4])^2))}
1504
1505
1506(* ::Subsubsection::Closed:: *)
1507(*Integrands of the form Coth[x] (a+b Coth[x]^n)^p*)
1508
1509
1510(* Integrands of the form Coth[x]*(a+b*Coth[x]^4)^m where m is a half-integer *)
1511{Coth[x]*Sqrt[a + b*Coth[x]^4], x, 8, Sqrt[a + b]*ArcTanh[(Sqrt[a + b*Coth[x]^4] + Sqrt[b]*Csch[x]^2)/Sqrt[a + b]] + (1/4)*((-Sqrt[b])*Coth[x]^2 - Sqrt[a + b*Coth[x]^4]) - a/(4*(Sqrt[b]*Coth[x]^2 + Sqrt[a + b*Coth[x]^4])) - (1/2)*Sqrt[b]*Log[Sqrt[b]*Coth[x]^2 + Sqrt[a + b*Coth[x]^4]]}
1512{Coth[x]/Sqrt[a + b*Coth[x]^4], x, 3, ArcTanh[(a + b*Coth[x]^2)/(Sqrt[a + b]*Sqrt[a + b*Coth[x]^4])]/(2*Sqrt[a + b])}
1513{Coth[x]/(a + b*Coth[x]^4)^(3/2), x, 11, ArcTanh[(Sqrt[a + b*Coth[x]^4] + Sqrt[b]*Csch[x]^2)/Sqrt[a + b]]/(a + b)^(3/2) - Sqrt[b]/((a + b)*(a + (Sqrt[b]*Coth[x]^2 + Sqrt[a + b*Coth[x]^4])^2)) - (Sqrt[b]*Coth[x]^2 + Sqrt[a + b*Coth[x]^4])/((a + b)*(a + (Sqrt[b]*Coth[x]^2 + Sqrt[a + b*Coth[x]^4])^2))}
1514
1515
1516(* ::Subsection::Closed:: *)
1517(*Integrands of the form x^q Hyper[x]^m (a+b Hyper[x]^n)^p*)
1518(**)
1519
1520
1521{x*Cosh[x]/(a + b*Sinh[x])^2, x, 2, -((2*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b*Sqrt[a^2 + b^2])) - x/(b*(a + b*Sinh[x]))}
1522{x*Cosh[x]/(a + b*Sinh[x])^3, x, 3, -((a*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b*(a^2 + b^2)^(3/2))) - x/(2*b*(a + b*Sinh[x])^2) - Cosh[x]/(2*(a^2 + b^2)*(a + b*Sinh[x]))}
1523
1524
1525{x*Sinh[x]/(a + b*Cosh[x])^2, x, 2, (2*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(b*Sqrt[a^2 - b^2]) - x/(b*(a + b*Cosh[x]))}
1526{x*Sinh[x]/(a + b*Cosh[x])^3, x, 3, (a*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(b*(a^2 - b^2)^(3/2)) - x/(2*b*(a + b*Cosh[x])^2) - Sinh[x]/(2*(a^2 - b^2)*(a + b*Cosh[x]))}
1527
1528
1529{x*Sech[x]^2/(a + b*Tanh[x])^2, x, 2, (a*x)/(b*(a^2 - b^2)) - Log[a*Cosh[x] + b*Sinh[x]]/(a^2 - b^2) - x/(b*(a + b*Tanh[x]))}
1530{x*Csch[x]^2/(a + b*Coth[x])^2, x, 2, -((a*x)/(b*(a^2 - b^2))) + x/(b*(a + b*Coth[x])) + Log[b*Cosh[x] + a*Sinh[x]]/(a^2 - b^2)}
1531
1532
1533{(Cosh[a + b*x]*(-2 + Sinh[a + b*x]^2))/x, x, 13, (-(9/4))*Cosh[a]*CoshIntegral[b*x] + (1/4)*Cosh[3*a]*CoshIntegral[3*b*x] - (9/4)*Sinh[a]*SinhIntegral[b*x] + (1/4)*Sinh[3*a]*SinhIntegral[3*b*x]}
1534{((2 + Cosh[a + b*x]^2)*Sinh[a + b*x])/x, x, 13, (9/4)*CoshIntegral[b*x]*Sinh[a] + (1/4)*CoshIntegral[3*b*x]*Sinh[3*a] + (9/4)*Cosh[a]*SinhIntegral[b*x] + (1/4)*Cosh[3*a]*SinhIntegral[3*b*x]}
1535
1536
1537(* ::Subsection::Closed:: *)
1538(*Integrands of the form (A+B Hyper[x]) (a+b Hyper[x])^n*)
1539
1540
1541(* ::Subsubsection::Closed:: *)
1542(*Integrands of the form (A+B Hyper[x]) (a+b Sinh[x])^n*)
1543
1544
1545(* Integrands of the form (A+B*Sinh[x])*(a+b*Sinh[x])^n where n is an integer *)
1546{(A + B*Sinh[x])/(a + b*Sinh[x]), x, 2, (B*x)/b - (2*(A*b - a*B)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b*Sqrt[a^2 + b^2])}
1547{(A + B*Sinh[x])/(a + b*Sinh[x])^2, x, 2, -((2*(a*A + b*B)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(3/2)) - ((A*b - a*B)*Cosh[x])/((a^2 + b^2)*(a + b*Sinh[x]))}
1548{(A + B*Sinh[x])/(a + b*Sinh[x])^3, x, 3, -(((2*a^2*A - b*(A*b - 3*a*B))*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(5/2)) - ((A*b - a*B)*Cosh[x])/(2*(a^2 + b^2)*(a + b*Sinh[x])^2) - ((2*b^2*B + a*(3*A*b - a*B))*Cosh[x])/(2*(a^2 + b^2)^2*(a + b*Sinh[x]))}
1549{(A + B*Sinh[x])/(a + b*Sinh[x])^4, x, 4, -(((b*(2*a^2*B - b*(5*a*A + 3*b*B)) - 2*a*(2*A*b^2 - a*(3*a*A + 5*b*B)))*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(3*(a^2 + b^2)^(7/2))) - ((A*b - a*B)*Cosh[x])/(3*(a^2 + b^2)*(a + b*Sinh[x])^3) + ((2*a^2*B - b*(5*a*A + 3*b*B))*Cosh[x])/(6*(a^2 + b^2)^2*(a + b*Sinh[x])^2) + ((a*(2*a^2*B - b*(5*a*A + 3*b*B)) + 2*b*(2*A*b^2 - a*(3*a*A + 5*b*B)))*Cosh[x])/(6*(a^2 + b^2)^3*(a + b*Sinh[x]))}
1550
1551{(A + B*Sinh[x])/(I + Sinh[x]), x, 2, B*x - ((A - I*B)*Cosh[x])/(1 - I*Sinh[x])}
1552{(A + B*Sinh[x])/(I + Sinh[x])^2, x, 4, (I*A*Cosh[x])/(3*(1 - I*Sinh[x])) - (2*B*Cosh[x])/(3*(1 - I*Sinh[x])) - (I*(A - I*B)*Cosh[x])/(3*(I + Sinh[x])^2)}
1553{(A + B*Sinh[x])/(I + Sinh[x])^3, x, 6, (2*A*Cosh[x])/(15*(1 - I*Sinh[x])) + (I*B*Cosh[x])/(5*(1 - I*Sinh[x])) - (I*(A - I*B)*Cosh[x])/(5*(I + Sinh[x])^3) - (2*A*Cosh[x])/(15*(I + Sinh[x])^2) - (I*B*Cosh[x])/(5*(I + Sinh[x])^2)}
1554{(A + B*Sinh[x])/(I + Sinh[x])^4, x, 8, -((2*I*A*Cosh[x])/(35*(1 - I*Sinh[x]))) + (8*B*Cosh[x])/(105*(1 - I*Sinh[x])) - (I*(A - I*B)*Cosh[x])/(7*(I + Sinh[x])^4) - (3*A*Cosh[x])/(35*(I + Sinh[x])^3) - (4*I*B*Cosh[x])/(35*(I + Sinh[x])^3) + (2*I*A*Cosh[x])/(35*(I + Sinh[x])^2) - (8*B*Cosh[x])/(105*(I + Sinh[x])^2)}
1555
1556{(A + B*Sinh[x])/(I - Sinh[x]), x, 2, (-B)*x + ((A + I*B)*Cosh[x])/(1 + I*Sinh[x])}
1557{(A + B*Sinh[x])/(I - Sinh[x])^2, x, 4, (I*(A + I*B)*Cosh[x])/(3*(I - Sinh[x])^2) - (I*A*Cosh[x])/(3*(1 + I*Sinh[x])) - (2*B*Cosh[x])/(3*(1 + I*Sinh[x]))}
1558{(A + B*Sinh[x])/(I - Sinh[x])^3, x, 6, (I*(A + I*B)*Cosh[x])/(5*(I - Sinh[x])^3) + (2*A*Cosh[x])/(15*(I - Sinh[x])^2) - (I*B*Cosh[x])/(5*(I - Sinh[x])^2) - (2*A*Cosh[x])/(15*(1 + I*Sinh[x])) + (I*B*Cosh[x])/(5*(1 + I*Sinh[x]))}
1559{(A + B*Sinh[x])/(I - Sinh[x])^4, x, 8, (I*(A + I*B)*Cosh[x])/(7*(I - Sinh[x])^4) + (3*A*Cosh[x])/(35*(I - Sinh[x])^3) - (4*I*B*Cosh[x])/(35*(I - Sinh[x])^3) - (2*I*A*Cosh[x])/(35*(I - Sinh[x])^2) - (8*B*Cosh[x])/(105*(I - Sinh[x])^2) + (2*I*A*Cosh[x])/(35*(1 + I*Sinh[x])) + (8*B*Cosh[x])/(105*(1 + I*Sinh[x]))}
1560
1561{(b*B/a + B*Sinh[x])/(a + b*Sinh[x]), x, 2, (B*x)/b + (2*(a - b^2/a)*B*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b*Sqrt[a^2 + b^2])}
1562{(a*B/b + B*Sinh[x])/(a + b*Sinh[x]), x, 2, (B*x)/b}
1563
1564{(a - b*Sinh[x])/(b + a*Sinh[x])^2, x, 1, -(Cosh[x]/(b + a*Sinh[x]))}
1565{(2 - Sinh[x])/(2 + Sinh[x]), x, 2, -x - (8*ArcTanh[(1 - 2*Tanh[x/2])/Sqrt[5]])/Sqrt[5]}
1566
1567
1568(* Integrands of the form (A+B*Sinh[x])*(a+b*Sinh[x])^n where n is a half-integer *)
1569{(A + B*Sinh[x])*(a + b*Sinh[x])^(5/2), x, 8, (2/105)*(15*a^2*B + b*(56*a*A - 25*b*B))*Cosh[x]*Sqrt[a + b*Sinh[x]] + (2/35)*(7*A*b + 5*a*B)*Cosh[x]*(a + b*Sinh[x])^(3/2) + (2/7)*B*Cosh[x]*(a + b*Sinh[x])^(5/2) + (2*I*(a*(15*a^2*B + b*(56*a*A - 25*b*B)) - 3*b*(21*A*b^2 - 5*a*(7*a*A - 8*b*B)))*EllipticE[Pi/4 - (I*x)/2, -((2*I*b)/(a - I*b))]*Sqrt[a + b*Sinh[x]])/(105*b*Sqrt[(a + b*Sinh[x])/(a - I*b)]) - (2*I*(a^2 + b^2)*(15*a^2*B + b*(56*a*A - 25*b*B))*EllipticF[Pi/4 - (I*x)/2, -((2*I*b)/(a - I*b))]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(105*b*Sqrt[a + b*Sinh[x]])}
1570{(A + B*Sinh[x])*(a + b*Sinh[x])^(3/2), x, 7, (2/15)*(5*A*b + 3*a*B)*Cosh[x]*Sqrt[a + b*Sinh[x]] + (2/5)*B*Cosh[x]*(a + b*Sinh[x])^(3/2) - (2*I*(9*b^2*B - a*(20*A*b + 3*a*B))*EllipticE[Pi/4 - (I*x)/2, -((2*I*b)/(a - I*b))]*Sqrt[a + b*Sinh[x]])/(15*b*Sqrt[(a + b*Sinh[x])/(a - I*b)]) + (2*I*(a*(9*b^2*B - a*(20*A*b + 3*a*B)) + b*(15*a^2*A - b*(5*A*b + 12*a*B)))*EllipticF[Pi/4 - (I*x)/2, -((2*I*b)/(a - I*b))]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(15*b*Sqrt[a + b*Sinh[x]])}
1571{(A + B*Sinh[x])*(a + b*Sinh[x])^(1/2), x, 6, (2/3)*B*Cosh[x]*Sqrt[a + b*Sinh[x]] + (2*I*(3*A*b + a*B)*EllipticE[Pi/4 - (I*x)/2, -((2*I*b)/(a - I*b))]*Sqrt[a + b*Sinh[x]])/(3*b*Sqrt[(a + b*Sinh[x])/(a - I*b)]) - (2*I*(a^2 + b^2)*B*EllipticF[Pi/4 - (I*x)/2, -((2*I*b)/(a - I*b))]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(3*b*Sqrt[a + b*Sinh[x]])}
1572{(A + B*Sinh[x])/(a + b*Sinh[x])^(1/2), x, 5, (2*I*B*EllipticE[Pi/4 - (I*x)/2, -((2*I*b)/(a - I*b))]*Sqrt[a + b*Sinh[x]])/(b*Sqrt[(a + b*Sinh[x])/(a - I*b)]) + (2*I*(A*b - a*B)*EllipticF[Pi/4 - (I*x)/2, -((2*I*b)/(a - I*b))]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(b*Sqrt[a + b*Sinh[x]])}
1573{(A + B*Sinh[x])/(a + b*Sinh[x])^(3/2), x, 6, -((2*(A*b - a*B)*Cosh[x])/((a^2 + b^2)*Sqrt[a + b*Sinh[x]])) + (2*I*(A*b - a*B)*EllipticE[Pi/4 - (I*x)/2, -((2*I*b)/(a - I*b))]*Sqrt[a + b*Sinh[x]])/(b*(a^2 + b^2)*Sqrt[(a + b*Sinh[x])/(a - I*b)]) + (2*I*B*EllipticF[Pi/4 - (I*x)/2, -((2*I*b)/(a - I*b))]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(b*Sqrt[a + b*Sinh[x]])}
1574{(A + B*Sinh[x])/(a + b*Sinh[x])^(5/2), x, 7, -((2*(A*b - a*B)*Cosh[x])/(3*(a^2 + b^2)*(a + b*Sinh[x])^(3/2))) - (2*(3*b^2*B + a*(4*A*b - a*B))*Cosh[x])/(3*(a^2 + b^2)^2*Sqrt[a + b*Sinh[x]]) + (2*I*(3*b^2*B + a*(4*A*b - a*B))*EllipticE[Pi/4 - (I*x)/2, -((2*I*b)/(a - I*b))]*Sqrt[a + b*Sinh[x]])/(3*b*(a^2 + b^2)^2*Sqrt[(a + b*Sinh[x])/(a - I*b)]) + (2*I*(b*(3*a^2*A - b*(A*b - 4*a*B)) - a*(3*b^2*B + a*(4*A*b - a*B)))*EllipticF[Pi/4 - (I*x)/2, -((2*I*b)/(a - I*b))]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(3*b*(a^2 + b^2)^2*Sqrt[a + b*Sinh[x]])}
1575
1576{(A + B*Sinh[x])*(a + a*I*Sinh[x])^(5/2), x, 8, (64*I*a^3*A*Cosh[x])/(15*Sqrt[a + I*a*Sinh[x]]) + (64*a^3*B*Cosh[x])/(21*Sqrt[a + I*a*Sinh[x]]) + (16/15)*I*a^2*A*Cosh[x]*Sqrt[a + I*a*Sinh[x]] + (16/21)*a^2*B*Cosh[x]*Sqrt[a + I*a*Sinh[x]] + (2/5)*I*a*A*Cosh[x]*(a + I*a*Sinh[x])^(3/2) + (2/7)*a*B*Cosh[x]*(a + I*a*Sinh[x])^(3/2) + (2/7)*B*Cosh[x]*(a + I*a*Sinh[x])^(5/2)}
1577{(A + B*Sinh[x])*(a + a*I*Sinh[x])^(3/2), x, 6, (8*I*a^2*A*Cosh[x])/(3*Sqrt[a + I*a*Sinh[x]]) + (8*a^2*B*Cosh[x])/(5*Sqrt[a + I*a*Sinh[x]]) + (2/3)*I*a*A*Cosh[x]*Sqrt[a + I*a*Sinh[x]] + (2/5)*a*B*Cosh[x]*Sqrt[a + I*a*Sinh[x]] + (2/5)*B*Cosh[x]*(a + I*a*Sinh[x])^(3/2)}
1578{(A + B*Sinh[x])*(a + a*I*Sinh[x])^(1/2), x, 4, (2*I*a*A*Cosh[x])/Sqrt[a + I*a*Sinh[x]] + (2*a*B*Cosh[x])/(3*Sqrt[a + I*a*Sinh[x]]) + (2/3)*B*Cosh[x]*Sqrt[a + I*a*Sinh[x]]}
1579{(A + B*Sinh[x])/(a + a*I*Sinh[x])^(1/2), x, 3, (2*(I*A - B)*ArcTanh[Sin[Pi/4 - (I*x)/2]]*Cos[Pi/4 - (I*x)/2])/Sqrt[a + I*a*Sinh[x]] + (2*B*Cosh[x])/Sqrt[a + I*a*Sinh[x]]}
1580{(A + B*Sinh[x])/(a + a*I*Sinh[x])^(3/2), x, 4, ((I*A - B)*Cosh[x])/(2*(a + I*a*Sinh[x])^(3/2)) + (I*A*ArcTanh[Sin[Pi/4 - (I*x)/2]]*Cos[Pi/4 - (I*x)/2])/(2*a*Sqrt[a + I*a*Sinh[x]]) + (3*B*ArcTanh[Sin[Pi/4 - (I*x)/2]]*Cos[Pi/4 - (I*x)/2])/(2*a*Sqrt[a + I*a*Sinh[x]])}
1581{(A + B*Sinh[x])/(a + a*I*Sinh[x])^(5/2), x, 6, ((I*A - B)*Cosh[x])/(4*(a + I*a*Sinh[x])^(5/2)) + (3*I*A*Cosh[x])/(16*a*(a + I*a*Sinh[x])^(3/2)) + (5*B*Cosh[x])/(16*a*(a + I*a*Sinh[x])^(3/2)) + (3*I*A*ArcTanh[Sin[Pi/4 - (I*x)/2]]*Cos[Pi/4 - (I*x)/2])/(16*a^2*Sqrt[a + I*a*Sinh[x]]) + (5*B*ArcTanh[Sin[Pi/4 - (I*x)/2]]*Cos[Pi/4 - (I*x)/2])/(16*a^2*Sqrt[a + I*a*Sinh[x]])}
1582
1583
1584(* Integrands of the form (A+B*Cosh[x])*(a+b*Sinh[x])^n where n is an integer *)
1585{(A + B*Cosh[x])/(a + b*Sinh[x]), x, 5, -((2*A*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2]) + (B*Log[a + b*Sinh[x]])/b}
1586
1587{(A + B*Cosh[x])/(I + Sinh[x]), x, 5, B*Log[I + Sinh[x]] - (A*Cosh[x])/(1 - I*Sinh[x])}
1588{(A + B*Cosh[x])/(I - Sinh[x]), x, 5, (-B)*Log[I - Sinh[x]] + (A*Cosh[x])/(1 + I*Sinh[x])}
1589
1590
1591{(A + B*Tanh[x])/(a + b*Sinh[x]), x, 10, (b*B*ArcTan[Sinh[x]])/(a^2 + b^2) - (2*A*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2] + (a*B*Log[Cosh[x]^2])/(2*(a^2 + b^2)) - (a*B*Log[a + b*Sinh[x]])/(a^2 + b^2)}
1592{(A + B*Coth[x])/(a + b*Sinh[x]), x, 5, -((2*A*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2]) + (B*Log[Sinh[x]])/a - (B*Log[a + b*Sinh[x]])/a}
1593{(A + B*Sech[x])/(a + b*Sinh[x]), x, 10, (a*B*ArcTan[Sinh[x]])/(a^2 + b^2) - (2*A*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/Sqrt[a^2 + b^2] - (b*B*Log[Cosh[x]^2])/(2*(a^2 + b^2)) + (b*B*Log[a + b*Sinh[x]])/(a^2 + b^2)}
1594{(A + B*Csch[x])/(a + b*Sinh[x]), x, 4, -((B*ArcCoth[Cosh[x]])/a) - (2*(a*A - b*B)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a*Sqrt[a^2 + b^2])}
1595
1596
1597(* ::Subsubsection::Closed:: *)
1598(*Integrands of the form (A+B Hyper[x]) (a+b Cosh[x])^n*)
1599
1600
1601(* Integrands of the form (A+B*Cosh[x])*(a+b*Cosh[x])^n where n is an integer *)
1602{(A + B*Cosh[x])/(a + b*Cosh[x]), x, 2, (B*x)/b + (2*(A*b - a*B)*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(b*Sqrt[a^2 - b^2])}
1603{(A + B*Cosh[x])/(a + b*Cosh[x])^2, x, 2, (2*(a*A - b*B)*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(3/2) - ((A*b - a*B)*Sinh[x])/((a^2 - b^2)*(a + b*Cosh[x]))}
1604{(A + B*Cosh[x])/(a + b*Cosh[x])^3, x, 3, ((2*a^2*A + b*(A*b - 3*a*B))*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(5/2) - ((A*b - a*B)*Sinh[x])/(2*(a^2 - b^2)*(a + b*Cosh[x])^2) + ((2*b^2*B - a*(3*A*b - a*B))*Sinh[x])/(2*(a^2 - b^2)^2*(a + b*Cosh[x]))}
1605{(A + B*Cosh[x])/(a + b*Cosh[x])^4, x, 4, ((2*a*(2*A*b^2 + a*(3*a*A - 5*b*B)) - b*(2*a^2*B - b*(5*a*A - 3*b*B)))*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(3*(a^2 - b^2)^(7/2)) - ((A*b - a*B)*Sinh[x])/(3*(a^2 - b^2)*(a + b*Cosh[x])^3) + ((2*a^2*B - b*(5*a*A - 3*b*B))*Sinh[x])/(6*(a^2 - b^2)^2*(a + b*Cosh[x])^2) - ((2*b*(2*A*b^2 + a*(3*a*A - 5*b*B)) - a*(2*a^2*B - b*(5*a*A - 3*b*B)))*Sinh[x])/(6*(a^2 - b^2)^3*(a + b*Cosh[x]))}
1606
1607{(A + B*Cosh[x])/(1 + Cosh[x]), x, 2, B*x + ((A - B)*Sinh[x])/(1 + Cosh[x])}
1608{(A + B*Cosh[x])/(1 + Cosh[x])^2, x, 4, ((A - B)*Sinh[x])/(3*(1 + Cosh[x])^2) + (A*Sinh[x])/(3*(1 + Cosh[x])) + (2*B*Sinh[x])/(3*(1 + Cosh[x]))}
1609{(A + B*Cosh[x])/(1 + Cosh[x])^3, x, 6, ((A - B)*Sinh[x])/(5*(1 + Cosh[x])^3) + (2*A*Sinh[x])/(15*(1 + Cosh[x])^2) + (B*Sinh[x])/(5*(1 + Cosh[x])^2) + (2*A*Sinh[x])/(15*(1 + Cosh[x])) + (B*Sinh[x])/(5*(1 + Cosh[x]))}
1610{(A + B*Cosh[x])/(1 + Cosh[x])^4, x, 8, ((A - B)*Sinh[x])/(7*(1 + Cosh[x])^4) + (3*A*Sinh[x])/(35*(1 + Cosh[x])^3) + (4*B*Sinh[x])/(35*(1 + Cosh[x])^3) + (2*A*Sinh[x])/(35*(1 + Cosh[x])^2) + (8*B*Sinh[x])/(105*(1 + Cosh[x])^2) + (2*A*Sinh[x])/(35*(1 + Cosh[x])) + (8*B*Sinh[x])/(105*(1 + Cosh[x]))}
1611
1612{(A + B*Cosh[x])/(1 - Cosh[x]), x, 2, (-B)*x - ((A + B)*Sinh[x])/(1 - Cosh[x])}
1613{(A + B*Cosh[x])/(1 - Cosh[x])^2, x, 4, -(((A + B)*Sinh[x])/(3*(1 - Cosh[x])^2)) - (A*Sinh[x])/(3*(1 - Cosh[x])) + (2*B*Sinh[x])/(3*(1 - Cosh[x]))}
1614{(A + B*Cosh[x])/(1 - Cosh[x])^3, x, 6, -(((A + B)*Sinh[x])/(5*(1 - Cosh[x])^3)) - (2*A*Sinh[x])/(15*(1 - Cosh[x])^2) + (B*Sinh[x])/(5*(1 - Cosh[x])^2) - (2*A*Sinh[x])/(15*(1 - Cosh[x])) + (B*Sinh[x])/(5*(1 - Cosh[x]))}
1615{(A + B*Cosh[x])/(1 - Cosh[x])^4, x, 8, -(((A + B)*Sinh[x])/(7*(1 - Cosh[x])^4)) - (3*A*Sinh[x])/(35*(1 - Cosh[x])^3) + (4*B*Sinh[x])/(35*(1 - Cosh[x])^3) - (2*A*Sinh[x])/(35*(1 - Cosh[x])^2) + (8*B*Sinh[x])/(105*(1 - Cosh[x])^2) - (2*A*Sinh[x])/(35*(1 - Cosh[x])) + (8*B*Sinh[x])/(105*(1 - Cosh[x]))}
1616
1617{(b*B/a + B*Cosh[x])/(a + b*Cosh[x]), x, 2, (B*x)/b - (2*Sqrt[a^2 - b^2]*B*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a*b)}
1618{(a*B/b + B*Cosh[x])/(a + b*Cosh[x]), x, 2, (B*x)/b}
1619
1620{(a + b*Cosh[x])/(b + a*Cosh[x])^2, x, 1, Sinh[x]/(b + a*Cosh[x])}
1621{(3 + Cosh[x])/(2 - Cosh[x]), x, 2, -x + (10*ArcTanh[Sqrt[3]*Tanh[x/2]])/Sqrt[3]}
1622
1623
1624(* Integrands of the form (A+B*Cosh[x])*(a+b*Cosh[x])^n where n is a half-integer *)
1625{(A + B*Cosh[x])*(a + b*Cosh[x])^(5/2), x, 8, -((2*I*(3*b*(21*A*b^2 + 5*a*(7*a*A + 8*b*B)) + a*(15*a^2*B + b*(56*a*A + 25*b*B)))*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(105*b*Sqrt[(a + b*Cosh[x])/(a + b)])) + (2*I*(a^2 - b^2)*(15*a^2*B + b*(56*a*A + 25*b*B))*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + b)])/(105*b*Sqrt[a + b*Cosh[x]]) + (2/105)*(15*a^2*B + b*(56*a*A + 25*b*B))*Sqrt[a + b*Cosh[x]]*Sinh[x] + (2/35)*(7*A*b + 5*a*B)*(a + b*Cosh[x])^(3/2)*Sinh[x] + (2/7)*B*(a + b*Cosh[x])^(5/2)*Sinh[x]}
1626{(A + B*Cosh[x])*(a + b*Cosh[x])^(3/2), x, 7, -((2*I*(9*b^2*B + a*(20*A*b + 3*a*B))*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(15*b*Sqrt[(a + b*Cosh[x])/(a + b)])) + (2*I*(a*(9*b^2*B + a*(20*A*b + 3*a*B)) - b*(15*a^2*A + b*(5*A*b + 12*a*B)))*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + b)])/(15*b*Sqrt[a + b*Cosh[x]]) + (2/15)*(5*A*b + 3*a*B)*Sqrt[a + b*Cosh[x]]*Sinh[x] + (2/5)*B*(a + b*Cosh[x])^(3/2)*Sinh[x]}
1627{(A + B*Cosh[x])*(a + b*Cosh[x])^(1/2), x, 6, -((2*I*(3*A*b + a*B)*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(3*b*Sqrt[(a + b*Cosh[x])/(a + b)])) + (2*I*(a^2 - b^2)*B*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + b)])/(3*b*Sqrt[a + b*Cosh[x]]) + (2/3)*B*Sqrt[a + b*Cosh[x]]*Sinh[x]}
1628{(A + B*Cosh[x])/(a + b*Cosh[x])^(1/2), x, 5, -((2*I*B*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(b*Sqrt[(a + b*Cosh[x])/(a + b)])) - (2*I*(A*b - a*B)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + b)])/(b*Sqrt[a + b*Cosh[x]])}
1629{(A + B*Cosh[x])/(a + b*Cosh[x])^(3/2), x, 6, -((2*I*(A*b - a*B)*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(b*(a^2 - b^2)*Sqrt[(a + b*Cosh[x])/(a + b)])) - (2*I*B*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + b)])/(b*Sqrt[a + b*Cosh[x]]) - (2*(A*b - a*B)*Sinh[x])/((a^2 - b^2)*Sqrt[a + b*Cosh[x]])}
1630{(A + B*Cosh[x])/(a + b*Cosh[x])^(5/2), x, 7, (2*I*(3*b^2*B - a*(4*A*b - a*B))*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(3*b*(a^2 - b^2)^2*Sqrt[(a + b*Cosh[x])/(a + b)]) - (2*I*(b*(3*a^2*A + b*(A*b - 4*a*B)) + a*(3*b^2*B - a*(4*A*b - a*B)))*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + b)])/(3*b*(a^2 - b^2)^2*Sqrt[a + b*Cosh[x]]) - (2*(A*b - a*B)*Sinh[x])/(3*(a^2 - b^2)*(a + b*Cosh[x])^(3/2)) + (2*(3*b^2*B - a*(4*A*b - a*B))*Sinh[x])/(3*(a^2 - b^2)^2*Sqrt[a + b*Cosh[x]])}
1631
1632{(A + B*Cosh[x])*(a + a*Cosh[x])^(5/2), x, 8, (64*a^3*A*Sinh[x])/(15*Sqrt[a + a*Cosh[x]]) + (64*a^3*B*Sinh[x])/(21*Sqrt[a + a*Cosh[x]]) + (16/15)*a^2*A*Sqrt[a + a*Cosh[x]]*Sinh[x] + (16/21)*a^2*B*Sqrt[a + a*Cosh[x]]*Sinh[x] + (2/5)*a*A*(a + a*Cosh[x])^(3/2)*Sinh[x] + (2/7)*a*B*(a + a*Cosh[x])^(3/2)*Sinh[x] + (2/7)*B*(a + a*Cosh[x])^(5/2)*Sinh[x]}
1633{(A + B*Cosh[x])*(a + a*Cosh[x])^(3/2), x, 6, (8*a^2*A*Sinh[x])/(3*Sqrt[a + a*Cosh[x]]) + (8*a^2*B*Sinh[x])/(5*Sqrt[a + a*Cosh[x]]) + (2/3)*a*A*Sqrt[a + a*Cosh[x]]*Sinh[x] + (2/5)*a*B*Sqrt[a + a*Cosh[x]]*Sinh[x] + (2/5)*B*(a + a*Cosh[x])^(3/2)*Sinh[x]}
1634{(A + B*Cosh[x])*(a + a*Cosh[x])^(1/2), x, 4, (2*a*A*Sinh[x])/Sqrt[a + a*Cosh[x]] + (2*a*B*Sinh[x])/(3*Sqrt[a + a*Cosh[x]]) + (2/3)*B*Sqrt[a + a*Cosh[x]]*Sinh[x]}
1635{(A + B*Cosh[x])/(a + a*Cosh[x])^(1/2), x, 3, (2*(A - B)*ArcTan[Sinh[x/2]]*Cosh[x/2])/Sqrt[a + a*Cosh[x]] + (2*B*Sinh[x])/Sqrt[a + a*Cosh[x]]}
1636{(A + B*Cosh[x])/(a + a*Cosh[x])^(3/2), x, 4, (A*ArcTan[Sinh[x/2]]*Cosh[x/2])/(2*a*Sqrt[a + a*Cosh[x]]) + (3*B*ArcTan[Sinh[x/2]]*Cosh[x/2])/(2*a*Sqrt[a + a*Cosh[x]]) + ((A - B)*Sinh[x])/(2*(a + a*Cosh[x])^(3/2))}
1637{(A + B*Cosh[x])/(a + a*Cosh[x])^(5/2), x, 6, (3*A*ArcTan[Sinh[x/2]]*Cosh[x/2])/(16*a^2*Sqrt[a + a*Cosh[x]]) + (5*B*ArcTan[Sinh[x/2]]*Cosh[x/2])/(16*a^2*Sqrt[a + a*Cosh[x]]) + ((A - B)*Sinh[x])/(4*(a + a*Cosh[x])^(5/2)) + (3*A*Sinh[x])/(16*a*(a + a*Cosh[x])^(3/2)) + (5*B*Sinh[x])/(16*a*(a + a*Cosh[x])^(3/2))}
1638
1639{(A + B*Cosh[x])*(a - a*Cosh[x])^(5/2), x, 8, -((64*a^3*A*Sinh[x])/(15*Sqrt[a - a*Cosh[x]])) + (64*a^3*B*Sinh[x])/(21*Sqrt[a - a*Cosh[x]]) - (16/15)*a^2*A*Sqrt[a - a*Cosh[x]]*Sinh[x] + (16/21)*a^2*B*Sqrt[a - a*Cosh[x]]*Sinh[x] - (2/5)*a*A*(a - a*Cosh[x])^(3/2)*Sinh[x] + (2/7)*a*B*(a - a*Cosh[x])^(3/2)*Sinh[x] + (2/7)*B*(a - a*Cosh[x])^(5/2)*Sinh[x]}
1640{(A + B*Cosh[x])*(a - a*Cosh[x])^(3/2), x, 6, -((8*a^2*A*Sinh[x])/(3*Sqrt[a - a*Cosh[x]])) + (8*a^2*B*Sinh[x])/(5*Sqrt[a - a*Cosh[x]]) - (2/3)*a*A*Sqrt[a - a*Cosh[x]]*Sinh[x] + (2/5)*a*B*Sqrt[a - a*Cosh[x]]*Sinh[x] + (2/5)*B*(a - a*Cosh[x])^(3/2)*Sinh[x]}
1641{(A + B*Cosh[x])*(a - a*Cosh[x])^(1/2), x, 4, -((2*a*A*Sinh[x])/Sqrt[a - a*Cosh[x]]) + (2*a*B*Sinh[x])/(3*Sqrt[a - a*Cosh[x]]) + (2/3)*B*Sqrt[a - a*Cosh[x]]*Sinh[x]}
1642{(A + B*Cosh[x])/(a - a*Cosh[x])^(1/2), x, 3, -((2*(A + B)*ArcTanh[Cosh[x/2]]*Sinh[x/2])/Sqrt[a - a*Cosh[x]]) + (2*B*Sinh[x])/Sqrt[a - a*Cosh[x]]}
1643{(A + B*Cosh[x])/(a - a*Cosh[x])^(3/2), x, 4, -((A*ArcTanh[Cosh[x/2]]*Sinh[x/2])/(2*a*Sqrt[a - a*Cosh[x]])) + (3*B*ArcTanh[Cosh[x/2]]*Sinh[x/2])/(2*a*Sqrt[a - a*Cosh[x]]) - ((A + B)*Sinh[x])/(2*(a - a*Cosh[x])^(3/2))}
1644{(A + B*Cosh[x])/(a - a*Cosh[x])^(5/2), x, 6, -((3*A*ArcTanh[Cosh[x/2]]*Sinh[x/2])/(16*a^2*Sqrt[a - a*Cosh[x]])) + (5*B*ArcTanh[Cosh[x/2]]*Sinh[x/2])/(16*a^2*Sqrt[a - a*Cosh[x]]) - ((A + B)*Sinh[x])/(4*(a - a*Cosh[x])^(5/2)) - (3*A*Sinh[x])/(16*a*(a - a*Cosh[x])^(3/2)) + (5*B*Sinh[x])/(16*a*(a - a*Cosh[x])^(3/2))}
1645
1646
1647(* Integrands of the form (A+B*Sinh[x])*(a+b*Cosh[x])^n where n is an integer *)
1648{(A + B*Sinh[x])/(a + b*Cosh[x]), x, 5, (2*A*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (B*Log[a + b*Cosh[x]])/b}
1649
1650{(A + B*Sinh[x])/(1 + Cosh[x]), x, 4, B*Log[1 + Cosh[x]] + (A*Sinh[x])/(1 + Cosh[x]), 2*B*Log[Cosh[x/2]] + (A*Sinh[x])/(1 + Cosh[x])}
1651{(A + B*Sinh[x])/(1 - Cosh[x]), x, 4, (-B)*Log[1 - Cosh[x]] - (A*Sinh[x])/(1 - Cosh[x]), -2*B*Log[Sinh[x/2]] - (A*Sinh[x])/(1 - Cosh[x])}
1652
1653
1654{(A + B*Tanh[x])/(a + b*Cosh[x]), x, 5, (2*A*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (B*Log[Cosh[x]])/a - (B*Log[a + b*Cosh[x]])/a}
1655{(A + B*Coth[x])/(a + b*Cosh[x]), x, 9, (2*A*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (B*Log[1 - Cosh[x]])/(2*(a + b)) + (B*Log[1 + Cosh[x]])/(2*(a - b)) - (a*B*Log[a + b*Cosh[x]])/(a^2 - b^2)}
1656{(A + B*Sech[x])/(a + b*Cosh[x]), x, 4, (B*ArcTan[Sinh[x]])/a + (2*(a*A - b*B)*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a*Sqrt[a^2 - b^2])}
1657{(A + B*Csch[x])/(a + b*Cosh[x]), x, 9, (2*A*ArcTanh[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] + (B*Log[1 - Cosh[x]])/(2*(a + b)) - (B*Log[1 + Cosh[x]])/(2*(a - b)) + (b*B*Log[a + b*Cosh[x]])/(a^2 - b^2)}
1658
1659
1660(* ::Subsection::Closed:: *)
1661(*Integrands of the form x^m (A+B Hyper[x]) (a+b Hyper[x])^n*)
1662
1663
1664{x*((b - a*Sinh[x])/(a + b*Sinh[x])^2), x, 3, Log[a + b*Sinh[x]]/b - (x*Cosh[x])/(a + b*Sinh[x])}
1665{x*((b + a*Cosh[x])/(a + b*Cosh[x])^2), x, 3, -(Log[a + b*Cosh[x]]/b) + (x*Sinh[x])/(a + b*Cosh[x])}
1666
1667
1668(* ::Subsection::Closed:: *)
1669(*Integrands of the form (A+B Hyper[x]^m) / (a+b Hyper[x]^n)*)
1670(**)
1671
1672
1673{(1 + Sinh[x]^2)/(1 - Sinh[x]^2), x, 5, -x + Sqrt[2]*ArcTanh[Coth[x]/Sqrt[2]]}
1674{(1 - Sinh[x]^2)/(1 + Sinh[x]^2), x, 4, -x + 2*Tanh[x]}
1675
1676
1677{(1 + Cosh[x]^2)/(1 - Cosh[x]^2), x, 5, -x + 2*Coth[x]}
1678{(1 - Cosh[x]^2)/(1 + Cosh[x]^2), x, 6, -x + Sqrt[2]*ArcTanh[Tanh[x]/Sqrt[2]]}
1679
1680
1681{(a + b*Sech[x])/(c + d*Cosh[x]), x, 4, (b*ArcTan[Sinh[x]])/c + (2*(a*c - b*d)*ArcTanh[((c - d)*Tanh[x/2])/Sqrt[c^2 - d^2]])/(c*Sqrt[c^2 - d^2])}
1682{(a + b*Csch[x])/(c + d*Sinh[x]), x, 4, -((b*ArcCoth[Cosh[x]])/c) - (2*(a*c - b*d)*ArcTanh[(d - c*Tanh[x/2])/Sqrt[c^2 + d^2]])/(c*Sqrt[c^2 + d^2])}
1683{(a + b*Sech[x]^2)/(c + d*Cosh[x]), x, 5, -((b*d*ArcTan[Sinh[x]])/c^2) + (2*(a*c^2 + b*d^2)*ArcTanh[((c - d)*Tanh[x/2])/Sqrt[c^2 - d^2]])/(c^2*Sqrt[c^2 - d^2]) + (b*Tanh[x])/c}
1684{(a + b*Csch[x]^2)/(c + d*Sinh[x]), x, 5, (b*d*ArcCoth[Cosh[x]])/c^2 - (2*(a*c^2 + b*d^2)*ArcTanh[(d - c*Tanh[x/2])/Sqrt[c^2 + d^2]])/(c^2*Sqrt[c^2 + d^2]) - (b*Coth[x])/c}
1685
1686
1687(* ::Subsection::Closed:: *)
1688(*Integrands of the form (a Hyper[c+d x] + b Hyper[c+d x])^n*)
1689
1690
1691(* ::Subsubsection::Closed:: *)
1692(*Integrands of the form (a Cosh[c+d x] + b Sinh[c+d x])^n*)
1693
1694
1695(* Integrands of the form (a*Cosh[c+d*x]+b*Sinh[c+d*x])^n *)
1696{(a*Cosh[x] + b*Sinh[x]), x, 3, b*Cosh[x] + a*Sinh[x]}
1697{(a*Cosh[x] + b*Sinh[x])^2, x, 2, (1/2)*(a^2 - b^2)*x + (1/2)*(b*Cosh[x] + a*Sinh[x])*(a*Cosh[x] + b*Sinh[x])}
1698{(a*Cosh[x] + b*Sinh[x])^3, x, 2, (a^2 - b^2)*(b*Cosh[x] + a*Sinh[x]) + (1/3)*(b*Cosh[x] + a*Sinh[x])^3}
1699{(a*Cosh[x] + b*Sinh[x])^4, x, 3, (3/8)*(a^2 - b^2)^2*x + (3/8)*(a^2 - b^2)*(b*Cosh[x] + a*Sinh[x])*(a*Cosh[x] + b*Sinh[x]) + (1/4)*(b*Cosh[x] + a*Sinh[x])*(a*Cosh[x] + b*Sinh[x])^3}
1700{(a*Cosh[x] + b*Sinh[x])^5, x, 3, (a^2 - b^2)^2*(b*Cosh[x] + a*Sinh[x]) + (2/3)*(a^2 - b^2)*(b*Cosh[x] + a*Sinh[x])^3 + (1/5)*(b*Cosh[x] + a*Sinh[x])^5}
1701
1702{1/(a*Cosh[x] + b*Sinh[x]), x, 1, (2*ArcTan[(b + a*Tanh[x/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2]}
1703{1/(a*Cosh[x] + b*Sinh[x])^2, x, 1, Sinh[x]/(a*(a*Cosh[x] + b*Sinh[x]))}
1704{1/(a*Cosh[x] + b*Sinh[x])^3, x, 2, ArcTan[(b + a*Tanh[x/2])/Sqrt[a^2 - b^2]]/(a^2 - b^2)^(3/2) + (b*Cosh[x] + a*Sinh[x])/(2*(a^2 - b^2)*(a*Cosh[x] + b*Sinh[x])^2)}
1705{1/(a*Cosh[x] + b*Sinh[x])^4, x, 2, (b*Cosh[x] + a*Sinh[x])/(3*(a^2 - b^2)*(a*Cosh[x] + b*Sinh[x])^3) + (2*Sinh[x])/(3*a*(a^2 - b^2)*(a*Cosh[x] + b*Sinh[x]))}
1706{1/(a*Cosh[x] + b*Sinh[x])^5, x, 3, (3*ArcTan[(b + a*Tanh[x/2])/Sqrt[a^2 - b^2]])/(4*(a^2 - b^2)^(5/2)) + (b*Cosh[x] + a*Sinh[x])/(4*(a^2 - b^2)*(a*Cosh[x] + b*Sinh[x])^4) + (3*(b*Cosh[x] + a*Sinh[x]))/(8*(a^2 - b^2)^2*(a*Cosh[x] + b*Sinh[x])^2)}
1707
1708{(a*Cosh[x] + b*Sinh[x])^(1/2), x, 1, (2*I*EllipticE[(1/4)*(Pi - 2*I*(x + I*ArcTan[I*b, a])), 2]*Sqrt[a*Cosh[x] + b*Sinh[x]])/Sqrt[-((a*Cosh[x] + b*Sinh[x])/Sqrt[a^2 - b^2])]}
1709{(a*Cosh[x] + b*Sinh[x])^(3/2), x, 2, (2/3)*(b*Cosh[x] + a*Sinh[x])*Sqrt[a*Cosh[x] + b*Sinh[x]] + (2*I*(a^2 - b^2)*EllipticF[(1/4)*(Pi - 2*I*(x + I*ArcTan[I*b, a])), 2]*Sqrt[-((a*Cosh[x] + b*Sinh[x])/Sqrt[a^2 - b^2])])/(3*Sqrt[a*Cosh[x] + b*Sinh[x]])}
1710{(a*Cosh[x] + b*Sinh[x])^(5/2), x, 2, (2/5)*(b*Cosh[x] + a*Sinh[x])*(a*Cosh[x] + b*Sinh[x])^(3/2) + (6*I*(a^2 - b^2)*EllipticE[(1/4)*(Pi - 2*I*(x + I*ArcTan[I*b, a])), 2]*Sqrt[a*Cosh[x] + b*Sinh[x]])/(5*Sqrt[-((a*Cosh[x] + b*Sinh[x])/Sqrt[a^2 - b^2])])}
1711
1712{1/(a*Cosh[x] + b*Sinh[x])^(1/2), x, 1, (2*I*EllipticF[(1/4)*(Pi - 2*I*(x + I*ArcTan[I*b, a])), 2]*Sqrt[-((a*Cosh[x] + b*Sinh[x])/Sqrt[a^2 - b^2])])/Sqrt[a*Cosh[x] + b*Sinh[x]]}
1713{1/(a*Cosh[x] + b*Sinh[x])^(3/2), x, 2, (2*(b*Cosh[x] + a*Sinh[x]))/((a^2 - b^2)*Sqrt[a*Cosh[x] + b*Sinh[x]]) - (2*I*EllipticE[(1/4)*(Pi - 2*I*(x + I*ArcTan[I*b, a])), 2]*Sqrt[a*Cosh[x] + b*Sinh[x]])/((a^2 - b^2)*Sqrt[-((a*Cosh[x] + b*Sinh[x])/Sqrt[a^2 - b^2])])}
1714{1/(a*Cosh[x] + b*Sinh[x])^(5/2), x, 2, (2*(b*Cosh[x] + a*Sinh[x]))/(3*(a^2 - b^2)*(a*Cosh[x] + b*Sinh[x])^(3/2)) + (2*I*EllipticF[(1/4)*(Pi - 2*I*(x + I*ArcTan[I*b, a])), 2]*Sqrt[-((a*Cosh[x] + b*Sinh[x])/Sqrt[a^2 - b^2])])/(3*(a^2 - b^2)*Sqrt[a*Cosh[x] + b*Sinh[x]])}
1715
1716
1717(* Integrands of the form (a*Cosh[c+d*x]+a*Sinh[c+d*x])^n *)
1718{(a*Cosh[c + d*x] + a*Sinh[c + d*x]), x, 3, (a*Cosh[c + d*x])/d + (a*Sinh[c + d*x])/d}
1719{(a*Cosh[c + d*x] + a*Sinh[c + d*x])^2, x, 1, (a^2*(Cosh[c + d*x] + Sinh[c + d*x])^2)/(2*d)}
1720{(a*Cosh[c + d*x] + a*Sinh[c + d*x])^3, x, 1, (a^3*(Cosh[c + d*x] + Sinh[c + d*x])^3)/(3*d)}
1721{(a*Cosh[c + d*x] + a*Sinh[c + d*x])^n, x, 1, (a*Cosh[c + d*x] + a*Sinh[c + d*x])^n/(d*n)}
1722
1723{1/(a*Cosh[c + d*x] + a*Sinh[c + d*x]), x, 1, -((Cosh[c + d*x] - Sinh[c + d*x])/(a*d))}
1724{1/(a*Cosh[c + d*x] + a*Sinh[c + d*x])^2, x, 1, -((Cosh[c + d*x] - Sinh[c + d*x])^2/(2*a^2*d))}
1725{1/(a*Cosh[c + d*x] + a*Sinh[c + d*x])^3, x, 1, -((Cosh[c + d*x] - Sinh[c + d*x])^3/(3*a^3*d))}
1726
1727{Sqrt[a*Cosh[c + d*x] + a*Sinh[c + d*x]], x, 1, (2*Sqrt[a*Cosh[c + d*x] + a*Sinh[c + d*x]])/d}
1728{1/Sqrt[a*Cosh[c + d*x] + a*Sinh[c + d*x]], x, 1, -(2/(d*Sqrt[a*Cosh[c + d*x] + a*Sinh[c + d*x]]))}
1729
1730
1731(* Integrands of the form (a*Cosh[c+d*x]-a*Sinh[c+d*x])^n *)
1732{(a*Cosh[c + d*x] - a*Sinh[c + d*x]), x, 3, -((a*Cosh[c + d*x])/d) + (a*Sinh[c + d*x])/d}
1733{(a*Cosh[c + d*x] - a*Sinh[c + d*x])^2, x, 1, -((a^2*(Cosh[c + d*x] - Sinh[c + d*x])^2)/(2*d))}
1734{(a*Cosh[c + d*x] - a*Sinh[c + d*x])^3, x, 1, -((a^3*(Cosh[c + d*x] - Sinh[c + d*x])^3)/(3*d))}
1735{(a*Cosh[c + d*x] - a*Sinh[c + d*x])^n, x, 1, -((a*Cosh[c + d*x] - a*Sinh[c + d*x])^n/(d*n))}
1736
1737{1/(a*Cosh[c + d*x] - a*Sinh[c + d*x]), x, 1, (Cosh[c + d*x] + Sinh[c + d*x])/(a*d)}
1738{1/(a*Cosh[c + d*x] - a*Sinh[c + d*x])^2, x, 1, (Cosh[c + d*x] + Sinh[c + d*x])^2/(2*a^2*d)}
1739{1/(a*Cosh[c + d*x] - a*Sinh[c + d*x])^3, x, 1, (Cosh[c + d*x] + Sinh[c + d*x])^3/(3*a^3*d)}
1740
1741{Sqrt[a*Cosh[c + d*x] - a*Sinh[c + d*x]], x, 1, -((2*Sqrt[a*Cosh[c + d*x] - a*Sinh[c + d*x]])/d)}
1742{1/Sqrt[a*Cosh[c + d*x] - a*Sinh[c + d*x]], x, 1, 2/(d*Sqrt[a*Cosh[c + d*x] - a*Sinh[c + d*x]])}
1743
1744
1745(* ::Subsubsection::Closed:: *)
1746(*Integrands of the form (a Sech[c+d x] + b Tanh[c+d x])^n*)
1747
1748
1749(* Integrands of the form (a*Sech[x]+b*Tanh[x])^n where n is an integer *)
1750{(a*Sech[x] + b*Tanh[x])^5, x, 14, (5/4)*a^3*b^2*ArcTan[Sinh[x]] + (3/8)*a*(a^4 + 5*b^4)*ArcTan[Sinh[x]] + (1/2)*b^5*Log[Cosh[x]^2] - b^3*(5*a^2 - b^2)*Sech[x]^2 - (1/4)*b*(5*a^4 - 10*a^2*b^2 + b^4)*Sech[x]^4 + (3/8)*a^5*Sech[x]*Tanh[x] + 5*a*b^2*(a^2/4 - (5*b^2)/8)*Sech[x]*Tanh[x] + (1/4)*a*(a^4 - 10*a^2*b^2 + 5*b^4)*Sech[x]^3*Tanh[x]}
1751{(a*Sech[x] + b*Tanh[x])^4, x, 11, b^4*x - 4*a*b^3*Sech[x] - (4/3)*a^3*b*Sech[x]^3 + (4/3)*a*b^3*Sech[x]^3 + a^4*Tanh[x] - b^4*Tanh[x] - (1/3)*a^4*Tanh[x]^3 + 2*a^2*b^2*Tanh[x]^3 - (1/3)*b^4*Tanh[x]^3}
1752{(a*Sech[x] + b*Tanh[x])^3, x, 10, (1/2)*a^3*ArcTan[Sinh[x]] + (3/2)*a*b^2*ArcTan[Sinh[x]] + (1/2)*b^3*Log[Cosh[x]^2] - (1/2)*b*(3*a^2 - b^2)*Sech[x]^2 + (1/2)*a*(a^2 - 3*b^2)*Sech[x]*Tanh[x]}
1753{(a*Sech[x] + b*Tanh[x])^2, x, 5, b^2*x - 2*a*b*Sech[x] + a^2*Tanh[x] - b^2*Tanh[x]}
1754{(a*Sech[x] + b*Tanh[x]), x, 3, a*ArcTan[Sinh[x]] + b*Log[Cosh[x]]}
1755{1/(a*Sech[x] + b*Tanh[x]), x, 3, Log[a + b*Sinh[x]]/b}
1756{1/(a*Sech[x] + b*Tanh[x])^2, x, 6, x/b^2 + (2*a*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^2*Sqrt[a^2 + b^2]) - Cosh[x]/(b*(a + b*Sinh[x]))}
1757{1/(a*Sech[x] + b*Tanh[x])^3, x, 7, Log[a + b*Sinh[x]]/b^3 - (a^2 + b^2)/(2*b^3*(a + b*Sinh[x])^2) + (2*a)/(b^3*(a + b*Sinh[x]))}
1758{1/(a*Sech[x] + b*Tanh[x])^4, x, 13, x/b^4 - (6*a^3*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^4*(a^2 + b^2)^(3/2)) - (5*a*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^2*(a^2 + b^2)^(3/2)) + (8*a*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(b^4*Sqrt[a^2 + b^2]) - ((a^2 + b^2)*Cosh[x])/(3*b^3*(a + b*Sinh[x])^3) + (7*a*Cosh[x])/(6*b^3*(a + b*Sinh[x])^2) - (11*a^2*Cosh[x])/(6*b^3*(a^2 + b^2)*(a + b*Sinh[x])) - (4*Cosh[x])/(3*b*(a^2 + b^2)*(a + b*Sinh[x]))}
1759{1/(a*Sech[x] + b*Tanh[x])^5, x, 9, Log[a + b*Sinh[x]]/b^5 - (a^2 + b^2)^2/(4*b^5*(a + b*Sinh[x])^4) + (4*a*(a^2 + b^2))/(3*b^5*(a + b*Sinh[x])^3) - (3*a^2 + b^2)/(b^5*(a + b*Sinh[x])^2) + (4*a)/(b^5*(a + b*Sinh[x]))}
1760
1761
1762(* Integrands of the form (Sech[x]+I*Tanh[x])^n where n is an integer *)
1763{(Sech[x] + I*Tanh[x])^5, x, 8, I*Log[1 - I*Sinh[x]] - (2*I)/(1 - I*Sinh[x])^2 + (4*I)/(1 - I*Sinh[x])}
1764{(Sech[x] + I*Tanh[x])^4, x, 3, x + 2*I*Tan[Pi/4 + (I*x)/2] - (2/3)*I*Tan[Pi/4 + (I*x)/2]^3}
1765{(Sech[x] + I*Tanh[x])^3, x, 7, (-I)*Log[I + Sinh[x]] + 2/(I + Sinh[x])}
1766{(Sech[x] + I*Tanh[x])^2, x, 3, -x - 2*I*Tan[Pi/4 + (I*x)/2]}
1767{(Sech[x] + I*Tanh[x]), x, 3, ArcTan[Sinh[x]] + I*Log[Cosh[x]]}
1768{1/(Sech[x] + I*Tanh[x]), x, 3, (-I)*Log[1 + I*Sinh[x]]}
1769{1/(Sech[x] + I*Tanh[x])^2, x, 3, -x + 2*I*Cot[Pi/4 + (I*x)/2]}
1770{1/(Sech[x] + I*Tanh[x])^3, x, 7, I*Log[-I + Sinh[x]] - 2/(I - Sinh[x])}
1771{1/(Sech[x] + I*Tanh[x])^4, x, 3, x - 2*I*Cot[Pi/4 + (I*x)/2] + (2/3)*I*Cot[Pi/4 + (I*x)/2]^3}
1772{1/(Sech[x] + I*Tanh[x])^5, x, 8, (-I)*Log[1 + I*Sinh[x]] + (2*I)/(1 + I*Sinh[x])^2 - (4*I)/(1 + I*Sinh[x])}
1773
1774
1775(* Integrands of the form (Sech[x]-I*Tanh[x])^n where n is an integer *)
1776{(Sech[x] - I*Tanh[x])^5, x, 8, (-I)*Log[1 + I*Sinh[x]] + (2*I)/(1 + I*Sinh[x])^2 - (4*I)/(1 + I*Sinh[x])}
1777{(Sech[x] - I*Tanh[x])^4, x, 3, x - 2*I*Tan[Pi/4 - (I*x)/2] + (2/3)*I*Tan[Pi/4 - (I*x)/2]^3}
1778{(Sech[x] - I*Tanh[x])^3, x, 7, I*Log[-I + Sinh[x]] - 2/(I - Sinh[x])}
1779{(Sech[x] - I*Tanh[x])^2, x, 3, -x + 2*I*Tan[Pi/4 - (I*x)/2]}
1780{(Sech[x] - I*Tanh[x]), x, 3, ArcTan[Sinh[x]] - I*Log[Cosh[x]]}
1781{1/(Sech[x] - I*Tanh[x]), x, 3, I*Log[1 - I*Sinh[x]]}
1782{1/(Sech[x] - I*Tanh[x])^2, x, 3, -x - 2*I*Cot[Pi/4 - (I*x)/2]}
1783{1/(Sech[x] - I*Tanh[x])^3, x, 7, (-I)*Log[I + Sinh[x]] + 2/(I + Sinh[x])}
1784{1/(Sech[x] - I*Tanh[x])^4, x, 3, x + 2*I*Cot[Pi/4 - (I*x)/2] - (2/3)*I*Cot[Pi/4 - (I*x)/2]^3}
1785{1/(Sech[x] - I*Tanh[x])^5, x, 8, I*Log[1 - I*Sinh[x]] - (2*I)/(1 - I*Sinh[x])^2 + (4*I)/(1 - I*Sinh[x])}
1786
1787
1788(* ::Subsubsection::Closed:: *)
1789(*Integrands of the form (a Coth[c+d x] + b Csch[c+d x])^n*)
1790
1791
1792(* Integrands of the form (a*Coth[x]+b*Csch[x])^n where n is an integer *)
1793{(a*Coth[x] + b*Csch[x]), x, 3, (-b)*ArcCoth[Cosh[x]] + a*Log[Sinh[x]]}
1794{(a*Coth[x] + b*Csch[x])^2, x, 5, a^2*x - a^2*Coth[x] - b^2*Coth[x] - 2*a*b*Csch[x]}
1795{(a*Coth[x] + b*Csch[x])^3, x, 8, (a + b)^3/(4*(1 - Cosh[x])) + (a - b)^3/(4*(1 + Cosh[x])) + (1/4)*(2*a - b)*(a + b)^2*Log[1 - Cosh[x]] + (1/4)*(a - b)^2*(2*a + b)*Log[1 + Cosh[x]]}
1796{(a*Coth[x] + b*Csch[x])^4, x, 11, a^4*x - a^4*Coth[x] + b^4*Coth[x] - (1/3)*a^4*Coth[x]^3 - 2*a^2*b^2*Coth[x]^3 - (1/3)*b^4*Coth[x]^3 - 4*a^3*b*Csch[x] - (4/3)*a^3*b*Csch[x]^3 - (4/3)*a*b^3*Csch[x]^3}
1797{(a*Coth[x] + b*Csch[x])^5, x, 10, -((a + b)^5/(16*(1 - Cosh[x])^2)) + ((7*a - 3*b)*(a + b)^4)/(16*(1 - Cosh[x])) - (a - b)^5/(16*(1 + Cosh[x])^2) + ((a - b)^4*(7*a + 3*b))/(16*(1 + Cosh[x])) + (1/16)*(a + b)^3*(8*a^2 - 9*a*b + 3*b^2)*Log[1 - Cosh[x]] + (1/16)*(a - b)^3*(8*a^2 + 9*a*b + 3*b^2)*Log[1 + Cosh[x]]}
1798
1799{1/(a*Coth[x] + b*Csch[x]), x, 3, Log[b + a*Cosh[x]]/a}
1800{1/(a*Coth[x] + b*Csch[x])^2, x, 6, x/a^2 - (2*b*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2*Sqrt[a^2 - b^2]) - Sinh[x]/(a*(b + a*Cosh[x]))}
1801{1/(a*Coth[x] + b*Csch[x])^3, x, 7, (a^2 - b^2)/(2*a^3*(b + a*Cosh[x])^2) + (2*b)/(a^3*(b + a*Cosh[x])) + Log[b + a*Cosh[x]]/a^3}
1802{1/(a*Coth[x] + b*Csch[x])^4, x, 13, x/a^4 + (5*b*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^2*(a^2 - b^2)^(3/2)) - (6*b^3*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^4*(a^2 - b^2)^(3/2)) - (8*b*ArcTan[((a - b)*Tanh[x/2])/Sqrt[a^2 - b^2]])/(a^4*Sqrt[a^2 - b^2]) + ((a^2 - b^2)*Sinh[x])/(3*a^3*(b + a*Cosh[x])^3) + (7*b*Sinh[x])/(6*a^3*(b + a*Cosh[x])^2) - (4*Sinh[x])/(3*a*(a^2 - b^2)*(b + a*Cosh[x])) + (11*b^2*Sinh[x])/(6*a^3*(a^2 - b^2)*(b + a*Cosh[x]))}
1803{1/(a*Coth[x] + b*Csch[x])^5, x, 9, -((a^2 - b^2)^2/(4*a^5*(b + a*Cosh[x])^4)) - (4*b*(a^2 - b^2))/(3*a^5*(b + a*Cosh[x])^3) + (a^2 - 3*b^2)/(a^5*(b + a*Cosh[x])^2) + (4*b)/(a^5*(b + a*Cosh[x])) + Log[b + a*Cosh[x]]/a^5}
1804
1805
1806(* Integrands of the form (Csch[x]+Coth[x])^n where n is an integer *)
1807{(Coth[x] + Csch[x]), x, 3, 2*Log[Sinh[x/2]], -ArcCoth[Cosh[x]] + Log[Sinh[x]]}
1808{(Coth[x] + Csch[x])^2, x, 2, x - 2*Coth[x/2]}
1809{(Coth[x] + Csch[x])^3, x, 6, 2/(1 - Cosh[x]) + Log[1 - Cosh[x]]}
1810{(Coth[x] + Csch[x])^4, x, 3, x - 2*Coth[x/2] - (2/3)*Coth[x/2]^3}
1811{(Coth[x] + Csch[x])^5, x, 7, -(2/(1 - Cosh[x])^2) + 4/(1 - Cosh[x]) + Log[1 - Cosh[x]]}
1812
1813{1/(Coth[x] + Csch[x]), x, 3, Log[1 + Cosh[x]]}
1814{1/(Coth[x] + Csch[x])^2, x, 2, x - 2*Tanh[x/2]}
1815{1/(Coth[x] + Csch[x])^3, x, 6, 2/(1 + Cosh[x]) + Log[1 + Cosh[x]]}
1816{1/(Coth[x] + Csch[x])^4, x, 3, x - 2*Tanh[x/2] - (2/3)*Tanh[x/2]^3}
1817{1/(Coth[x] + Csch[x])^5, x, 7, -(2/(1 + Cosh[x])^2) + 4/(1 + Cosh[x]) + Log[1 + Cosh[x]]}
1818
1819
1820(* Integrands of the form (Csch[x]-Coth[x])^n where n is an integer *)
1821{(-Coth[x] + Csch[x]), x, 3, -2*Log[Cosh[x/2]], -ArcCoth[Cosh[x]] - Log[Sinh[x]]}
1822{(-Coth[x] + Csch[x])^2, x, 2, x - 2*Tanh[x/2]}
1823{(-Coth[x] + Csch[x])^3, x, 6, -(2/(1 + Cosh[x])) - Log[1 + Cosh[x]]}
1824{(-Coth[x] + Csch[x])^4, x, 3, x - 2*Tanh[x/2] - (2/3)*Tanh[x/2]^3}
1825{(-Coth[x] + Csch[x])^5, x, 8, 2/(1 + Cosh[x])^2 - 4/(1 + Cosh[x]) - Log[1 + Cosh[x]]}
1826
1827{1/(-Coth[x] + Csch[x]), x, 3, -Log[1 - Cosh[x]]}
1828{1/(-Coth[x] + Csch[x])^2, x, 2, x - 2*Coth[x/2]}
1829{1/(-Coth[x] + Csch[x])^3, x, 7, -(2/(1 - Cosh[x])) - Log[1 - Cosh[x]]}
1830{1/(-Coth[x] + Csch[x])^4, x, 3, x - 2*Coth[x/2] - (2/3)*Coth[x/2]^3}
1831{1/(-Coth[x] + Csch[x])^5, x, 8, 2/(1 - Cosh[x])^2 - 4/(1 - Cosh[x]) - Log[1 - Cosh[x]]}
1832
1833
1834(* ::Subsubsection::Closed:: *)
1835(*Integrands of the form (a Csch[c+d x] + b Sinh[c+d x])^n*)
1836
1837
1838(* Integrands of the form (Csch[x]-Sinh[x])^n *)
1839(* Note that Csch[x]+Sinh[x] == Cosh[x]*Coth[x] *)
1840{(Csch[x] + Sinh[x]), x, 3, -ArcCoth[Cosh[x]] + Cosh[x]}
1841{(Csch[x] + Sinh[x])^2, x, 3, (3*x)/2 - (3*Coth[x])/2 + (1/2)*Cosh[x]^2*Coth[x]}
1842{(Csch[x] + Sinh[x])^3, x, 5, (-(5/2))*ArcCoth[Cosh[x]] + (5*Cosh[x])/2 - (5/6)*Cosh[x]*Coth[x]^2 + (1/3)*Cosh[x]^3*Coth[x]^2}
1843
1844{(Csch[x] + Sinh[x])^(1/2), x, 3, 2*Sqrt[Cosh[x]*Coth[x]]*Tanh[x]}
1845{(Csch[x] + Sinh[x])^(3/2), x, 4, (-(2/3))*(4 - Cosh[x]^2)*Sqrt[Cosh[x]*Coth[x]]*Sech[x]}
1846{(Csch[x] + Sinh[x])^(5/2), x, 5, (2/15)*Sqrt[Cosh[x]*Coth[x]]*(32 - (8 - 3*Cosh[x]^2)*Coth[x]^2)*Tanh[x]}
1847
1848
1849(* ::Subsubsection::Closed:: *)
1850(*Integrands of the form (a Sech[c+d x] + b Cosh[c+d x])^n*)
1851
1852
1853(* Integrands of the form (Sech[x]-Cosh[x])^n *)
1854(* Note that Sech[x]-Cosh[x] == -Sinh[x]*Tanh[x] *)
1855{(Sech[x] - Cosh[x]), x, 3, ArcTan[Sinh[x]] - Sinh[x]}
1856{(Sech[x] - Cosh[x])^2, x, 3, -((3*x)/2) + (3*Tanh[x])/2 + (1/2)*Sinh[x]^2*Tanh[x]}
1857{(Sech[x] - Cosh[x])^3, x, 6, (-(5/2))*ArcTan[Sinh[x]] + (5*Sinh[x])/2 - (5/6)*Sinh[x]*Tanh[x]^2 - (1/3)*Sinh[x]^3*Tanh[x]^2}
1858
1859{(Sech[x] - Cosh[x])^(1/2), x, 3, 2*Coth[x]*Sqrt[(-Sinh[x])*Tanh[x]]}
1860{(Sech[x] - Cosh[x])^(3/2), x, 4, (-(2/3))*Csch[x]*(4 + Sinh[x]^2)*Sqrt[(-Sinh[x])*Tanh[x]]}
1861{(Sech[x] - Cosh[x])^(5/2), x, 5, (-(2/15))*Coth[x]*Sqrt[(-Sinh[x])*Tanh[x]]*(32 - (8 + 3*Sinh[x]^2)*Tanh[x]^2)}
1862
1863
1864(* ::Subsubsection::Closed:: *)
1865(*Integrands of the form (a Sinh[c+d x] + b Tanh[c+d x])^n*)
1866
1867
1868{1/(Sinh[x] + Tanh[x]), x, 3, (1/2)*Log[Tanh[x/2]] + (1/4)*Tanh[x/2]^2}
1869{1/(Sinh[x] - Tanh[x]), x, 3, (-(1/4))*Coth[x/2]^2 + (1/2)*Log[Tanh[x/2]]}
1870
1871
1872(* ::Subsection::Closed:: *)
1873(*Integrands of the form (a Hyper[c+d x]^2 + b Hyper[c+d x]^2)^n*)
1874
1875
1876(* Integrands of the form 1/(Cosh[x]^2+/-Sinh[x]^2)^n where n is an integer *)
1877{1/(Cosh[x]^2 + Sinh[x]^2), x, 2, ArcTan[Tanh[x]], (1/2)*ArcTan[Sinh[2*x]]}
1878{1/(Cosh[x]^2 + Sinh[x]^2)^2, x, 2, (1/2)*Tanh[2*x]}
1879{1/(Cosh[x]^2 + Sinh[x]^2)^3, x, 3, (1/4)*ArcTan[Sinh[2*x]] + (1/4)*Sech[2*x]*Tanh[2*x]}
1880
1881{1/(Cosh[x]^2 - Sinh[x]^2), x, 2, x}
1882{1/(Cosh[x]^2 - Sinh[x]^2)^2, x, 2, x}
1883{1/(Cosh[x]^2 - Sinh[x]^2)^3, x, 2, x}
1884
1885
1886(* Integrands of the form 1/(Sech[x]^2+/-Tanh[x]^2)^n where n is an integer *)
1887{1/(Sech[x]^2 + Tanh[x]^2), x, 2, x}
1888{1/(Sech[x]^2 + Tanh[x]^2)^2, x, 2, x}
1889{1/(Sech[x]^2 + Tanh[x]^2)^3, x, 2, x}
1890
1891{1/(Sech[x]^2 - Tanh[x]^2), x, 5, -x + Sqrt[2]*ArcTanh[Coth[x]/Sqrt[2]]}
1892{1/(Sech[x]^2 - Tanh[x]^2)^2, x, 8, x - ArcTanh[Coth[x]/Sqrt[2]]/Sqrt[2] - Coth[x]/(2 - Coth[x]^2), x - 2*Sqrt[2]*ArcTanh[Coth[x]/Sqrt[2]] + (3*ArcTanh[Sqrt[2]*Tanh[x]])/Sqrt[2] + Sinh[2*x]/(3 - Cosh[2*x])}
1893{1/(Sech[x]^2 - Tanh[x]^2)^3, x, 12, -x + (7*ArcTanh[Coth[x]/Sqrt[2]])/(4*Sqrt[2]) - (15*Coth[x])/(2 - Coth[x]^2)^2 + (8*Coth[x]^3)/(2 - Coth[x]^2)^2 + (31*Coth[x])/(4*(2 - Coth[x]^2)), -x + 3*Sqrt[2]*ArcTanh[Coth[x]/Sqrt[2]] - (17*ArcTanh[Sqrt[2]*Tanh[x]])/(4*Sqrt[2]) + (2*Sinh[2*x])/(3 - Cosh[2*x])^2 - (3*Sinh[2*x])/(4*(3 - Cosh[2*x]))}
1894
1895
1896(* Integrands of the form 1/(Coth[x]^2+/-Csch[x]^2)^n where n is an integer *)
1897{1/(Coth[x]^2 + Csch[x]^2), x, 5, x - Sqrt[2]*ArcTanh[Tanh[x]/Sqrt[2]]}
1898{1/(Coth[x]^2 + Csch[x]^2)^2, x, 8, x - ArcTanh[Sqrt[2]*Coth[x]]/Sqrt[2] + Coth[x]/(1 - 2*Coth[x]^2), x - 2*Sqrt[2]*ArcTanh[Sqrt[2]*Coth[x]] + (3*ArcTanh[Tanh[x]/Sqrt[2]])/Sqrt[2] - Sinh[2*x]/(3 + Cosh[2*x])}
1899{1/(Coth[x]^2 + Csch[x]^2)^3, x, 12, x - (7*ArcTanh[Tanh[x]/Sqrt[2]])/(4*Sqrt[2]) - Tanh[x]/(2 - Tanh[x]^2)^2 + Tanh[x]/(4*(2 - Tanh[x]^2)), x - 3*Sqrt[2]*ArcTanh[Sqrt[2]*Coth[x]] + (17*ArcTanh[Tanh[x]/Sqrt[2]])/(4*Sqrt[2]) + (2*Sinh[2*x])/(3 + Cosh[2*x])^2 - (3*Sinh[2*x])/(4*(3 + Cosh[2*x]))}
1900
1901{1/(Coth[x]^2 - Csch[x]^2), x, 2, x}
1902{1/(Coth[x]^2 - Csch[x]^2)^2, x, 2, x}
1903{1/(Coth[x]^2 - Csch[x]^2)^3, x, 2, x}
1904
1905
1906(* ::Subsection::Closed:: *)
1907(*Integrands of the form (a + b Hyper[d+e x] + c Hyper[d+e x])^n*)
1908(**)
1909
1910
1911(* ::Subsubsection::Closed:: *)
1912(*Integrands of the form (a + b Cosh[d+e x] + c Sinh[d+e x])^n*)
1913
1914
1915(* Integrands of the form (a+b*Cosh[x]+c*Sinh[x])^n where n is an integer *)
1916{(a + b*Cosh[x] + c*Sinh[x])^3, x, 8, a^3*x + (3/2)*a*b^2*x - (3/2)*a*c^2*x + (2/3)*c*(4*a^2 + b^2 - c^2)*Cosh[x] + (5/3)*a*b*c*Cosh[x]^2 + (2/3)*b*(4*a^2 + b^2 - c^2)*Sinh[x] + (5/6)*a*b^2*Cosh[x]*Sinh[x] + (5/6)*a*c^2*Cosh[x]*Sinh[x] + (1/3)*(c*Cosh[x] + b*Sinh[x])*(a + b*Cosh[x] + c*Sinh[x])^2}
1917{(a + b*Cosh[x] + c*Sinh[x])^2, x, 4, a^2*x + (b^2*x)/2 - (c^2*x)/2 + (3/2)*a*c*Cosh[x] + (3/2)*a*b*Sinh[x] + (1/2)*(c*Cosh[x] + b*Sinh[x])*(a + b*Cosh[x] + c*Sinh[x])}
1918{(a + b*Cosh[x] + c*Sinh[x]), x, 3, a*x + c*Cosh[x] + b*Sinh[x]}
1919{1/(a + b*Cosh[x] + c*Sinh[x]), x, 1, -((2*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/Sqrt[a^2 - b^2 + c^2])}
1920{1/(a + b*Cosh[x] + c*Sinh[x])^2, x, 2, -((2*a*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/(a^2 - b^2 + c^2)^(3/2)) - (c*Cosh[x] + b*Sinh[x])/((a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x]))}
1921{1/(a + b*Cosh[x] + c*Sinh[x])^3, x, 3, -((c*Cosh[x] + b*Sinh[x])/(2*(a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])^2)) - (-((2*(-2*a^2 - b^2 + c^2)*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]])/(a^2 - b^2 + c^2)^(3/2)) - (-3*a*c*Cosh[x] - 3*a*b*Sinh[x])/((a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])))/(2*(a^2 - b^2 + c^2))}
1922{1/(a + b*Cosh[x] + c*Sinh[x])^4, x, 4, -((c*Cosh[x] + b*Sinh[x])/(3*(a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])^3)) - (6*a*(2*a^2 + 3*b^2 - 3*c^2)*ArcTanh[(c - (a - b)*Tanh[x/2])/Sqrt[a^2 - b^2 + c^2]] + (Sqrt[a^2 - b^2 + c^2]*(c*Cosh[x] + b*Sinh[x])*(5*a*(a^2 - b^2 + c^2) + (11*a^2 + 4*(b^2 - c^2))*(a + b*Cosh[x] + c*Sinh[x])))/(a + b*Cosh[x] + c*Sinh[x])^2)/(6*(a^2 - b^2 + c^2)^(7/2))}
1923
1924{(a + a*Cosh[x] + c*Sinh[x])^3, x, 10, (5*a^3*x)/2 - (3/2)*a*c^2*x + (2/3)*c*(5*a^2 - c^2)*Cosh[x] + (5/3)*a^2*c*Cosh[x]^2 + (2/3)*a*(5*a^2 - c^2)*Sinh[x] + (5/6)*a^3*Cosh[x]*Sinh[x] + (5/6)*a*c^2*Cosh[x]*Sinh[x] + (1/3)*(c*Cosh[x] + a*Sinh[x])*(a + a*Cosh[x] + c*Sinh[x])^2}
1925{(a + a*Cosh[x] + c*Sinh[x])^2, x, 4, (3*a^2*x)/2 - (c^2*x)/2 + (3/2)*a*c*Cosh[x] + (3/2)*a^2*Sinh[x] + (1/2)*(c*Cosh[x] + a*Sinh[x])*(a + a*Cosh[x] + c*Sinh[x])}
1926{(a + a*Cosh[x] + c*Sinh[x]), x, 3, a*x + c*Cosh[x] + a*Sinh[x]}
1927{1/(a + a*Cosh[x] + c*Sinh[x]), x, 1, Log[a + c*Tanh[x/2]]/c}
1928{1/(a + a*Cosh[x] + c*Sinh[x])^2, x, 2, (a*Log[a + c*Tanh[x/2]])/c^3 - (c*Cosh[x] + a*Sinh[x])/(c^2*(a + a*Cosh[x] + c*Sinh[x]))}
1929{1/(a + a*Cosh[x] + c*Sinh[x])^3, x, 3, -((c*Cosh[x] + a*Sinh[x])/(2*c^2*(a + a*Cosh[x] + c*Sinh[x])^2)) + ((3*a^2 - c^2)*Log[a + c*Tanh[x/2]] - (3*a*c*(c*Cosh[x] + a*Sinh[x]))/(a + a*Cosh[x] + c*Sinh[x]))/(2*c^5)}
1930{1/(a + a*Cosh[x] + c*Sinh[x])^4, x, 4, -((c*Cosh[x] + a*Sinh[x])/(3*c^2*(a + a*Cosh[x] + c*Sinh[x])^3)) + (3*a*(5*a^2 - 3*c^2)*Log[a + c*Tanh[x/2]] - (c*(c*Cosh[x] + a*Sinh[x])*(5*a*c^2 + (15*a^2 - 4*c^2)*(a + a*Cosh[x] + c*Sinh[x])))/(a + a*Cosh[x] + c*Sinh[x])^2)/(6*c^7)}
1931
1932{(Sqrt[b^2 - c^2]+b*Cosh[x]+c*Sinh[x])^4, x, 6, (35/8)*(b^2 - c^2)^2*x + (35/8)*c*(b^2 - c^2)^(3/2)*Cosh[x] + (35/8)*b*(b^2 - c^2)^(3/2)*Sinh[x] + (35/24)*(b^2 - c^2)*(c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]) + (7/12)*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^2 + (1/4)*(c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^3}
1933{(Sqrt[b^2 - c^2]+b*Cosh[x]+c*Sinh[x])^3, x, 5, (5/2)*(b^2 - c^2)^(3/2)*x + (5/2)*c*(b^2 - c^2)*Cosh[x] + (5/2)*b*(b^2 - c^2)*Sinh[x] + (5/6)*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]) + (1/3)*(c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^2}
1934{(Sqrt[b^2 - c^2]+b*Cosh[x]+c*Sinh[x])^2, x, 4, (3/2)*(b^2 - c^2)*x + (3/2)*c*Sqrt[b^2 - c^2]*Cosh[x] + (3/2)*b*Sqrt[b^2 - c^2]*Sinh[x] + (1/2)*(c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])}
1935{(Sqrt[b^2 - c^2]+b*Cosh[x]+c*Sinh[x]), x, 3, Sqrt[b^2 - c^2]*x + c*Cosh[x] + b*Sinh[x]}
1936{1/(Sqrt[b^2 - c^2]+b*Cosh[x]+c*Sinh[x]), x, 1, -((c + Sqrt[b^2 - c^2]*Sinh[x])/(c*(c*Cosh[x] + b*Sinh[x])))}
1937{1/(Sqrt[b^2 - c^2]+b*Cosh[x]+c*Sinh[x])^2, x, 2, (c*Cosh[x] + b*Sinh[x])/(3*Sqrt[b^2 - c^2]*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^2) - (c + Sqrt[b^2 - c^2]*Sinh[x])/(3*c*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x]))}
1938{1/(Sqrt[b^2 - c^2]+b*Cosh[x]+c*Sinh[x])^3, x, 3, (c*Cosh[x] + b*Sinh[x])/(5*Sqrt[b^2 - c^2]*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^3) + (2*(c*Cosh[x] + b*Sinh[x]))/(15*(b^2 - c^2)*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^2) - (2*(c + Sqrt[b^2 - c^2]*Sinh[x]))/(15*c*(b^2 - c^2)*(c*Cosh[x] + b*Sinh[x]))}
1939{1/(Sqrt[b^2 - c^2]+b*Cosh[x]+c*Sinh[x])^4, x, 4, (c*Cosh[x] + b*Sinh[x])/(7*Sqrt[b^2 - c^2]*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^4) + (3*(c*Cosh[x] + b*Sinh[x]))/(35*(b^2 - c^2)*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^3) + (2*(c*Cosh[x] + b*Sinh[x]))/(35*(b^2 - c^2)^(3/2)*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^2) - (2*(c + Sqrt[b^2 - c^2]*Sinh[x]))/(35*c*(b^2 - c^2)^(3/2)*(c*Cosh[x] + b*Sinh[x]))}
1940
1941
1942(* Integrands of the form (a+b*Cosh[x]+c*Sinh[x])^n where n is a half-integer *)
1943{(a + b*Cosh[x] + c*Sinh[x])^(5/2), x, 5, (16/15)*a*(c*Cosh[x] + b*Sinh[x])*Sqrt[a + b*Cosh[x] + c*Sinh[x]] + (2/5)*(c*Cosh[x] + b*Sinh[x])*(a + b*Cosh[x] + c*Sinh[x])^(3/2) + (2*I*(23*a^2 + 9*(b^2 - c^2))*EllipticE[(1/4)*(Pi - 2*I*(x + I*ArcTan[I*c, b])), 2/(1 - a/Sqrt[b^2 - c^2])]*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/(15*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a - Sqrt[b^2 - c^2])]) - (16*I*a*(a^2 - b^2 + c^2)*EllipticF[(1/4)*(Pi - 2*I*(x + I*ArcTan[I*c, b])), 2/(1 - a/Sqrt[b^2 - c^2])]*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a - Sqrt[b^2 - c^2])])/(15*Sqrt[a + b*Cosh[x] + c*Sinh[x]])}
1944{(a + b*Cosh[x] + c*Sinh[x])^(3/2), x, 4, (2/3)*(c*Cosh[x] + b*Sinh[x])*Sqrt[a + b*Cosh[x] + c*Sinh[x]] + (8*I*a*EllipticE[(1/4)*(Pi - 2*I*(x + I*ArcTan[I*c, b])), 2/(1 - a/Sqrt[b^2 - c^2])]*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/(3*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a - Sqrt[b^2 - c^2])]) - (2*I*(a^2 - b^2 + c^2)*EllipticF[(1/4)*(Pi - 2*I*(x + I*ArcTan[I*c, b])), 2/(1 - a/Sqrt[b^2 - c^2])]*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a - Sqrt[b^2 - c^2])])/(3*Sqrt[a + b*Cosh[x] + c*Sinh[x]])}
1945{(a + b*Cosh[x] + c*Sinh[x])^(1/2), x, 1, (2*I*EllipticE[(1/4)*(Pi - 2*I*(x + I*ArcTan[I*c, b])), 2/(1 - a/Sqrt[b^2 - c^2])]*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a - Sqrt[b^2 - c^2])]}
1946{1/(a + b*Cosh[x] + c*Sinh[x])^(1/2), x, 1, (2*I*EllipticF[(1/4)*(Pi - 2*I*(x + I*ArcTan[I*c, b])), 2/(1 - a/Sqrt[b^2 - c^2])]*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a - Sqrt[b^2 - c^2])])/Sqrt[a + b*Cosh[x] + c*Sinh[x]]}
1947{1/(a + b*Cosh[x] + c*Sinh[x])^(3/2), x, 2, -((2*(c*Cosh[x] + b*Sinh[x]))/((a^2 - b^2 + c^2)*Sqrt[a + b*Cosh[x] + c*Sinh[x]])) + (2*I*EllipticE[(1/4)*(Pi - 2*I*(x + I*ArcTan[I*c, b])), 2/(1 - a/Sqrt[b^2 - c^2])]*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/((a^2 - b^2 + c^2)*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a - Sqrt[b^2 - c^2])])}
1948{1/(a + b*Cosh[x] + c*Sinh[x])^(5/2), x, 5, -((2*(c*Cosh[x] + b*Sinh[x]))/(3*(a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])^(3/2))) - (1/(3*(a^2 - b^2 + c^2)))*(2*((2*(-2*a*c*Cosh[x] - 2*a*b*Sinh[x]))/((-a^2 + b^2 - c^2)*Sqrt[a + b*Cosh[x] + c*Sinh[x]]) + (4*I*a*EllipticE[(1/2)*(Pi/2 - I*(x + I*ArcTan[I*c, b])), 2/(1 - a/Sqrt[b^2 - c^2])]*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/((-a^2 + b^2 - c^2)*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a - Sqrt[b^2 - c^2])]) + (4*I*((-a^2)*b + (1/2)*b*((3*a^2)/2 + b^2/2 - c^2/2))*EllipticF[(1/2)*(Pi/2 - I*(x + I*ArcTan[I*c, b])), 2/(1 - a/Sqrt[b^2 - c^2])]*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a - Sqrt[b^2 - c^2])])/(b*(-a^2 + b^2 - c^2)*Sqrt[a + b*Cosh[x] + c*Sinh[x]])))}
1949(* {1/(a + b*Cosh[x] + c*Sinh[x])^(7/2), x, 6, -((2*(c*Cosh[x] + b*Sinh[x]))/(5*(a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])^(5/2))) - (1/(5*(a^2 - b^2 + c^2)))*(2*(-((2*(-4*a*c*Cosh[x] - 4*a*b*Sinh[x]))/(3*(a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])^(3/2))) - (4*I*(-2*a^2*b + (3/2)*b*(-((5*a^2)/2) - (3*b^2)/2 + (3*c^2)/2))*EllipticE[(1/2)*(Pi/2 - I*(x + I*ArcTan[I*c, b])), 2/(1 - a/Sqrt[b^2 - c^2])]*Sqrt[a + b*Cosh[x] + c*Sinh[x]])/(3*b*(-a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a - Sqrt[b^2 - c^2])]) - (8*I*((1/2)*b*(-2*a*b^2 + 2*a*c^2 + (3/2)*a*(-((5*a^2)/2) - (3*b^2)/2 + (3*c^2)/2)) - (1/2)*a*(-2*a^2*b + (3/2)*b*(-((5*a^2)/2) - (3*b^2)/2 + (3*c^2)/2)))*EllipticF[(1/2)*(Pi/2 - I*(x + I*ArcTan[I*c, b])), 2/(1 - a/Sqrt[b^2 - c^2])]*Sqrt[(a + b*Cosh[x] + c*Sinh[x])/(a - Sqrt[b^2 - c^2])])/(3*b*(-a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*Sqrt[a + b*Cosh[x] + c*Sinh[x]]) - (4*((-(-2*a^2*c + (3/2)*c*(-((5*a^2)/2) - (3*b^2)/2 + (3*c^2)/2)))*Cosh[x] + (2*a^2*b - (3/2)*b*(-((5*a^2)/2) - (3*b^2)/2 + (3*c^2)/2))*Sinh[x]))/(3*(-a^2 + b^2 - c^2)*(a^2 - b^2 + c^2)*Sqrt[a + b*Cosh[x] + c*Sinh[x]])))} *)
1950
1951
1952(* Integrands of the form (a+b*Cosh[x]+c*Sinh[x])^n where n is a half-integer and a^2=b^2-c^2 *)
1953{(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(5/2), x, 3, (64*(b^2 - c^2)*(c*Cosh[x] + b*Sinh[x]))/(15*Sqrt[Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]]) + (16/15)*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x])*Sqrt[Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]] + (2/5)*(c*Cosh[x] + b*Sinh[x])*(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2)}
1954{(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2), x, 2, (8*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x]))/(3*Sqrt[Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]]) + (2/3)*(c*Cosh[x] + b*Sinh[x])*Sqrt[Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]]}
1955{(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(1/2), x, 1, (2*(c*Cosh[x] + b*Sinh[x]))/Sqrt[Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]]}
1956(* {1/(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(1/2), x, 0, 0}
1957{1/(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2), x, 0, 0}
1958{1/(Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(5/2), x, 0, 0} *)
1959
1960{(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(5/2), x, 3, (64*(b^2 - c^2)*(c*Cosh[x] + b*Sinh[x]))/(15*Sqrt[-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]]) - (16/15)*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x])*Sqrt[-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]] + (2/5)*(c*Cosh[x] + b*Sinh[x])*(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2)}
1961{(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2), x, 2, -((8*Sqrt[b^2 - c^2]*(c*Cosh[x] + b*Sinh[x]))/(3*Sqrt[-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]])) + (2/3)*(c*Cosh[x] + b*Sinh[x])*Sqrt[-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]]}
1962{(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(1/2), x, 1, (2*(c*Cosh[x] + b*Sinh[x]))/Sqrt[-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]]}
1963{1/(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(1/2), x, 1, (I*Sqrt[2]*EllipticF[(1/4)*(Pi - 2*I*(x + I*ArcTan[I*c, b])), 1]*Sqrt[(Sqrt[b^2 - c^2] - b*Cosh[x] - c*Sinh[x])/Sqrt[b^2 - c^2]])/Sqrt[-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]]}
1964{1/(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2), x, 2, -((c*Cosh[x] + b*Sinh[x])/(2*Sqrt[b^2 - c^2]*(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2))) - (I*EllipticF[(1/4)*(Pi - 2*I*(x + I*ArcTan[I*c, b])), 1]*Sqrt[(Sqrt[b^2 - c^2] - b*Cosh[x] - c*Sinh[x])/Sqrt[b^2 - c^2]])/(2*Sqrt[2]*Sqrt[b^2 - c^2]*Sqrt[-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]])}
1965{1/(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(5/2), x, 3, -((c*Cosh[x] + b*Sinh[x])/(4*Sqrt[b^2 - c^2]*(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(5/2))) + (3*(c*Cosh[x] + b*Sinh[x]))/(16*(b^2 - c^2)*(-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x])^(3/2)) + (3*I*EllipticF[(1/4)*(Pi - 2*I*(x + I*ArcTan[I*c, b])), 1]*Sqrt[(Sqrt[b^2 - c^2] - b*Cosh[x] - c*Sinh[x])/Sqrt[b^2 - c^2]])/(16*Sqrt[2]*(b^2 - c^2)*Sqrt[-Sqrt[b^2 - c^2] + b*Cosh[x] + c*Sinh[x]])}
1966
1967
1968(* ::Subsubsection::Closed:: *)
1969(*Integrands of the form (a + b Tanh[d+e x] + c Sech[d+e x])^n*)
1970
1971
1972{1/(a + b*Tanh[x] + c*Sech[x]), x, -8, (a*x)/(a^2 - b^2) - (2*a*c*ArcTan[(b + (a - c)*Tanh[x/2])/Sqrt[a^2 - b^2 - c^2]])/((a^2 - b^2)*Sqrt[a^2 - b^2 - c^2]) - (b*Log[c + a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)}
1973{1/(a + b*Coth[x] + c*Csch[x]), x, -8, (a*x)/(a^2 - b^2) - (2*a*c*ArcTan[(a + (b - c)*Tanh[x/2])/Sqrt[-a^2 + b^2 - c^2]])/((a^2 - b^2)*Sqrt[-a^2 + b^2 - c^2]) - (b*Log[c + b*Cosh[x] + a*Sinh[x]])/(a^2 - b^2)}
1974
1975
1976(* ::Subsection::Closed:: *)
1977(*Miscellaneous integrands involving two hyperbolic functions*)
1978
1979
1980{1/(a + b*Sinh[x]*Cosh[x]), x, 2, -((2*ArcTanh[(b - 2*a*Tanh[x])/Sqrt[4*a^2 + b^2]])/Sqrt[4*a^2 + b^2])}
1981{x/(a + b*Sinh[x]*Cosh[x]), x, 9, (x*Log[1 + (b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2])])/Sqrt[4*a^2 + b^2] - (x*Log[1 + (b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2])])/Sqrt[4*a^2 + b^2] + PolyLog[2, -((b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2]))]/(2*Sqrt[4*a^2 + b^2]) - PolyLog[2, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))]/(2*Sqrt[4*a^2 + b^2])}
1982{x^2/(a + b*Sinh[x]*Cosh[x]), x, 11, (x^2*Log[1 + (b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2])])/Sqrt[4*a^2 + b^2] - (x^2*Log[1 + (b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2])])/Sqrt[4*a^2 + b^2] + (x*PolyLog[2, -((b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2]))])/Sqrt[4*a^2 + b^2] - (x*PolyLog[2, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))])/Sqrt[4*a^2 + b^2] - PolyLog[3, -((b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2]))]/(2*Sqrt[4*a^2 + b^2]) + PolyLog[3, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))]/(2*Sqrt[4*a^2 + b^2])}
1983{x^3/(a + b*Sinh[x]*Cosh[x]), x, 13, (x^3*Log[1 + (b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2])])/Sqrt[4*a^2 + b^2] - (x^3*Log[1 + (b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2])])/Sqrt[4*a^2 + b^2] + (3*x^2*PolyLog[2, -((b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2]))])/(2*Sqrt[4*a^2 + b^2]) - (3*x^2*PolyLog[2, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))])/(2*Sqrt[4*a^2 + b^2]) - (3*x*PolyLog[3, -((b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2]))])/(2*Sqrt[4*a^2 + b^2]) + (3*x*PolyLog[3, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))])/(2*Sqrt[4*a^2 + b^2]) + (3*PolyLog[4, -((b*E^(2*x))/(2*a - Sqrt[4*a^2 + b^2]))])/(4*Sqrt[4*a^2 + b^2]) - (3*PolyLog[4, -((b*E^(2*x))/(2*a + Sqrt[4*a^2 + b^2]))])/(4*Sqrt[4*a^2 + b^2])}
1984
1985
1986{Sqrt[a + b*Sinh[x]*Cosh[x]], x, 3, (I*EllipticE[Pi/4 - I*x, -((2*I*b)/(2*a - I*b))]*Sqrt[2*a + b*Sinh[2*x]])/(Sqrt[2]*Sqrt[(2*a + b*Sinh[2*x])/(2*a - I*b)])}
1987{1/Sqrt[a + b*Sinh[x]*Cosh[x]], x, 3, (I*Sqrt[2]*EllipticF[Pi/4 - I*x, -((2*I*b)/(2*a - I*b))]*Sqrt[(2*a + b*Sinh[2*x])/(2*a - I*b)])/Sqrt[2*a + b*Sinh[2*x]]}
1988
1989
1990(* Integrands of the form x*Cosh[2*x]*Sech[x]^n where n is an integer *)
1991{x*Cosh[2*x]*Sech[x], x, -1, -2*Cosh[x] + I*x*Log[1 - I/E^x] - I*x*Log[1 + I/E^x] + I*PolyLog[2, -I/E^x] - I*PolyLog[2, I/E^x] + 2*x*Sinh[x]}
1992{x*Cosh[2*x]*Sech[x]^2, x, 7, x^2+Log[Cosh[x]]-x*Tanh[x]}
1993{x*Cosh[2*x]*Sech[x]^3, x, -1, 3*x*ArcTan[E^x] - (3/2)*I*PolyLog[2, (-I)*E^x] + (3/2)*I*PolyLog[2, I*E^x] - Sech[x]/2 - (1/2)*x*Sech[x]*Tanh[x]}
1994
1995
1996(* Integrands of the form Hyper[x]^2/(a+b*Hyper[2*x]) *)
1997{Sinh[x]^2/(a + b*Sinh[2*x]), x, 4, ArcTanh[(b - a*Tanh[x])/Sqrt[a^2 + b^2]]/(2*Sqrt[a^2 + b^2]) + Log[a + b*Sinh[2*x]]/(4*b)}
1998{Cosh[x]^2/(a + b*Sinh[2*x]), x, 4, -(ArcTanh[(b - a*Tanh[x])/Sqrt[a^2 + b^2]]/(2*Sqrt[a^2 + b^2])) + Log[a + b*Sinh[2*x]]/(4*b)}
1999
2000{Sinh[x]^2/(a + b*Cosh[2*x]), x, 4, x/(2*b) - ArcTanh[((a - b)*Tanh[x])/Sqrt[a^2 - b^2]]/(2*Sqrt[a^2 - b^2]) - (a*ArcTanh[((a - b)*Tanh[x])/Sqrt[a^2 - b^2]])/(2*b*Sqrt[a^2 - b^2])}
2001{Cosh[x]^2/(a + b*Cosh[2*x]), x, 4, x/(2*b) + ArcTanh[((a - b)*Tanh[x])/Sqrt[a^2 - b^2]]/(2*Sqrt[a^2 - b^2]) - (a*ArcTanh[((a - b)*Tanh[x])/Sqrt[a^2 - b^2]])/(2*b*Sqrt[a^2 - b^2])}
2002
2003
2004{Tanh[c+d*x]/Sqrt[a*Sinh[c+d*x]^2], x, 2, (ArcTan[Sinh[c + d*x]]*Sinh[c + d*x])/(d*Sqrt[a*Sinh[c + d*x]^2])}
2005{Coth[c+d*x]/Sqrt[a*Cosh[c+d*x]^2], x, 2, -((ArcCoth[Cosh[c + d*x]]*Cosh[c + d*x])/(d*Sqrt[a*Cosh[c + d*x]^2]))}
2006
2007
2008{(Sech[Sqrt[x]]*Tanh[Sqrt[x]])/Sqrt[x], x, 2, -2*Sech[Sqrt[x]]}
2009
2010
2011{Sqrt[Csch[x]]*(x*Cosh[x] - 4*Sech[x]*Tanh[x]), x, 10, (2*x)/Sqrt[Csch[x]] - (4*Sech[x])/Csch[x]^(3/2)}
2012