1
2comment
3
4  Testing some integrals of inverse trigonometric and hyperbolic functions
5
6end;
7
8
9int(asin(x),x);
10
11
12                     2
13asin(x)*x + sqrt( - x  + 1)
14
15
16int(acos(x),x);
17
18
19                     2
20acos(x)*x - sqrt( - x  + 1)
21
22
23int(asinh(x),x);
24
25
26                   2
27asinh(x)*x - sqrt(x  + 1)
28
29
30int(acosh(x),x);
31
32
33                   2
34acosh(x)*x - sqrt(x  - 1)
35
36
37int(acos(x-1),x);
38
39
40                                       2
41acos(x - 1)*x - acos(x - 1) - sqrt( - x  + 2*x)
42
43
44int(acos(x)*x,x);
45
46
47            2                      2
48 2*acos(x)*x  + asin(x) - sqrt( - x  + 1)*x
49--------------------------------------------
50                     4
51
52
53int(x^2*asin(3*x),x);
54
55
56               3                2       2                2
57 81*asin(3*x)*x  + 9*sqrt( - 9*x  + 1)*x  + 2*sqrt( - 9*x  + 1)
58----------------------------------------------------------------
59                              243
60
61
62int(x^2*acos(3*x),x);
63
64
65               3                2       2                2
66 81*acos(3*x)*x  - 9*sqrt( - 9*x  + 1)*x  - 2*sqrt( - 9*x  + 1)
67----------------------------------------------------------------
68                              243
69
70
71int(x^2*asinh(3*x),x);
72
73
74                3             2       2             2
75 81*asinh(3*x)*x  - 9*sqrt(9*x  + 1)*x  + 2*sqrt(9*x  + 1)
76-----------------------------------------------------------
77                            243
78
79
80int(x^2*acosh(3*x),x);
81
82
83                3             2       2             2
84 81*acosh(3*x)*x  - 9*sqrt(9*x  - 1)*x  - 2*sqrt(9*x  - 1)
85-----------------------------------------------------------
86                            243
87
88
89int(asin(3*x)/x^3,x);
90
91
92                             2
93  - asin(3*x) - 3*sqrt( - 9*x  + 1)*x
94--------------------------------------
95                    2
96                 2*x
97
98
99int(acos(3*x)/x^3,x);
100
101
102                             2
103  - acos(3*x) + 3*sqrt( - 9*x  + 1)*x
104--------------------------------------
105                    2
106                 2*x
107
108
109int(asinh(3*x)/x^3,x);
110
111
112                           2             2
113  - asinh(3*x) - 3*sqrt(9*x  + 1)*x - 9*x
114-------------------------------------------
115                      2
116                   2*x
117
118
119int(acosh(3*x)/x^3,x);
120
121
122                           2             2
123  - acosh(3*x) - 3*sqrt(9*x  - 1)*x - 9*x
124-------------------------------------------
125                      2
126                   2*x
127
128
129
130% The following is correct, but not optimal
131
132int(x*acos(x-1),x);
133
134
135                2
136(2*acos(x - 1)*x  - sqrt(x)*sqrt( - x + 2)*x - 3*sqrt(x)*sqrt( - x + 2)
137
138           sqrt( - x + 2) + sqrt(x)*i
139  - 6*log(----------------------------)*i)/4
140                    sqrt(2)
141
142
143df(ws,x);
144
145
146(x*(2*sqrt(x)*sqrt( - x + 2)*acos(x - 1)*x
147
148     - 4*sqrt(x)*sqrt( - x + 2)*acos(x - 1)
149
150                                       2           2
151     + sqrt(x)*sqrt( - x + 2)*sqrt( - x  + 2*x) + x  - 2*x))/(2*sqrt(x)
152
153   *sqrt( - x + 2)*(x - 2))
154
155
156int(x*acos(3*x-2),x);
157
158
159                   2                                  2
160(18*acos(3*x - 2)*x  + 9*asin(3*x - 2) - 3*sqrt( - 3*x  + 4*x - 1)*sqrt(3)*x
161
162                 2
163  - 6*sqrt( - 3*x  + 4*x - 1)*sqrt(3) + 8)/36
164
165
166df(ws,x);
167
168
169acos(3*x - 2)*x
170
171
172end;
173
174Tested on x86_64-pc-windows CSL
175Time (counter 1): 0 ms  plus GC time: 16 ms
176
177End of Lisp run after 0.00+0.06 seconds
178real 0.20
179user 0.06
180sys 0.03
181