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Searched refs:Cosh (Results 1 – 25 of 1340) sorted by relevance

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/dports/math/reduce/Reduce-svn5758-src/packages/rubi_red/tests4/HyperbolicFunctions/
H A DHyperbolicCosineFunctions.m74 …Cosh[x]^3)^(5/2),x, 6, -((26*I*a^2*Sqrt[a*Cosh[x]^3]*EllipticF[(I*x)/2, 2])/(77*Cosh[x]^(3/2))) + …
77 {1/(a*Cosh[x]^3)^(1/2), x, 3, (2*I*Cosh[x]^(3/2)*EllipticE[(I*x)/2, 2])/Sqrt[a*Cosh[x]^3] + (2*Cosh…
86 …Cosh[x]^4)^(5/2),x, 6, (63/256)*a^2*x*Sqrt[a*Cosh[x]^4]*Sech[x]^2 + (21/128)*a^2*Cosh[x]*Sqrt[a*Co…
87 …Cosh[x]^4)^(3/2),x, 4, (5/16)*a*x*Sqrt[a*Cosh[x]^4]*Sech[x]^2 + (5/24)*a*Cosh[x]*Sqrt[a*Cosh[x]^4]…
89 {1/(a*Cosh[x]^4)^(1/2), x, 2, (Cosh[x]*Sinh[x])/Sqrt[a*Cosh[x]^4]}
114 {Sinh[x]^3/(1 + Cosh[x])^2, x, 4, Cosh[x] - 2*Log[1 + Cosh[x]]}
115 {Sinh[x]^3/(1 - Cosh[x])^2, x, 4, Cosh[x] + 2*Log[1 - Cosh[x]]}
127 {Sinh[x]^7/(a + a*Cosh[x]), x, 4, -(Cosh[x]/a) + Cosh[x]^2/(2*a) + (2*Cosh[x]^3)/(3*a) - Cosh[x]^4/…
129 {Sinh[x]^5/(a + a*Cosh[x]), x, 4, Cosh[x]/a - Cosh[x]^2/(2*a) - Cosh[x]^3/(3*a) + Cosh[x]^4/(4*a)}
131 {Sinh[x]^3/(a + a*Cosh[x]), x, 3, -(Cosh[x]/a) + Cosh[x]^2/(2*a)}
[all …]
H A Dcosh(c+d x)^m (a+b cosh(c+d x))^n.m4 (*Cosh[c+d x]^m (a+b Cosh[c+d x])^n*)
8 (*Integrands of the form Cosh[c+d x]^m (a+b Cosh[c+d x])^n*)
350 …Cosh[c + d*x]]/Sqrt[2 - 3*Cosh[c + d*x]], x, 3, -((2*Sqrt[-((2 - 3*Cosh[c + d*x])/(1 + Cosh[c + d*…
354 …Cosh[c + d*x]]/Sqrt[3 - 2*Cosh[c + d*x]], x, 3, -((2*Sqrt[(3 - 2*Cosh[c + d*x])/(1 + Cosh[c + d*x]…
356 …Cosh[c + d*x]]/Sqrt[-3 + 2*Cosh[c + d*x]], x, 3, -((2*Sqrt[(3 - 2*Cosh[c + d*x])/(1 + Cosh[c + d*x…
372 {Cosh[c + d*x]^(7/3)/(a + b*Cosh[c + d*x]), x, 0, Defer[Int][Cosh[c + d*x]^(7/3)/(a + b*Cosh[c + d*…
373 {Cosh[c + d*x]^(5/3)/(a + b*Cosh[c + d*x]), x, 0, Defer[Int][Cosh[c + d*x]^(5/3)/(a + b*Cosh[c + d*…
374 {Cosh[c + d*x]^(4/3)/(a + b*Cosh[c + d*x]), x, 0, Defer[Int][Cosh[c + d*x]^(4/3)/(a + b*Cosh[c + d*…
375 {Cosh[c + d*x]^(2/3)/(a + b*Cosh[c + d*x]), x, 0, Defer[Int][Cosh[c + d*x]^(2/3)/(a + b*Cosh[c + d*…
376 {Cosh[c + d*x]^(1/3)/(a + b*Cosh[c + d*x]), x, 0, Defer[Int][Cosh[c + d*x]^(1/3)/(a + b*Cosh[c + d*…
[all …]
H A DTwoHyperbolicFunctions.m249 {Cosh[a + b*x]*Coth[c + b*x], x, 3, -((ArcTanh[Cosh[c + b*x]]*Cosh[a - c])/b) + Cosh[a + b*x]/b}
785 {Cosh[2*x]*Sinh[x], x, 3, -(Cosh[x]/2) + (1/6)*Cosh[3*x]}
786 {Cosh[3*x]*Sinh[x], x, 3, (-(1/4))*Cosh[2*x] + (1/8)*Cosh[4*x]}
787 {Cosh[4*x]*Sinh[x], x, 3, (-(1/6))*Cosh[3*x] + (1/10)*Cosh[5*x]}
826 {Sinh[3*x]*Cosh[x], x, 3, (1/4)*Cosh[2*x] + (1/8)*Cosh[4*x]}
827 {Sinh[4*x]*Cosh[x], x, 3, (1/6)*Cosh[3*x] + (1/10)*Cosh[5*x]}
844 {Coth[2*x]*Cosh[x], x, 4, (-(1/2))*ArcTanh[Cosh[x]] + Cosh[x]}
845 {Coth[3*x]*Cosh[x], x, 3, (-(1/3))*ArcTanh[(3*Cosh[x])/(1 + 2*Cosh[x]^2)] + Cosh[x]}
846 {Coth[4*x]*Cosh[x], x, 6, (-(1/4))*ArcTanh[Cosh[x]] - ArcTanh[Sqrt[2]*Cosh[x]]/(2*Sqrt[2]) + Cosh[x…
848 {Coth[6*x]*Cosh[x], x, 9, (-(1/6))*ArcTanh[Cosh[x]] - (1/6)*ArcTanh[2*Cosh[x]] - ArcTanh[(2*Cosh[x]…
[all …]
H A DHyperbolicFunctions.m24 {Cosh[x]/(a*Cosh[x] + b*Sinh[x]), x, 2, (a*x)/(a^2 - b^2) - (b*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b…
97 …Cosh[x])/(b*Cosh[x] + c*Sinh[x])^3, x, 4, (A*ArcTan[(c*Cosh[x] + b*Sinh[x])/Sqrt[b^2 - c^2]])/(2*(…
132 {(a + b*Cosh[x])/(b^2 + 2*a*b*Cosh[x] + a^2*Cosh[x]^2), x, 2, Sinh[x]/(b + a*Cosh[x])}
145 (* {(-Cosh[x] + x*Sinh[x])/(x - Cosh[x])^2, x, 0, x/(x - Cosh[x])} *)
258 {(Cosh[x] - Sinh[x])/(Cosh[x] + Sinh[x]), x, 4, -(Cosh[x]/(Cosh[x] + Sinh[x]))}
259 {(Cosh[x] - I*Sinh[x])/(Cosh[x] + I*Sinh[x]), x, 1, (-I)*Log[Cosh[x] + I*Sinh[x]]}
280 {(b^2 - c^2 + a*b*Cosh[x] + a*c*Sinh[x])/(a + b*Cosh[x] + c*Sinh[x])^2, x, 1, (c*Cosh[x] + b*Sinh[x…
458 {Cosh[x]^3*(a + b*Cosh[x]^2)^3*Sinh[x], x, 4, -((a*(a + b*Cosh[x]^2)^4)/(8*b^2)) + (a + b*Cosh[x]^2…
469 {(Cosh[x] - Sinh[x])/(Cosh[x] + Sinh[x]), x, 4, -(Cosh[x]/(Cosh[x] + Sinh[x]))}
472 {(1 + Sinh[x]^2)/(1 + Cosh[x] + Sinh[x]), x, -13, (3*x)/4 + Cosh[x]/2 - Cosh[x]^2/8 - Log[1 + Cosh[…
[all …]
H A DHyperbolicSineFunctions.m17 {Sinh[a + b*x]^3, x, 2, -(Cosh[a + b*x]/b) + Cosh[a + b*x]^3/(3*b)}
19 {Sinh[a + b*x]^5, x, 2, Cosh[a + b*x]/b - (2*Cosh[a + b*x]^3)/(3*b) + Cosh[a + b*x]^5/(5*b)}
146 {Sinh[x]^4/(I + Sinh[x]), x, 4, (3*I*x)/2 - 4*Cosh[x] + (4*Cosh[x]^3)/3 - (3/2)*I*Cosh[x]*Sinh[x] -…
254 {Cosh[x]^8/(I + Sinh[x]), x, 4, -((5*I*x)/16) + Cosh[x]^7/7 - (5/16)*I*Cosh[x]*Sinh[x] - (5/24)*I*C…
256 {Cosh[x]^6/(I + Sinh[x]), x, 3, -((3*I*x)/8) + Cosh[x]^5/5 - (3/8)*I*Cosh[x]*Sinh[x] - (1/4)*I*Cosh…
258 {Cosh[x]^4/(I + Sinh[x]), x, 2, -((I*x)/2) + Cosh[x]^3/3 - (1/2)*I*Cosh[x]*Sinh[x]}
260 {Cosh[x]^2/(I + Sinh[x]), x, 2, (-I)*x + Cosh[x]}
269 {Cosh[x]^6/(I + Sinh[x])^2, x, 3, -((5*x)/8) - (5/8)*Cosh[x]*Sinh[x] - (5/12)*Cosh[x]^3*Sinh[x] + (…
271 {Cosh[x]^4/(I + Sinh[x])^2, x, 2, -((3*x)/2) - (3/2)*Cosh[x]*Sinh[x] + (2*Cosh[x]^3)/(I + Sinh[x])}
625 {Coth[x]/Sqrt[1 + Sinh[x]^2], x, 2, -((ArcTanh[Cosh[x]]*Cosh[x])/Sqrt[Cosh[x]^2])}
[all …]
/dports/math/reduce/Reduce-svn5758-src/packages/rubi_red/tests/HyperbolicFunctions/
H A DHyperbolicCosineFunctions.m99 {1/(a*Cosh[x]^2)^(1/2), x, 2, (ArcTan[Sinh[x]]*Cosh[x])/Sqrt[a*Cosh[x]^2]}
105 …Cosh[x]^3)^(5/2),x, 6, -((2*a^2*Sqrt[a*Cosh[x]^3]*(195*I*EllipticF[(I*x)/2, 2] - 195*Sqrt[Cosh[x]]…
108 {1/(a*Cosh[x]^3)^(1/2), x, 3, (2*Cosh[x]*(I*Sqrt[Cosh[x]]*EllipticE[(I*x)/2, 2] + Sinh[x]))/Sqrt[a*…
114 …Cosh[x]^4)^(5/2),x, 6, (63/256)*a^2*x*Sqrt[a*Cosh[x]^4]*Sech[x]^2 + (21/128)*a^2*Cosh[x]*Sqrt[a*Co…
115 …Cosh[x]^4)^(3/2),x, 4, (5/16)*a*x*Sqrt[a*Cosh[x]^4]*Sech[x]^2 + (5/24)*a*Cosh[x]*Sqrt[a*Cosh[x]^4]…
117 {1/(a*Cosh[x]^4)^(1/2), x, 2, (Cosh[x]*Sinh[x])/Sqrt[a*Cosh[x]^4]}
216 …Cosh[x])^(3/2), x, 9, (-(1280/9))*a*Sqrt[a + a*Cosh[x]] - 16*a*x^2*Sqrt[a + a*Cosh[x]] - (64/27)*a…
218 …Cosh[x])^(3/2), x, 4, (-(16/3))*a*Sqrt[a + a*Cosh[x]] - (8/9)*a*Cosh[x/2]^2*Sqrt[a + a*Cosh[x]] + …
538 {x/Cosh[x]^(3/2) + x*Sqrt[Cosh[x]], x, 2, -4*Sqrt[Cosh[x]] + (2*x*Sinh[x])/Sqrt[Cosh[x]]}
539 {x/Cosh[x]^(5/2) - x/(3*Sqrt[Cosh[x]]), x, 2, 4/(3*Sqrt[Cosh[x]]) + (2*x*Sinh[x])/(3*Cosh[x]^(3/2))}
[all …]
H A DTwoHyperbolicFunctions.m244 {Cosh[a + b*x]*Coth[c + b*x], x, 3, -((ArcCoth[Cosh[c + b*x]]*Cosh[a - c])/b) + Cosh[a + b*x]/b}
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]}
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]}
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]}
542 …Cosh[x], x, 8, ArcTanh[(1 - 4*Cosh[x])/Sqrt[5]]/(2*Sqrt[5]) - (1/5)*ArcTanh[Cosh[x]] - ArcTanh[(1 …
683 {Cosh[x]/(a + a*Cosh[x]), x, 2, x/a - Sinh[x]/(a*(1 + Cosh[x]))}
[all …]
H A DHyperbolicFunctions.m26 {Cosh[x]/(a*Cosh[x] + b*Sinh[x]), x, 1, (a*x)/(a^2 - b^2) - (b*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b…
74 {Cosh[x]/(a*Cosh[x] + b*Sinh[x])^3, x, 3, b/(2*(a^2 - b^2)*(a*Cosh[x] + b*Sinh[x])^2) + Sinh[x]/((a…
140 (* Integrands of the form Cosh[x]^m/(a+b*Cosh[x]*c*Cosh[x]^2) where m is an integer *)
146 {(a + b*Cosh[x])/(b^2 + 2*a*b*Cosh[x] + a^2*Cosh[x]^2), x, 2, Sinh[x]/(b + a*Cosh[x])}
160 (* {(-Cosh[x] + x*Sinh[x])/(x - Cosh[x])^2, x, 0, x/(x - Cosh[x])} *)
276 {(Cosh[x] - Sinh[x])/(Cosh[x] + Sinh[x]), x, 2, (-(1/2))*(Cosh[x] - Sinh[x])^2}
277 {(Cosh[x] - I*Sinh[x])/(Cosh[x] + I*Sinh[x]), x, 1, (-I)*Log[Cosh[x] + I*Sinh[x]]}
476 {Cosh[x]^3*(a + b*Cosh[x]^2)^3*Sinh[x], x, 4, -((a*(a + b*Cosh[x]^2)^4)/(40*b^2)) + (Cosh[x]^2*(a +…
488 {(Cosh[x] - Sinh[x])/(Cosh[x] + Sinh[x]), x, 2, (-(1/2))*(Cosh[x] - Sinh[x])^2}
491 {(1 + Sinh[x]^2)/(1 + Cosh[x] + Sinh[x]), x, -13, (3*x)/4 + Cosh[x]/2 - Cosh[x]^2/8 - Log[1 + Cosh[…
[all …]
H A DHyperbolicSineFunctions.m11 {Sinh[a + b*x], x, 1, Cosh[a + b*x]/b}
13 {Sinh[a + b*x]^3, x, 2, -(Cosh[a + b*x]/b) + Cosh[a + b*x]^3/(3*b)}
15 {Sinh[a + b*x]^5, x, 2, Cosh[a + b*x]/b - (2*Cosh[a + b*x]^3)/(3*b) + Cosh[a + b*x]^5/(5*b)}
58 {x*Sinh[a + b*x^2]^7, x, 3, -(Cosh[a + b*x^2]/(2*b)) + Cosh[a + b*x^2]^3/(2*b) - (3*Cosh[a + b*x^2]…
104 …Cosh[c + d*x])/d - (4*a*b^3*Cosh[c + d*x])/d + (4*a*b^3*Cosh[c + d*x]^3)/(3*d) + (3*a^2*b^2*Cosh[c…
105 …a^3*x - (3/2)*a*b^2*x + (3*a^2*b*Cosh[c + d*x])/d - (b^3*Cosh[c + d*x])/d + (b^3*Cosh[c + d*x]^3)/…
285 {1/Sqrt[1 + Sinh[x]^2], x, 3, (ArcTan[Sinh[x]]*Cosh[x])/Sqrt[Cosh[x]^2]}
288 {1/Sqrt[-1 - Sinh[x]^2], x, 3, (ArcTan[Sinh[x]]*Cosh[x])/Sqrt[-Cosh[x]^2]}
435 {x*Sinh[a + b*x^2]^3, x, 3, -(Cosh[a + b*x^2]/(2*b)) + Cosh[a + b*x^2]^3/(6*b)}
497 {Sinh[a + b*Log[c*x^n]]^5/x, x, 3, Cosh[a + b*Log[c*x^n]]/(b*n) - (2*Cosh[a + b*Log[c*x^n]]^3)/(3*b…
[all …]
/dports/math/reduce/Reduce-svn5758-src/contrib/rubi-red/rubi4/IntegrationProblems/MathematicaSyntaxFiles/HyperbolicFunctions/
H A DHyperbolicCosineFunctions.m99 {1/(a*Cosh[x]^2)^(1/2), x, 2, (ArcTan[Sinh[x]]*Cosh[x])/Sqrt[a*Cosh[x]^2]}
105 …Cosh[x]^3)^(5/2),x, 6, -((2*a^2*Sqrt[a*Cosh[x]^3]*(195*I*EllipticF[(I*x)/2, 2] - 195*Sqrt[Cosh[x]]…
108 {1/(a*Cosh[x]^3)^(1/2), x, 3, (2*Cosh[x]*(I*Sqrt[Cosh[x]]*EllipticE[(I*x)/2, 2] + Sinh[x]))/Sqrt[a*…
114 …Cosh[x]^4)^(5/2),x, 6, (63/256)*a^2*x*Sqrt[a*Cosh[x]^4]*Sech[x]^2 + (21/128)*a^2*Cosh[x]*Sqrt[a*Co…
115 …Cosh[x]^4)^(3/2),x, 4, (5/16)*a*x*Sqrt[a*Cosh[x]^4]*Sech[x]^2 + (5/24)*a*Cosh[x]*Sqrt[a*Cosh[x]^4]…
117 {1/(a*Cosh[x]^4)^(1/2), x, 2, (Cosh[x]*Sinh[x])/Sqrt[a*Cosh[x]^4]}
216 …Cosh[x])^(3/2), x, 9, (-(1280/9))*a*Sqrt[a + a*Cosh[x]] - 16*a*x^2*Sqrt[a + a*Cosh[x]] - (64/27)*a…
218 …Cosh[x])^(3/2), x, 4, (-(16/3))*a*Sqrt[a + a*Cosh[x]] - (8/9)*a*Cosh[x/2]^2*Sqrt[a + a*Cosh[x]] + …
538 {x/Cosh[x]^(3/2) + x*Sqrt[Cosh[x]], x, 2, -4*Sqrt[Cosh[x]] + (2*x*Sinh[x])/Sqrt[Cosh[x]]}
539 {x/Cosh[x]^(5/2) - x/(3*Sqrt[Cosh[x]]), x, 2, 4/(3*Sqrt[Cosh[x]]) + (2*x*Sinh[x])/(3*Cosh[x]^(3/2))}
[all …]
H A DTwoHyperbolicFunctions.m244 {Cosh[a + b*x]*Coth[c + b*x], x, 3, -((ArcCoth[Cosh[c + b*x]]*Cosh[a - c])/b) + Cosh[a + b*x]/b}
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]}
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]}
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]}
542 …Cosh[x], x, 8, ArcTanh[(1 - 4*Cosh[x])/Sqrt[5]]/(2*Sqrt[5]) - (1/5)*ArcTanh[Cosh[x]] - ArcTanh[(1 …
683 {Cosh[x]/(a + a*Cosh[x]), x, 2, x/a - Sinh[x]/(a*(1 + Cosh[x]))}
[all …]
H A DHyperbolicFunctions.m26 {Cosh[x]/(a*Cosh[x] + b*Sinh[x]), x, 1, (a*x)/(a^2 - b^2) - (b*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b…
74 {Cosh[x]/(a*Cosh[x] + b*Sinh[x])^3, x, 3, b/(2*(a^2 - b^2)*(a*Cosh[x] + b*Sinh[x])^2) + Sinh[x]/((a…
140 (* Integrands of the form Cosh[x]^m/(a+b*Cosh[x]*c*Cosh[x]^2) where m is an integer *)
146 {(a + b*Cosh[x])/(b^2 + 2*a*b*Cosh[x] + a^2*Cosh[x]^2), x, 2, Sinh[x]/(b + a*Cosh[x])}
160 (* {(-Cosh[x] + x*Sinh[x])/(x - Cosh[x])^2, x, 0, x/(x - Cosh[x])} *)
276 {(Cosh[x] - Sinh[x])/(Cosh[x] + Sinh[x]), x, 2, (-(1/2))*(Cosh[x] - Sinh[x])^2}
277 {(Cosh[x] - I*Sinh[x])/(Cosh[x] + I*Sinh[x]), x, 1, (-I)*Log[Cosh[x] + I*Sinh[x]]}
476 {Cosh[x]^3*(a + b*Cosh[x]^2)^3*Sinh[x], x, 4, -((a*(a + b*Cosh[x]^2)^4)/(40*b^2)) + (Cosh[x]^2*(a +…
488 {(Cosh[x] - Sinh[x])/(Cosh[x] + Sinh[x]), x, 2, (-(1/2))*(Cosh[x] - Sinh[x])^2}
491 {(1 + Sinh[x]^2)/(1 + Cosh[x] + Sinh[x]), x, -13, (3*x)/4 + Cosh[x]/2 - Cosh[x]^2/8 - Log[1 + Cosh[…
[all …]
/dports/math/reduce/Reduce-svn5758-src/contrib/rubi-red/mathematicatests/
H A DHyperbolicCosineFunctions.m99 {1/(a*Cosh[x]^2)^(1/2), x, 2, (ArcTan[Sinh[x]]*Cosh[x])/Sqrt[a*Cosh[x]^2]}
105 …Cosh[x]^3)^(5/2),x, 6, -((2*a^2*Sqrt[a*Cosh[x]^3]*(195*I*EllipticF[(I*x)/2, 2] - 195*Sqrt[Cosh[x]]…
108 {1/(a*Cosh[x]^3)^(1/2), x, 3, (2*Cosh[x]*(I*Sqrt[Cosh[x]]*EllipticE[(I*x)/2, 2] + Sinh[x]))/Sqrt[a*…
114 …Cosh[x]^4)^(5/2),x, 6, (63/256)*a^2*x*Sqrt[a*Cosh[x]^4]*Sech[x]^2 + (21/128)*a^2*Cosh[x]*Sqrt[a*Co…
115 …Cosh[x]^4)^(3/2),x, 4, (5/16)*a*x*Sqrt[a*Cosh[x]^4]*Sech[x]^2 + (5/24)*a*Cosh[x]*Sqrt[a*Cosh[x]^4]…
117 {1/(a*Cosh[x]^4)^(1/2), x, 2, (Cosh[x]*Sinh[x])/Sqrt[a*Cosh[x]^4]}
216 …Cosh[x])^(3/2), x, 9, (-(1280/9))*a*Sqrt[a + a*Cosh[x]] - 16*a*x^2*Sqrt[a + a*Cosh[x]] - (64/27)*a…
218 …Cosh[x])^(3/2), x, 4, (-(16/3))*a*Sqrt[a + a*Cosh[x]] - (8/9)*a*Cosh[x/2]^2*Sqrt[a + a*Cosh[x]] + …
538 {x/Cosh[x]^(3/2) + x*Sqrt[Cosh[x]], x, 2, -4*Sqrt[Cosh[x]] + (2*x*Sinh[x])/Sqrt[Cosh[x]]}
539 {x/Cosh[x]^(5/2) - x/(3*Sqrt[Cosh[x]]), x, 2, 4/(3*Sqrt[Cosh[x]]) + (2*x*Sinh[x])/(3*Cosh[x]^(3/2))}
[all …]
H A DTwoHyperbolicFunctions.m244 {Cosh[a + b*x]*Coth[c + b*x], x, 3, -((ArcCoth[Cosh[c + b*x]]*Cosh[a - c])/b) + Cosh[a + b*x]/b}
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]}
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]}
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]}
542 …Cosh[x], x, 8, ArcTanh[(1 - 4*Cosh[x])/Sqrt[5]]/(2*Sqrt[5]) - (1/5)*ArcTanh[Cosh[x]] - ArcTanh[(1 …
683 {Cosh[x]/(a + a*Cosh[x]), x, 2, x/a - Sinh[x]/(a*(1 + Cosh[x]))}
[all …]
H A DHyperbolicIntegralFunctions.m20 {x^2*SinhIntegral[b*x], x, 5, -((2*Cosh[b*x])/(3*b^3)) - (x^2*Cosh[b*x])/(3*b) + (2*x*Sinh[b*x])/(3…
43 …tegral[a + b*x], x, 11, -((2*Cosh[a + b*x])/(3*b^3)) - (a^2*Cosh[a + b*x])/(3*b^3) + (a*x*Cosh[a +…
44 …Cosh[a + b*x])/(2*b^4) + (a^3*Cosh[a + b*x])/(4*b^4) - (3*x*Cosh[a + b*x])/(2*b^3) - (a^2*x*Cosh[a…
70 …Cosh[b*x]*SinhIntegral[b*x]/x^3, x, 14, b/(4*x) - (b*Cosh[2*b*x])/(2*x) + (1/2)*b^2*Int[(Cosh[b*x]…
72 {Cosh[b*x]*SinhIntegral[b*x]/x, x, 0, Int[(Cosh[b*x]*SinhIntegral[b*x])/x, x]}
74 {x*Cosh[b*x]*SinhIntegral[b*x], x, 8, x/(2*b) - (Cosh[b*x]*Sinh[b*x])/(2*b^2) - ((Cosh[b*x] - b*x*S…
76 …Cosh[b*x]*SinhIntegral[b*x], x, 17, (4*x)/b^3 + x^3/(6*b) - (4*Cosh[b*x]*Sinh[b*x])/b^4 - (x^2*Cos…
94 {Cosh[a + b*x]*SinhIntegral[c + d*x]/x, x, 0, Int[(Cosh[a + b*x]*SinhIntegral[c + d*x])/x, x]}
155 …Cosh[b*x]*CoshIntegral[b*x]/x^3, x, 16, -(Cosh[b*x]^2/(4*x^2)) + (1/4)*b^2*CoshIntegral[b*x]^2 + b…
156 {Cosh[b*x]*CoshIntegral[b*x]/x^2, x, 7, -(1/(2*x)) - Cosh[2*b*x]/(2*x) - (Cosh[b*x]*CoshIntegral[b*…
[all …]
H A DHyperbolicFunctions.m26 {Cosh[x]/(a*Cosh[x] + b*Sinh[x]), x, 1, (a*x)/(a^2 - b^2) - (b*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b…
74 {Cosh[x]/(a*Cosh[x] + b*Sinh[x])^3, x, 3, b/(2*(a^2 - b^2)*(a*Cosh[x] + b*Sinh[x])^2) + Sinh[x]/((a…
140 (* Integrands of the form Cosh[x]^m/(a+b*Cosh[x]*c*Cosh[x]^2) where m is an integer *)
146 {(a + b*Cosh[x])/(b^2 + 2*a*b*Cosh[x] + a^2*Cosh[x]^2), x, 2, Sinh[x]/(b + a*Cosh[x])}
160 (* {(-Cosh[x] + x*Sinh[x])/(x - Cosh[x])^2, x, 0, x/(x - Cosh[x])} *)
276 {(Cosh[x] - Sinh[x])/(Cosh[x] + Sinh[x]), x, 2, (-(1/2))*(Cosh[x] - Sinh[x])^2}
277 {(Cosh[x] - I*Sinh[x])/(Cosh[x] + I*Sinh[x]), x, 1, (-I)*Log[Cosh[x] + I*Sinh[x]]}
476 {Cosh[x]^3*(a + b*Cosh[x]^2)^3*Sinh[x], x, 4, -((a*(a + b*Cosh[x]^2)^4)/(40*b^2)) + (Cosh[x]^2*(a +…
488 {(Cosh[x] - Sinh[x])/(Cosh[x] + Sinh[x]), x, 2, (-(1/2))*(Cosh[x] - Sinh[x])^2}
491 {(1 + Sinh[x]^2)/(1 + Cosh[x] + Sinh[x]), x, -13, (3*x)/4 + Cosh[x]/2 - Cosh[x]^2/8 - Log[1 + Cosh[…
[all …]
H A DHyperbolicSineFunctions.m11 {Sinh[a + b*x], x, 1, Cosh[a + b*x]/b}
13 {Sinh[a + b*x]^3, x, 2, -(Cosh[a + b*x]/b) + Cosh[a + b*x]^3/(3*b)}
15 {Sinh[a + b*x]^5, x, 2, Cosh[a + b*x]/b - (2*Cosh[a + b*x]^3)/(3*b) + Cosh[a + b*x]^5/(5*b)}
58 {x*Sinh[a + b*x^2]^7, x, 3, -(Cosh[a + b*x^2]/(2*b)) + Cosh[a + b*x^2]^3/(2*b) - (3*Cosh[a + b*x^2]…
104 …Cosh[c + d*x])/d - (4*a*b^3*Cosh[c + d*x])/d + (4*a*b^3*Cosh[c + d*x]^3)/(3*d) + (3*a^2*b^2*Cosh[c…
105 …a^3*x - (3/2)*a*b^2*x + (3*a^2*b*Cosh[c + d*x])/d - (b^3*Cosh[c + d*x])/d + (b^3*Cosh[c + d*x]^3)/…
285 {1/Sqrt[1 + Sinh[x]^2], x, 3, (ArcTan[Sinh[x]]*Cosh[x])/Sqrt[Cosh[x]^2]}
288 {1/Sqrt[-1 - Sinh[x]^2], x, 3, (ArcTan[Sinh[x]]*Cosh[x])/Sqrt[-Cosh[x]^2]}
435 {x*Sinh[a + b*x^2]^3, x, 3, -(Cosh[a + b*x^2]/(2*b)) + Cosh[a + b*x^2]^3/(6*b)}
497 {Sinh[a + b*Log[c*x^n]]^5/x, x, 3, Cosh[a + b*Log[c*x^n]]/(b*n) - (2*Cosh[a + b*Log[c*x^n]]^3)/(3*b…
[all …]
/dports/math/reduce/Reduce-svn5758-src/packages/rubi_red/rubi-rules/HyperbolicFunctions/
H A DRationalFunctionsOfHyperbolicSinesAndCosines.m101 (*(((b Cosh[c+d x]+a Sinh[c+d x]) (a Cosh[c+d x]+b Sinh[c+d x])^(n-1))/(d n))+*)
111 (b*Cosh[c+d*x]+a*Sinh[c+d*x])*(a*Cosh[c+d*x]+b*Sinh[c+d*x])^(n-1)/(d*n) +
158 (*Basis: (Cosh[z] Sinh[z])/(a Cosh[z]+b Sinh[z])=-((b Cosh[z])/(a^2-b^2))+(a Sinh[z])/(a^2-b^2)+(a …
193 (*Basis: Sinh[z]^2/(a Cosh[z]+b Sinh[z])=-((b Sinh[z])/(a^2-b^2))+(a Cosh[z])/(a^2-b^2)-a^2/((a^2-b…
227 (*Basis: Cosh[z]^2/(a Cosh[z]+b Sinh[z])=(a Cosh[z])/(a^2-b^2)-(b Sinh[z])/(a^2-b^2)-b^2/((a^2-b^2)…
244 Int[u_.*Cosh[c_.+d_.*x_]^n_/(a_.*Cosh[c_.+d_.*x_]+b_.*Sinh[c_.+d_.*x_]),x_Symbol] :=
247 Dist[b^2/(a^2-b^2),Int[u*Cosh[c+d*x]^(n-2)/(a*Cosh[c+d*x]+b*Sinh[c+d*x]),x]] /;
408 2*(c*Cosh[d+e*x]+b*Sinh[d+e*x])/(e*Sqrt[a+b*Cosh[d+e*x]+c*Sinh[d+e*x]]) /;
582 (c*Cosh[d+e*x]+b*Sinh[d+e*x])*(a+b*Cosh[d+e*x]+c*Sinh[d+e*x])^(n-1)/(e*n) +
611 (c*Cosh[d+e*x]+b*Sinh[d+e*x])*(a+b*Cosh[d+e*x]+c*Sinh[d+e*x])^(n-1)/(e*n) +
[all …]
H A DTwoHyperbolicFunctions.m31 Int[x_*(A_+B_.*Cosh[c_.+d_.*x_])/(a_+b_.*Cosh[c_.+d_.*x_])^2,x_Symbol] :=
431 (*Basis: Sinh[v] Sinh[w]=1/2 Cosh[v+w]-1/2 Cosh[v-w]*)
454 (*Basis: Cosh[v] Cosh[w]=1/2 Cosh[v-w]+1/2 Cosh[v+w]*)
458 Int[u_.*Cosh[v_]*Cosh[w_],x_Symbol] :=
473 (*Basis: Sinh[v] Tanh[w]=Cosh[v]-Cosh[v-w] Sech[w]*)
490 Int[u*Cosh[v]*Tanh[w]^(n-1),x] - Cosh[v-w]*Int[u*Sech[w]*Tanh[w]^(n-1),x] /;
495 (*Basis: Cosh[v] Coth[w]=Sinh[v]+Cosh[v-w] Csch[w]*)
575 (*Basis: Cosh[v]*Csch[w]=Cosh[v-w]*Coth[w]+Sinh[v-w]*)
615 (*Basis: Cosh[v] Sech[w]=Sinh[v-w] Tanh[w]+Cosh[v-w]*)
735 …ment]\[DoubleStruckCapitalZ], then Sinh[z]^m Cosh[z]^n=Cosh[z]^n (-1+Cosh[z]^2)^((m-1)/2) Derivati…
[all …]
H A DHyperbolicSineFunctions.m147 (*Basis: 1+Cosh[z]=2 Cosh[z/2]^2*)
207 (*Basis: 1+Cosh[z]=2 Cosh[z/2]^2*)
214 (*SqrtBox[\(a + a\ Cosh[z]\)], \(Cosh[*)
232 Dist[(2*a)^(n-1/2)*Sqrt[a+b*Cosh[c+d*x]]/Cosh[c/2+d*x/2],Int[x^m*Cosh[c/2+d*x/2]^(2*n),x]] /;
301 Dist[b/(a^2-b^2),Int[x*(b+a*Cosh[c+d*x])/(a+b*Cosh[c+d*x])^2,x]] /;
374 (*Basis: Cosh[z]^2=1/2 (1+Cosh[2 z])*)
418 (*Basis: Cosh[z]^2=1/2 (1+Cosh[2 z])*)
546 x*Cosh[a+b*x^n] -
604 (*Basis: Sinh[w+z]=Sinh[w] Cosh[z]+Cosh[w] Sinh[z]*)
628 Dist[Cosh[a],Int[Cosh[b*x^n]/x,x]] +
[all …]
/dports/math/reduce/Reduce-svn5758-src/packages/rubi_red/tests/SpecialFunctions/
H A DHyperbolicIntegralFunctions.m20 {x^2*SinhIntegral[b*x], x, 5, -((2*Cosh[b*x])/(3*b^3)) - (x^2*Cosh[b*x])/(3*b) + (2*x*Sinh[b*x])/(3…
43 …tegral[a + b*x], x, 11, -((2*Cosh[a + b*x])/(3*b^3)) - (a^2*Cosh[a + b*x])/(3*b^3) + (a*x*Cosh[a +…
44 …Cosh[a + b*x])/(2*b^4) + (a^3*Cosh[a + b*x])/(4*b^4) - (3*x*Cosh[a + b*x])/(2*b^3) - (a^2*x*Cosh[a…
70 …Cosh[b*x]*SinhIntegral[b*x]/x^3, x, 14, b/(4*x) - (b*Cosh[2*b*x])/(2*x) + (1/2)*b^2*Int[(Cosh[b*x]…
72 {Cosh[b*x]*SinhIntegral[b*x]/x, x, 0, Int[(Cosh[b*x]*SinhIntegral[b*x])/x, x]}
74 {x*Cosh[b*x]*SinhIntegral[b*x], x, 8, x/(2*b) - (Cosh[b*x]*Sinh[b*x])/(2*b^2) - ((Cosh[b*x] - b*x*S…
76 …Cosh[b*x]*SinhIntegral[b*x], x, 17, (4*x)/b^3 + x^3/(6*b) - (4*Cosh[b*x]*Sinh[b*x])/b^4 - (x^2*Cos…
94 {Cosh[a + b*x]*SinhIntegral[c + d*x]/x, x, 0, Int[(Cosh[a + b*x]*SinhIntegral[c + d*x])/x, x]}
155 …Cosh[b*x]*CoshIntegral[b*x]/x^3, x, 16, -(Cosh[b*x]^2/(4*x^2)) + (1/4)*b^2*CoshIntegral[b*x]^2 + b…
156 {Cosh[b*x]*CoshIntegral[b*x]/x^2, x, 7, -(1/(2*x)) - Cosh[2*b*x]/(2*x) - (Cosh[b*x]*CoshIntegral[b*…
[all …]
/dports/math/reduce/Reduce-svn5758-src/contrib/rubi-red/rubi4/IntegrationProblems/MathematicaSyntaxFiles/SpecialFunctions/
H A DHyperbolicIntegralFunctions.m20 {x^2*SinhIntegral[b*x], x, 5, -((2*Cosh[b*x])/(3*b^3)) - (x^2*Cosh[b*x])/(3*b) + (2*x*Sinh[b*x])/(3…
43 …tegral[a + b*x], x, 11, -((2*Cosh[a + b*x])/(3*b^3)) - (a^2*Cosh[a + b*x])/(3*b^3) + (a*x*Cosh[a +…
44 …Cosh[a + b*x])/(2*b^4) + (a^3*Cosh[a + b*x])/(4*b^4) - (3*x*Cosh[a + b*x])/(2*b^3) - (a^2*x*Cosh[a…
70 …Cosh[b*x]*SinhIntegral[b*x]/x^3, x, 14, b/(4*x) - (b*Cosh[2*b*x])/(2*x) + (1/2)*b^2*Int[(Cosh[b*x]…
72 {Cosh[b*x]*SinhIntegral[b*x]/x, x, 0, Int[(Cosh[b*x]*SinhIntegral[b*x])/x, x]}
74 {x*Cosh[b*x]*SinhIntegral[b*x], x, 8, x/(2*b) - (Cosh[b*x]*Sinh[b*x])/(2*b^2) - ((Cosh[b*x] - b*x*S…
76 …Cosh[b*x]*SinhIntegral[b*x], x, 17, (4*x)/b^3 + x^3/(6*b) - (4*Cosh[b*x]*Sinh[b*x])/b^4 - (x^2*Cos…
94 {Cosh[a + b*x]*SinhIntegral[c + d*x]/x, x, 0, Int[(Cosh[a + b*x]*SinhIntegral[c + d*x])/x, x]}
155 …Cosh[b*x]*CoshIntegral[b*x]/x^3, x, 16, -(Cosh[b*x]^2/(4*x^2)) + (1/4)*b^2*CoshIntegral[b*x]^2 + b…
156 {Cosh[b*x]*CoshIntegral[b*x]/x^2, x, 7, -(1/(2*x)) - Cosh[2*b*x]/(2*x) - (Cosh[b*x]*CoshIntegral[b*…
[all …]
/dports/math/reduce/Reduce-svn5758-src/packages/rubi_red/tests4/SpecialFunctions/
H A DHyperbolicIntegralFunctions.m18 {x^2*SinhIntegral[b*x], x, 5, -((2*Cosh[b*x])/(3*b^3)) - (x^2*Cosh[b*x])/(3*b) + (2*x*Sinh[b*x])/(3…
43 …Cosh[a + b*x])/(2*b^4) + (a^3*Cosh[a + b*x])/(4*b^4) - (3*x*Cosh[a + b*x])/(2*b^3) - (a^2*x*Cosh[a…
44 …tegral[a + b*x], x, 10, -((2*Cosh[a + b*x])/(3*b^3)) - (a^2*Cosh[a + b*x])/(3*b^3) + (a*x*Cosh[a +…
76 …Cosh[b*x]*SinhIntegral[b*x]/x^3, x, 12, -((b*Cosh[2*b*x])/(4*x)) + (1/2)*b^2*Int[(Cosh[b*x]*SinhIn…
78 {Cosh[b*x]*SinhIntegral[b*x]/x, x, 0, Int[(Cosh[b*x]*SinhIntegral[b*x])/x, x]}
80 {x*Cosh[b*x]*SinhIntegral[b*x], x, 8, x/(2*b) - (Cosh[b*x]*Sinh[b*x])/(2*b^2) - (Cosh[b*x]*SinhInte…
82 …Cosh[b*x]*SinhIntegral[b*x], x, 17, (4*x)/b^3 + x^3/(6*b) - (4*Cosh[b*x]*Sinh[b*x])/b^4 - (x^2*Cos…
100 {Cosh[a + b*x]*SinhIntegral[c + d*x]/x, x, 0, Int[(Cosh[a + b*x]*SinhIntegral[c + d*x])/x, x]}
167 …Cosh[b*x]*CoshIntegral[b*x]/x^3, x, 14, -(Cosh[b*x]^2/(4*x^2)) - (Cosh[b*x]*CoshIntegral[b*x])/(2*…
168 {Cosh[b*x]*CoshIntegral[b*x]/x^2, x, 5, -(Cosh[b*x]^2/x) - (Cosh[b*x]*CoshIntegral[b*x])/x + b*Int[…
[all …]
/dports/math/reduce/Reduce-svn5758-src/packages/rubi_red/tests4/MiscellaneousFunctions/
H A DzTrigProblems.m44 …Cosh[x])^(5/2), x, 8, -((2*I*(23*a^2 + 9*b^2)*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)…
45 …Cosh[x])^(3/2), x, 5, -((8*I*a*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(3*Sqrt[(a +…
47 {1/(a + b*Cosh[x])^(1/2), x, 2, -((2*I*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[(I*x)/2, (2*b)/(a + …
48 …Cosh[x])^(3/2), x, 5, -((2*I*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/((a^2 - b^2)*S…
49 …Cosh[x])^(5/2), x, 7, -((8*I*a*Sqrt[a + b*Cosh[x]]*EllipticE[(I*x)/2, (2*b)/(a + b)])/(3*(a^2 - b^…
55 {1/(a + a*Cosh[c + d*x])^(1/2), x, 1, (2*ArcTan[Sinh[(1/2)*(c + d*x)]]*Cosh[(1/2)*(c + d*x)])/(d*Sq…
56 …a*Cosh[c + d*x])^(3/2), x, 2, (ArcTan[Sinh[(1/2)*(c + d*x)]]*Cosh[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a …
57 …Cosh[c + d*x])^(5/2), x, 3, (3*ArcTan[Sinh[(1/2)*(c + d*x)]]*Cosh[(1/2)*(c + d*x)])/(16*a^2*d*Sqrt…
62 {1/(a - a*Cosh[c + d*x])^(1/2), x, 1, -((2*ArcTanh[Cosh[(1/2)*(c + d*x)]]*Sinh[(1/2)*(c + d*x)])/(d…
63 …Cosh[c + d*x])^(3/2), x, 2, -((ArcTanh[Cosh[(1/2)*(c + d*x)]]*Sinh[(1/2)*(c + d*x)])/(2*a*d*Sqrt[a…
[all …]
H A DTestGradnew.m860 {Cosh[a x+b] Cosh[c x+d],x},
863 {Cosh[a x+b] Cosh[a x+d],x},
868 {Cosh[a x] Cosh[b x] Cosh[c x],x},
899 {Cosh[2 x] Cosh[x]^(-1),x},
900 {Cosh[2 x] Cosh[x]^(-2),x},
901 {Cosh[2 x] Cosh[x]^(-3),x},
911 {Cosh[3 x] Cosh[x]^(-1),x},
912 {Cosh[3 x] Cosh[x]^(-2),x},
913 {Cosh[3 x] Cosh[x]^(-3),x},
957 {(A+Cosh[x]^2+2B Sinh[x] Cosh[x]+C Sinh[x]^2)(a Cosh[x]^2+2b Sinh[x] Cosh[x]+
[all …]

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