Date: Nov 15, 2013 3:53 PM
Author: Nasser Abbasi
Subject: Re: The A. F. Timofeev symbolic integration test suite

On 11/15/2013 10:14 AM, clicliclic@freenet.de wrote:

>I am a bit curious what Mathematica and Maple
> return for integral 69 from Timofeev's Chapter 8.
>
> Martin.
>

Hi Martin;
Here it is, M V9.01 windows 7, and Maple 17.02, windows 7

--------------------
integrand = (Tanh[x] - Cosh[2 x]) Cosh[x] /
(( Sinh[2 x] + Sinh[x]^2) Sqrt[Sinh[2 x]])

optimalAnti =
Sqrt[2] ArcTan[Sech[x] Sqrt[Cosh[x] Sinh[x]]] +
1/6 ArcTan[Sinh[x]/Sqrt[Sinh[2 x]]] -
1/2 Sqrt[2] ArcTanh[Sech[x] Sqrt[Cosh[x] Sinh[x]]] +
Cosh[x]/Sqrt[Sinh[2 x]] (*copied from ref. Albert Rich, page 18*)

Integrate[integrand, x]

-((Coth[x] Sqrt[Sinh[2 x]] (-Cosh[2 x] + Tanh[x]))/(
Cosh[x] + Cosh[3 x] -
2 Sinh[x])) + (Cosh[
x] (-(6 (-1)^(1/4) Sqrt[
1 + Coth[x/
2]^2] (EllipticF[
I ArcSinh[(-1)^(1/4)/Sqrt[Tanh[x/2]]], -1] -
EllipticPi[-(-1)^(1/6),
I ArcSinh[(-1)^(1/4)/Sqrt[Tanh[x/2]]], -1] -
EllipticPi[-(-1)^(5/6),
I ArcSinh[(-1)^(1/4)/Sqrt[Tanh[x/2]]], -1]) Sqrt[
Sinh[2 x]] Sqrt[Tanh[x/2]] Sqrt[
Tanh[x/2] + Tanh[x/2]^3])/((1 + Cosh[x]) Sqrt[
Sinh[2 x]/(1 + Cosh[x])^2] (1 + Tanh[x/2]^2)) + (16 (-1)^(
5/12) ((3 - 3 I Sqrt[3]) EllipticPi[-I,
I ArcSinh[(-1)^(1/4) Sqrt[Tanh[x/2]]], -1] +
2 ((-1 + (-1)^(1/3)) EllipticPi[I,
ArcSin[(-1)^(3/4) Sqrt[Tanh[x/2]]], -1] +
1/2 I (I + Sqrt[3]) EllipticPi[-(-1)^(1/6),
I ArcSinh[(-1)^(1/4) Sqrt[
Tanh[x/2]]], -1] + (-1 + (-1)^(
1/3)) EllipticPi[-(-1)^(5/6),
I ArcSinh[(-1)^(1/4) Sqrt[Tanh[x/2]]], -1])) Sinh[
2 x]^(3/2) Sqrt[
Tanh[x/2] +
Tanh[x/2]^3])/(3 (-I + Sqrt[3]) (1 + Cosh[x])^3 (Sinh[
2 x]/(1 + Cosh[x])^2)^(3/2) Sqrt[Tanh[x/2]] Sqrt[
1 + Tanh[x/2]^2])) (-Cosh[2 x] + Tanh[x]))/(2 (Cosh[x] +
Cosh[3 x] - 2 Sinh[x]))
------------------------

Maple:
restart;
integrand:=(tanh(x) - cosh(2*x))*cosh(x)/(( sinh(2*x)+sinh(x)^2)*sqrt(sinh(2*x)));
int(integrand,x);

-((3/4)*I)*(-I*(tanh((1/2)*x)+I))^(1/2)*2^(1/2)*(I*(tanh((1/2)*x)-I))^(1/2)
*(I*tanh((1/2)*x))^(1/2)*EllipticF((-I*(tanh((1/2)*x)+I))^(1/2), (1/2)
*2^(1/2))/(tanh((1/2)*x)^3+tanh((1/2)*x))^(1/2)+(-1+I)*(-I*(tanh((1/2)*x)+I))^(1/2)
*2^(1/2)*(I*(tanh((1/2)*x)-I))^(1/2)*(I*tanh((1/2)*x))^(1/2)*
EllipticPi((-I*(tanh((1/2)*x)+I))^(1/2), 1/2-(1/2)*I,
(1/2)*2^(1/2))/(tanh((1/2)*x)^3+tanh((1/2)*x))^(1/2)+(-1/3-(1/3)*I)
*(-I*(tanh((1/2)*x)+I))^(1/2)*2^(1/2)*(I*(tanh((1/2)*x)-I))^(1/2)
*(I*tanh((1/2)*x))^(1/2)*EllipticPi((-I*(tanh((1/2)*x)+I))^(1/2),
1/2+(1/2)*I, (1/2)*2^(1/2))/(tanh((1/2)*x)^3+tanh((1/2)*x))^(1/2)+(1/12)*
(sum(-I*_alpha*(-I*(tanh((1/2)*x)+I))^(1/2)*2^(1/2)*(I*(tanh((1/2)*x)-I))^(1/2)*
(I*tanh((1/2)*x))^(1/2)*(I*_alpha+1+I)*EllipticPi((-I*(tanh((1/2)*x)+I))^(1/2),
_alpha+1-I, (1/2)*2^(1/2))/(tanh((1/2)*x)^3+tanh((1/2)*x))^(1/2),
_alpha = RootOf(_Z^2+_Z+1)))+(1/2)*(tanh((1/2)*x)^2+1)/(tanh((1/2)*x)*
(tanh((1/2)*x)^2+1))^(1/2)

Rubi 4.2
-------------
ShowSteps := False;
Int[integrand, x]

(Sqrt[I*Sinh[2*x]]*Int[(Cosh[x]*Cosh[2*x])/((-Sinh[x]^2 - Sinh[2*x])*
Sqrt[I*Sinh[2*x]]), x])/Sqrt[Sinh[2*x]] +
I*Int[(I*Sinh[x])/((-Sinh[x]^2 - Sinh[2*x])*Sqrt[Sinh[2*x]]), x]