Date: Nov 16, 2013 7:47 AM
Author: clicliclic@freenet.de
Subject: Re: The A. F. Timofeev symbolic integration test suite


"Nasser M. Abbasi" schrieb:
>
> 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.
> >

>
> 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]


Good grief!

Martin.