```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.
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