```Date: Nov 7, 2013 7:15 AM
Author: clicliclic@freenet.de
Subject: Re: The A. F. Timofeev symbolic integration test suite

did schrieb:> > On Thursday, 7 November 2013 03:33:19 UTC+1, Nasser M. Abbasi  wrote:> >> > On 11/6/2013 6:16 PM, Albert Rich wrote:> > >> > > I am currently transcribing the 109 integration examples in Chapter 8 of Timofeev's book.> > > However I am unable to figure out what Timofeev intended for examples #66 and #69 on> > > page 366.  The integrands for #66 and #69 appear to be> > >> > > (cosh(x)^2 - sinh(x)^3) / (cosh[x)^3 + sinh(x)^3)> > >> > > and> > >> > > (tanh(x)-cosh(2x))*cosh(x)/((sinh(2x)+sinh(x)^2)*sqrt(sinh(2*x)))> > >> > > respectively, but the antiderivatives shown for them are not valid.> > > Can anybody out there help me out?> > >> >> > Yes, there is an error/typo somewhere. difference between> > the derivative of the anti derivative and the integrand is  not> > zero and not even linear constant difference. This is from> > the 1948 edition. May be there is a newer edition than this> > to check?> >> > #66> > restart;> > integrand:=(cosh(x)^2 - sinh(x)^3) / (cosh(x)^3 + sinh(x)^3):> > anti:= 1/(3*(1+tanh(x)))+ 4/(3*sqrt(3))*arctan( (2*tanh(x)-1)/sqrt(3)):> > check:=simplify(integrand-diff(anti,x));> >> >   (1/3)*(3*cosh(x)^3-6*cosh(x)^2*sinh(x)+3*cosh(x)^2-4*cosh(x)> >   +2*sinh(x)) /(cosh(x)^3+cosh(x)^2*sinh(x)-sinh(x))> >> > plot(%,x=-Pi..Pi);> >> >> > #69> > restart;> > integrand:=(tanh(x)-cosh(2*x))*cosh(x)/((sinh(2*x)+sinh(x)^2)*sqrt(sinh(2*x))):> > anti:= 1/sqrt(2*tanh(x))+sqrt(2)/6*log( (1-sqrt(tanh(x)))/(1+sqrt(tanh(x))) )> > +sqrt(2)*arctan(tanh(x))+1/6 * arctan(sqrt(tanh(x)/2)):> > check:=simplify(integrand-diff(anti,x)); %too large to show> > plot(%,x=-Pi..Pi);> >> > A possible fix for #66 is:> > restart:> integrand:=(cosh(x)^3 - sinh(x)^3) / (cosh(x)^3 + sinh(x)^3);> anti:= -1/(3*(1+tanh(x))) + 4/(3*sqrt(3))*arctan( (2*tanh(x)-1)/sqrt(3));> check:=simplify(integrand-diff(anti,x));> I agree with this fix of Timofeev's example #66 from Chapter VIII. Itamounts to negating the first term of the antiderivative and changingthe exponent 2 in the integrand to 3.Regarding example #69 from Chapter VIII, I propose to change the 3rdterm of the antiderivative from SQRT(2)*ATAN(TANH(x)) to SQRT(2)*ATAN(SQRT(TANH(x))); this makes the antiderivative valid where theradicands are positive. A compact antiderivative for the entire complexplane is:COSH(x)/SQRT(SINH(2*x)) - SQRT(2)/3*ATANH(SQRT(SINH(2*x))/(SQRT(2)*COSH(x))) + SQRT(2)*ATAN(SQRT(SINH(2*x))/(SQRT(2)*COSH(x))) + 1/6*ATAN(SINH(x)/SQRT(SINH(2*x)))Martin.
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