Date: Nov 7, 2013 12:31 AM
Author: did
Subject: Re: The A. F. Timofeev symbolic integration test suite

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?
>
> >
>
> > Albert
>
> >
>
>
>
> 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);
>
>
>
> --Nasser


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));

Did