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. It
amounts to negating the first term of the antiderivative and changing
the exponent 2 in the integrand to 3.

Regarding example #69 from Chapter VIII, I propose to change the 3rd
term of the antiderivative from SQRT(2)*ATAN(TANH(x)) to SQRT(2)*
ATAN(SQRT(TANH(x))); this makes the antiderivative valid where the
radicands are positive. A compact antiderivative for the entire complex
plane 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.