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Topic: first anniversary of the IITS
Replies: 21   Last Post: Jan 10, 2015 4:01 PM

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 clicliclic@freenet.de Posts: 1,245 Registered: 4/26/08
Re: first anniversary of the IITS
Posted: Jan 9, 2015 1:06 PM

Albert Rich schrieb:
>
> On Friday, December 19, 2014 9:13:37 AM UTC-10, clicl...@freenet.de wrote:
>

> >> [...] A systematic check against the evaluations in Timofeev's book,
> >> however, remains to be made.

> >
> > This referred to the trigonometric integration examples in Chapter 5
> > of Timofeev's 1948 book, as provided in digitized form at
> > <http://www.apmaths.uwo.ca/~arich/>. Such a check has now been made,
> > with the result that seven integrands need to be corrected because they
> > were typeset incorrectly or transcribed incorrectly or both.

>
> [...]
>
> I revised the 7 problems in the Timofeev test suite you flagged for
> correction pretty much as you suggested.
>

> > While the next two integrals and their evaluations are correct, the
> > evaluations can be improved in some respects.
> >
> > 1. Either of the following evaluations of Example 19.m (p. 220) is
> > shorter than the version currently shown, the two differ in being
> > discontinuous where SIN(x)^2 = 1 and where COS(x)^2 = 1, respectively:
> >
> > INT(SIN(x)^(2*m)*COS(x)^(2*m),x)=SIN(x)^(2*m+1)*COS(x)^(2*m+1)/(~
> > 2*m+1)*F21(1,1+2*m,3/2+m,SIN(x)^2)=-SIN(x)^(2*m+1)*COS(x)^(2*m+1~
> > )/(2*m+1)*F21(1,1+2*m,3/2+m,COS(x)^2)

>
> Although shorter, and probably correct, I am unable to get Mathematica
> to verify these antiderivatives.
>

> > 2. The evaluation of Example 45 (p. 254) can be continuitized in the
> > customary manner:
> >
> > INT(1/(4-3*COS(x)^2+5*SIN(x)^2),x)=x/3+1/3*ATAN(2*SIN(x)*COS(x)/(2*SIN(x)^2+1))

>
> I tweaked Rubi to give the continuous antiderivative you suggested.
>

> > The remaining 29 items involve minor variations from the versions
> > currently shown, such as a modified term order in the integrand and/or
> > different trigonometric functions in the evaluation; they are offered
> > here for inspection without much further comment.

>
> [...]
>
> The only suggestion I incorporated was to example 54 (p. 260).
> Although some of your suggestions resulted in slightly more compact
> results, they were not really simpler.
>

> > I expect repercussions of the above on Rubi to be small.
>
> Yes the repercussions were small, but resulted in a few nice
> improvements. The revised Timofeev test suite is now available at
> http://www.apmaths.uwo.ca/~arich.
>

Looking at Example 114 again, I find that both the integrand involving
1 + (1-8*TAN(x)^2)^(1/3) and that involving 1 - (1-8*TAN(x)^2)^(1/3)