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.