```Date: Sep 25, 2013 5:15 PM
Author: Albert D. Rich
Subject: Re: Rubi 4.1 and the Timofeev test suite

On Wednesday, September 25, 2013 9:06:12 AM UTC-10, clicl...@freenet.de wrote:> You have been too fast for Peter! But it looks somebody with a knack for> mathematical puzzle solving is needed now. I am still in the process of> digesting the Chapter 9 examples. Here are some more suggestions:> > In example #12 replace #i*ATAN(x + #i*SQRT(1-x^2)) by ATANH(#i*x -> SQRT(1-x^2)), and similarly in #19 replace #i*ATAN(x - #i*SQRT(1-x^2))> by ATANH(#i*x + SQRT(1-x^2)). This saves one imaginary unit, and also> makes the ATANH argument a common subexpression, as also found in> examples #37 and #43.> > In example #49 convert ATANH to ATAN and collapse the piecewise> constants. This gives the more natural and simpler evaluation:> > INT(ASIN(SQRT((x - a)/(x + a))), x) =>  - 2*a*(SQRT((x - a)/(x + a))/SQRT(2*a/(x + a)))>  + x*ASIN(SQRT((x - a)/(x + a)))>  + a*ATAN(SQRT((x - a)/(x + a))/SQRT(2*a/(x + a)))> > The ATANH argument of the old evaluation is complex when the radicand> (x - a)/(x + a) is positive; such a result doesn't deserve full points.> > In example #55 replace SQRT(1 - x^2) by SQRT(1 - x)*SQRT(1 + x) and> simplify, which results in:> > INT(ASIN(x)/(1 - x)^(5/2), x) = - SQRT(1 + x)/(3*(1 - x))>  + 2*ASIN(x)/(3*(1 - x)^(3/2)) - SQRT(2)/6*ATANH(SQRT(1 + x)/SQRT(2))> > And in example #56 move part of the piecewise prefactor into the ATANH> argument and simplify (x + 1)*SQRT(x - 1)/SQRT(x^2 - 1) to SQRT(x^2 -> 1)/SQRT(x - 1) throughout the evaluation:> > INT((x - 1)^(5/2)*ACSC(x), x) = 2/7*(x - 1)^(7/2)*ACSC(x)>  + 4/105*(x/SQRT(x^2))*(83 - 19*x + 3*x^2)*(SQRT(x^2 - 1)/SQRT(x - 1))>  + 4/7*(x/SQRT(x^2))*ATANH(SQRT(x^2 - 1)/SQRT(x - 1))Hello Martin,Your perfectionist credentials remain impeccable!  Chapter 9 of the Timofeev test suite, revised as you suggested, is now available athttp://www.apmaths.uwo.ca/~arich/TimofeevChapter9TestResults.pdfAloha,Albert
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