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 at

http://www.apmaths.uwo.ca/~arich/TimofeevChapter9TestResults.pdf

Aloha,
Albert