Date: Sep 24, 2013 1:21 AM
Author: daly@axiom-developer.org
Subject: Re: Rubi 4.1 and the Timofeev test suite

On Sunday, September 22, 2013 3:02:31 AM UTC-4, da...@axiom-developer.org wrote:
> On Saturday, September 21, 2013 11:38:33 AM UTC-7, clicl...@freenet.de wrote:
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> > Albert Rich schrieb:
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> > > Chapter 9 of Timofeev's book discusses the integration of expressions
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> > > involving inverse trig functions. I transcribed the problems in the
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> > > chapter and tested various systems on them. The test results,
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> > > problems and answers are available as a pdf file at
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> > > http://www.apmaths.uwo.ca/~arich/TimofeevChapter9TestResults.pdf
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> > > Note the dramatic improvement of the recently released version 4.2 of
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> > > Rubi over previous versions.
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> > > The 61 problems from Chapter 9 are also available in machine readable
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> > > form at
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> > > http://www.apmaths.uwo.ca/~arich/
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> > > expressed in Axiom, Maple, Mathematica and Maxima syntax. The
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> > > antiderivatives in these test file are only a first approximation at
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> > > being optimal. Please let me know if you find and would like to
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> > > contribute significantly better ones.
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> > This should help CAS programmers help improve their integrators :).
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> > Strange that Derive 6.10 can handle example #53 while Rubi 4.2 cannot
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> > :(. And Rubi 4.0 even returned incorrect results for examples #30 and
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> > #35.
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> > I suggest to replace ArcSin[1/x] by ArcCsc[x] throughout the file, and
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> > ArcCos[1/x] by ArcSec[x]. Sqrt[-1+x^2]/Sqrt[1-1/x^2] appears repeatedly
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> > and can be simplified to Sqrt[x^2].
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> > I will insert the Chapter 9 data into our performance table once the
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> > FriCAS results are also known.
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> > And now everybody is waiting for Peter Luschny to finish Chapter 7 (120
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> > trigonometric integrands). But who is going to transcribe the examples
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> > from Chapter 8 (109 exponential and hyperbolic integrands)?
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> > Martin.
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> On the subject of improving integrators I'm rather partial to
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> algorithmic solutions, which as Rich has shown, are lagging
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> behind his enormous effort at pattern matching.
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> One interesting study would be to collect all of the integrals
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> that involve a single function in the answer such as a sqrt- or
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> atan-only solution (along with the usual +-*/ terms). Risch
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> developed a theory that used only log terms in the answer. Is
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> there an equivalent for answers with only sqrt terms? atan terms?
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> I tend to refer to these as mono-result integrals.
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> If the original functions of mono-result integrals are graphed
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> is there some common observations to be made? Is it possible to
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> predict when a single-function answer will result?
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> Another observation is that b^2+4ac kind of terms occur a lot.
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> Is there any theory here? Does anybody have a grad student looking
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> for a math phd? I'd find it interesting to look at this question
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> but I'll be generating the CATS files from Rich's work for the
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> next few months. Ideally there would be a Risch-like algorithm
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> for mono-result integrals.
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> Tim Daly


Richard Fateman was kind enough to point out that this is
already covered by Guy Cherry's thesis. Apparently I'm
behind on my reading :-)