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