Albert Rich schrieb: > > Chapter 5 of Timofeev's book discusses the integration of expressions > involving trig functions. I transcribed the 121 problems in the > chapter and attempted to find optimal antiderivatives for them. They > are available as a pdf file at > > http://www.apmaths.uwo.ca/~arich/TimofeevChapter5IntegrationProblems.pdf > > The problems along with all the problems in the odd-numbered chapters > of Timofeev book are also available in machine readable form at > > http://www.apmaths.uwo.ca/~arich/ > > expressed in Axiom, Maple, Mathematica and Maxima syntax. > > Timofeev book contains numerous typographical errors, and he does not > seem to distinguish between (z^m)^(1/n) and z^(m/n). Also some of the > problems require huge, non-elementary antiderivatives which were > surely not his intent. If based on Timofeev proposed antiderivative > you can determine his true intentions or you know of a more optimal > antiderivative, please let me know so I can revise the test suite. >
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:
I saw Peter on his website giving polylogarithm antiderivatives for examples #8 and #9 from Chapter 7, but his expressions (presumably from Sage/Maxima) are not valid on the entire complex plane. In addition to being shorter, the following however are: