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
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.