Hello,
i have seen a web site where symbolic integration using reduction to
standard forms via pattern maching is presented in a very favourable light
compared to using systematic algorithms like Risch algorithm. See
http://www.apmaths.uwo.ca/~arich/
This could be of interest for maxima.
For example the "algebraic function highlights" here:
http://www.apmaths.uwo.ca/~arich/TestResultHighlights/AlgebraicFunctionHighlights.pdf#pagemode=bookmarks
show very interesting examples, notably the last one (which my version of
maxima is not able to integrate at all).
--
Michel Talon