I recently helped someone with a contour integration problem, you may find my calculation at the following link. It occurred to me to test this integral with Maple and Mathematica.
produces an unusable output from Mathematica. Maple is a little better, but does not recognize the simple end result.
seems to hang Maple, while Mathematica gives up after a long computation.
Finally, the integral
gets a correct answer from Maple while Mathematica produces something more cryptic.
I hope these observations are useful and that perhaps the case where Maple hangs can be fixed. As I said in an earlier message there is nothing wrong with adding a (sophisticated) table lookup to Maple's set of internal integration routines to catch cases like these.