There are two interesting threads about definite integration on the usenet group comp.soft-sys.math.maple posted in the past few days.
The first thread is more about the wrong result coming out of Maple's int command when computing via its `MeijerG` method. Motivating examples there include
int(cos(t)/(1+t^k), t=-infinity..infinity); # for k=10,12,14,...
The poster demonstrates using residues to obtain a terse result.
In the second thread the poster examines the ideas that the solutions involving RootOfs (as returned by the `ftoc` method) may suffer from numerical difficulties during floating-point evaluation and are unnecessarily complicated when re-expressed in terms of more standard functions (ie. allvalues, then simplification...).
For those interested, an existing `contour` method exists within `int`. It is not very strong however. One can examine its source using commands such as,
See also Methods of contour integration on wikipedia.