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Apparently Maxima includes this, and we didn't even know it! See for instance here and the official docs.

(%i1) residue (s/(s**2+a**2), s, a*%i);
(%o1)                           1/2

In Sage, that would be doable.

sage: f(x)=1/x
sage: f.maxima_methods().residue(x,0)

Well, we should wrap this!

This does not solve the problem of finding poles. Maple apparently has this, though perhaps not for such an example as the original one.