Hm, well the Cosine transform looks a bit crazy because of the
strange 3-valued signum, but it actually says: roots in the
upper plane, the rest involves roots of unity and some exp.

For Numerics I gave some fast and accurate solution.

Concerning Maple's possible bug versus Mathematica:

That is difficult to judge. First we do not *know* whether MMA
uses some symbolics inbetween (Maple does not). And second it
is for sure, that Maple uses only 10 Digits (in Your setting),
while MMA uses much more.

Anyway: I also would expect Maple to get it (may be with some
higher Digits, issuing warnings). And it does not know the
generic solution (from Residue calculus). But you have 2 ways
which work to get the numerical value.

And as already said: do *not* use evalf(lOWERCASE int(...)).