I'm considering multiply both top and bottom by , then let which I believe converts to a familiar function (but you may also have to integrate by parts).
You can also use complex line integration. See here for a very similar sort of integral.
There is only a minor detail: the integral proposed by parkhid is indefinite and its solution is a family of functions... the integral of Your example is definite and its solution [if it exists...] is a real [or complex] number...