WolframAlpha balks at that integral (as does my local copy of Mathematica, as you'd expect). Are you sure the inverse FT has a closed-form solution? It sure doesn't look like it.

Also, I should point out that the two integrals you claim are equal are only equal if the first one is real. If you take the real part of the first integral, you get the second one.

[EDIT]: I think you also inadvertently changed the limits of integration in the second integral without multiplying by two to take advantage of the symmetry.