If I am to make a suggestion (I haven't tried it out), you should try to utilize the residue theorem from complex analysis.
Given a length "L" along the real axis, you can make a closed contour by say, considering a half-circle with radius L/2, with the origin at the real axis at L/2.
Then let "L" go to infinity, and use the residue theorem.
Hopefully, the integral along the other part of the contour is easy to evaluate.
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