$$ \int_{-\infty}^{\infty}d\omega\frac{e^{i \omega t}e^{i \omega R}}{{((\omega-\Omega-iA^2(1+e^{i \omega R}))((\omega-\Omega-iA^2(1-e^{i \omega R}))}} $$
All constants are positve real numbers.
During my work i came across this very interesting integral. My plan was to specify the poles of the integrand which are given in terms of Lambert's W-Function, close the integration contour in the appropriate half-plane and use residue theorem to get the result.
The problem now is that we obtain a very nasty sum in the end which i think has to be evaluated numerically. Therefore i wanted to ask the community may have a better idea how to tackle this problem.