Recently I've become aware of the result$$\int_{-\infty}^\infty \frac{dx}{(e^x -x)^2 +\pi^2}=\frac{1}{1+W(1)}$$ where $W(z)$ is the Lambert W Function. However, I do not know of any methods to prove this result. If anyone could help me out, I would be very grateful. I also wonder if there are similar integrals that give other values of $W(z)$.
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It seem to me you should consider $x$ to be complex at first, and consider line integral. (Then you can use representation $e^{x}=cos(\frac{x}{i})+i sin(\frac{x}{i})=cos(ix)-i sin(ix)$.) I'm not sure this will help. – kolobokish Nov 21 '18 at 18:03