How to prove $\int_{-\infty}^\infty \frac{dx}{(e^x -x)^2 +\pi^2}=\frac{1}{1+W(1)}$?

Question

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 $W(z)$.