Integral of $\int_0^\infty\frac{1}{e^{x}-x} dx$ I am curious as to whether a closed form exists for the following integral:  
$$\int_0^\infty\frac{1}{e^{x}-x} dx$$
I have tried a few elemetary methods on it, but I believe this integral (if it has a solution) can only be solved through complex analysis which I have no working knowledge of. Wolfram does not return any closed form. I'm not sure if it is much use, but this integral appears to be equivalent to $$\int_0^\infty\frac{x}{e^{x}-x} dx$$ Maybe this integral is related to the gamma function like $\int_0^\infty\frac{x^n}{e^{x}-1} dx$?

Here is an approach.
$$ \int_0^\infty\frac{1}{e^{x}-x} dx = \sum_{k=0}^{\infty} \int_{0}^{\infty}x^k e^{-kx-x}dx = \sum_{k=0}^{\infty}\frac{k!}{(k+1)^{k+1}}\sim 1.359098277. $$
See related techniques. 
