It is possible to verify or show numerically that
$\displaystyle \int^{\infty}_{-\infty}W(e^{x-e^{x}})dx=\frac{\pi^2}{12}$
$W(x)$ is the Lambert W function.
This integral does not seem to have a solution in terms of elementary or standard functions.
Is it possible to prove or solve it analytically, or to find a symbolic solution?
The substitution $y=e^x$ obtains your integral as $\int_0^\infty y^{-1}W(ye^{-y})dy$. Writing$$W(z)=\sum_{n\ge 1} w_n z^n$ with $w_n:=(-1)^{n-1}\frac{n^{n-1}}{n!},$$we can rewrite this integral as$$\sum_n\dfrac{w_n\Gamma(n)}{n^n}=\sum_n (-1)^{n-1} n^{-2}=\dfrac{1}{2}\zeta(2).$$