I've been struggling a while now trying to find an alternate form for $$G=\int_0^{\infty}{x^{1/x-x}dx}$$ in function of known constants, (like $e$,$\pi$,etc.)
Having looked at the problem, I don't think I have the knowledge to solve this problem.
Worth noting:
$$ G \approx 1.32073040087 $$
$$ G = \int_0^1{x^{1/x-x}d\left(x-\frac{1}{x}\right)},\tiny \left(\forall f(x)=f(1/x):\int_0^{\infty}{f(x)dx}=\int_0^1{f(x)d\left(x-\frac{1}{x}\right)}\right ) $$
$$ G = \int_{-\infty}^{\infty}{e^{u(1-2\sinh(u))}du} , \small x=e^u $$