Q:

Evaluate: $\int_{0}^{\infty}\left[\frac{\ln(1+x)}{1+x}\right]^n\mathrm dx$

I am looking for the closed form, but I can't...
$$\int_{0}^{\infty}\left[\frac{\ln(1+x)}{1+x}\right]^n\mathrm dx=G(n)$$
$n\ge2$
We have: $G(2)=2$, $G(3)=\frac{3}{8}$,$G(4)=\frac{8}{81}$, $G(5)=\frac{15}{512}$, and so on,...
$$\int_{0}^{\infty}u^n e^{u(1-n)}\mathrm du\tag1$$
Is there an easy of calculating $(1)$

A:

$$
\int_{0}^{\infty}u^n e^{u(1-n)}\mathrm du = \frac{1}{(n-1)^{n+1}} \int_0^\infty e^{-t} t^{n} \;dt  = \frac{\Gamma(n+1)}{(n-1)^{n+1}} = \frac{n!}{(n-1)^{n+1}}
$$

