Looking for the closed form of this integral
$$\int_{0}^{\infty}\frac{\ln^{n}(x)\ln(1+x)}{x(1+x^2)}\mathrm dx=G(n)$$
$n:=1,3,5,7,...$
Wolfram can evaluate it but I can't generalise into a closed form.
Can anyone help, please.
$$\int_{0}^{\infty}\frac{\ln^{3}(x)\ln(1+x)}{x(1+x^2)}\mathrm dx=\frac{1}{256}[-24\pi^2\zeta(3)-45\zeta(5)-7\pi^4\ln(4)]$$