$$ I = int_ {0} ^ { infty} ln ^ 2 (x) ln (1 + x) ln ^ 2 left (1 + frac {1} {x} right) frac { mathrm dx} {x} $$

$ ln (1 + x) = x- frac {x ^ 2} {2} + frac {x ^ 3} {3} – cdots $

$$ int_ {0} ^ { infty} left (1- frac {x} {2} + frac {x ^ 2} {3} + cdots right) left[ln(x)lnleft(1+frac{1}{x}right)right]^ 2 mathrm dx $$

This integral takes the form of $$ J = int_ {0} ^ { infty} x ^ n left[ln(x)lnleft(1+frac{1}{x}right)right]^ 2 mathrm dx, n ge0 $$

$$ u = left[ln(x)lnleft(1+frac{1}{x}right)right]^ 2 $$

$$ u ^ {# 1} = frac {2 ln (x) ln (1 + 1 / x) left[(1+x)ln(1+1/x)-ln(x)right]} {x (1 + x)} $$

$$ v = frac {x ^ {n + 1}} {n + 1} $$

$$ J = frac {x ^ {n + 1}} {n + 1} left[ln(x)lnleft(1+frac{1}{x}right)right]^ 2- frac {2} {n + 1} int_ {0} ^ { infty} x ^ {n + 1} cdot frac { ln (x) ln (1 + 1 / x) left[(1+x)ln(1+1/x)-ln(x)right]} {x (1 + x)} mathrm dx $$

Wow … it gets too hard, I'm totally lost, no help.