what is the value

$\begin{aligned} I & = \int_0^1 \left(\frac {\ln x}{1-x}\right)^2 dx & \small \blue{\text{Let }-u = \ln x \implies -du = \frac {dx}x} \\ & = \int_0^\infty \frac {u^2 e^{-u}}{(1-e^{-u})^2} du & \small \blue{\text{Multiply up and down by }e^{2u}} \\ & = \int_0^\infty \frac {u^2 e^u}{(e^u - 1)^2} du & \small \blue{\text{by integration by parts}} \\ & = - \frac {u^2}{e^u - 1} \bigg|_0^\infty + \int_0^\infty \frac {2u}{e^u-1} du \\ & = - \blue{\lim_{u \to \infty} \frac {u^2}{e^u-1}} + 0 + 2 \red{\zeta(2) \Gamma(2)} & \small \blue{\text{A }\infty/\infty \text{ case, L'Hôpital's rule applies}} \\ & = - \blue{\lim_{u \to \infty} \frac {2u}{e^u}} + 2 \cdot \red{\frac {\pi^2}6 \cdot 1!} & \small \red{\text{Riemann zeta function }\zeta (2) = \sum_{k=1}^\infty \frac 1{k^2} = \frac \pi 6} \\ & = - \blue{\lim_{u \to \infty} \frac {2}{e^u}} + \frac {\pi^2}3 & \small \red{\text{Gamma function }\Gamma (n) = (n-1)!} \\ & = \blue 0 + \frac \pi 3 \approx \boxed{3.29} \end{aligned}$

References:

• Riemann zeta function
• Gamma function