Derive $Im \text{ Li}_3\left(\frac{1+i}2\right)= \frac12\int_0^1 \frac{\ln^2(1-x)}{1+x^2}dx$

Question

I would like to have a derivation of the following close-form

$$Im \ \text{Li}_3\left(\frac{1+i}2\right)= \frac12\int_0^1 \frac{\ln^2(1-x)}{1+x^2}dx$$

I expect that I could just rely on the recursive representations of the polylogarithmic functions below

$$\text{Li}_3 \left(\frac{1+i}2\right)= \int_0^{\frac{1+i}2} \frac{\text{Li}_2(t)}t\ dt = -\int_0^{\frac{1+i}2} \frac{dt}t\ \int_0^t\frac{\ln(1-s)}s ds $$

and manipulate the RHS, (e.g. IBP, substitution, etc.) to arrive at the desired integral, with no luck yet. Would like to get some pointers in deriving it.

Answer 1

Useful formula \begin{eqnarray*} \int_0^1 \frac{a (\ln(x))^2}{1-ax} dx = 2 \operatorname{Li}_3(a). \end{eqnarray*} Sub $x=1-u$ and partial fractions ... should give the result?

Also see FDP's answer here for something similar Evaluate $\int\limits_0^\infty \frac{\ln^2(1+x)}{1+x^2}\ dx$