calculus - Calculating in closed form $int_0^1 log(x)left(frac{operatorname{Li}_2left( x right)}{sqrt{1-x^2}}right)^2 ,dx$

What real tools excepting the ones provided here Closed-form of $\int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx$ would you like to recommend? I'm not against them, they might be great, but it seems they didn't lead anywhere for the version $\displaystyle \int_0^1 \frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}} \,dx$. Perhaps we can find an approach that covers both cases, also

$$\int_0^1 \log(x) \left(\frac{\operatorname{Li}_2\left( x \right)}{\sqrt{1-x^2}}\right)^2 \,dx$$
that I would like to calculate.

Might we possibly expect a nice closed form as in the previous case? What do you propose?

EDIT: Thanks David, I had to modify it a bit to fix the convergence issue. Also, for the previous question there is already a 300 points bounty offered for a full solution with all steps clearly explained.