I've been exploring interesting integrals on this site and fell into a rabbit hole on one particular user with a special history of answering integrals (giving only answers). I enjoyed reading the interesting solutions of other users who tried to match their solution with the given answer.
That said, I found one question requesting the answers to 2 integrals which is answered but doesn't have a solution from any other user. On searching, I couldn't find any other solution but here are some related questions. I thought there might be sufficient interest from others who might want to provide an full solution to this question. If it's possible, perhaps link this solution to the original question as well?
That's my motivation for why I want the question solved. Unfortunately, I don't have near enough expertise to solve it. The original question has asked that only one integral be posted at a time so here's the second one (the first one has been posted).
$$\int_0^1\frac{\ln^2(1-x)}{x}\text{Li}_2\left(\frac{1 + x}{2} \right)dx$$
Here's also the attached answer from the user : $$\begin{align*}{\large\int}_0^1\frac{\ln^2(1-x)\ \operatorname{Li}_2\left(\frac{1+x}2\right)}xdx&=\frac{81\zeta(5)}{32}+\frac{5\pi^2}{16}\zeta(3)-\frac{\zeta(3)}8\ln^22+\frac1{15}\ln^52\\ -\frac{\pi^2}{18}\ln^32-\frac{\pi^4}{15}\ln2+2{Li}_5\left(\tfrac12\right)+2\operatorname{Li}_4\left(\tfrac12\right)\ln2. \end{align*}$$
