TITLE: How to evaluate $\int_{0}^{1}{\frac{{{\ln }^{2}}\left( 1-x \right){{\ln }^{2}}\left( 1+x \right)}{1+x}dx}$
QUESTION [18 upvotes]: I want to evaluate $$\int_{0}^{1}{\frac{{{\ln }^{2}}\left( 1-x \right){{\ln }^{2}}\left( 1+x \right)}{1+x}dx}$$
I run this integral on Maple, It does converge. How we get a closed form?
Is that related to polylogs? $\operatorname{Li}_{5}\left(\frac{1}{2}\right)$

REPLY [3 votes]: The result is
-4〖Li〗_5 (1/2)+4ζ(3) 〖Log〗^2 2-(2π^2)/9 〖Log〗^3 2- π^2/3 ζ(3)-π^4/20 Log2+ 7/30 〖Log〗^5 2+ 63/8 ζ(5)
 =0,418709830998418751408037243448…