TITLE: Integral $\int_{0}^{1} \frac{\ln^2 x \ln^2 (1+x)\ln^2(1-x)}{x^2}dx$
QUESTION [8 upvotes]: Due to curiosity and also since I evaluated lower degree sums like these but this one is too hard to manipulate I am eager to know does this have a closed form ?
I broke it into the series $\displaystyle \sum_{m,n\ge 1}(-1)^{m+n}\frac{{\rm H}_m{\rm H}_n}{(m+1)(n+1)} \frac{2}{(2n-1)^3} $ , but does this help ?

REPLY [6 votes]: See this paper. One have

$\scriptsize \int_{0}^{1} \frac{\ln^2 x \ln^2 (1+x)\ln^2(1-x)}{x^2}dx=8 \zeta(\bar5,1)-\frac{8}{3} \pi ^2 \text{Li}_4\left(\frac{1}{2}\right)+8 \text{Li}_4\left(\frac{1}{2}\right)+24 \text{Li}_5\left(\frac{1}{2}\right)+16 \text{Li}_6\left(\frac{1}{2}\right)+8 \text{Li}_4\left(\frac{1}{2}\right) \log ^2(2)+24 \text{Li}_4\left(\frac{1}{2}\right) \log (2)+16 \text{Li}_5\left(\frac{1}{2}\right) \log (2)+\frac{19 \pi ^2 \zeta (3)}{6}-62 \zeta (5)-\frac{227 \zeta (3)^2}{8}-7 \zeta (3) \log ^3(2)-\frac{35}{2} \zeta (3) \log ^2(2)+\frac{35}{6} \pi ^2 \zeta (3) \log (2)-9 \zeta (3) \log (2)-\frac{217}{2} \zeta (5) \log (2)+\frac{73 \pi ^6}{1260}+\frac{\pi ^4}{90}+\frac{2 \log ^6(2)}{9}+\frac{4 \log ^5(2)}{5}-\frac{5}{18} \pi ^2 \log ^4(2)-\frac{5 \log ^4(2)}{3}+\frac{2}{3} \pi ^2 \log ^3(2)+\frac{13}{36} \pi ^4 \log ^2(2)+\pi ^2 \log ^2(2)+\frac{1}{6} \pi ^4 \log (2)$

Where $\zeta(\bar5,1)$ is a Multiple Zeta Value. Bonus:

$\scriptsize \int_0^1 \frac{\log ^3(1-x) \log ^2(x) \log ^2(x+1)}{x^2} \, dx=-12 \zeta(\bar5,1)+36 \zeta(\bar5,1,1)-6 \zeta(5,\bar1,1)+54 \log (2) \zeta(\bar5,1)+156 \text{Li}_4\left(\frac{1}{2}\right) \zeta (3)-8 \pi ^2 \text{Li}_4\left(\frac{1}{2}\right)-34 \pi ^2 \text{Li}_5\left(\frac{1}{2}\right)-48 \text{Li}_5\left(\frac{1}{2}\right)+144 \text{Li}_6\left(\frac{1}{2}\right)-96 \text{Li}_7\left(\frac{1}{2}\right)-22 \pi ^2 \text{Li}_4\left(\frac{1}{2}\right) \log (2)-51 \zeta (3)^2-\frac{71 \pi ^4 \zeta (3)}{320}+\frac{9 \pi ^2 \zeta (3)}{4}+\frac{4461 \pi ^2 \zeta (5)}{64}-3 \zeta (5)-\frac{1755 \zeta (7)}{4}-\frac{1}{2} \zeta (3) \log ^4(2)-28 \zeta (3) \log ^3(2)+\frac{95}{8} \pi ^2 \zeta (3) \log ^2(2)-24 \zeta (3) \log ^2(2)-\frac{3999}{16} \zeta (5) \log ^2(2)-\frac{1563}{16} \zeta (3)^2 \log (2)+\frac{31}{2} \pi ^2 \zeta (3) \log (2)-\frac{279}{2} \zeta (5) \log (2)+\frac{\pi ^6}{112}+\frac{2 \log ^7(2)}{105}+\frac{\log ^6(2)}{5}-\frac{7}{10} \pi ^2 \log ^5(2)-\frac{6 \log ^5(2)}{5}+\frac{2}{3} \pi ^2 \log ^4(2)+\frac{121}{180} \pi ^4 \log ^3(2)+\frac{4}{3} \pi ^2 \log ^3(2)+\frac{29}{60} \pi ^4 \log ^2(2)+\frac{1321 \pi ^6 \log (2)}{10080}+\frac{1}{15} \pi ^4 \log (2)$

The latter will be extremely difficult via ordinary means. See the second link of my answer here for algorithmic reference. One may verify their correctness by numerical methods.