Playing with Wolfram Alpha and inspired in [1] (I refers it if someone know how relates my problem with some of problems involving the Apéry constant in this reference, but the relation doesn't seem explicit), defining $$I_n:=-\int_0^1\frac{\log(1+x^{2n})\log x}{x}dx$$ for integers $n\geq 1$, I can calculate, as I am saying with Wolfram Alpha (but I don't know how get the indefinite integrals) $I_1$, $I_2$, $I_3$ and $I_4$. And as a conjecture $$I_8=\frac{6\zeta(3)}{8\cdot 16^2}.$$
Motivation. I would like to do a comparison with the sequence $I_1$, $I_2$, $I_3$, $I_4$ and $I_8$.
Question. If do you know that this problem was solved in the literature please add a comment: can you evaluate in a closed-form $I_5$? Many thanks.
References:
[1] Walther Janous , Around's Apéry's constant, J. Ineq. Pure and Appl. Math. 7(1) Art. 35 (2006).