When I was playing with Wolfram Alpha about the integral $$\int_0^1\frac{x^{s-1}}{1+x^2}dx$$ and its derivatives, since I know the relationship between the Apéry's constant and particular values of the polygamma function, and since I presume that this way will be known, I found playing with the code a closed-form for this $$\int_0^1\frac{\log^2(x)}{1+x^3}dx$$ see this code
integrate 1/(1+x^3)(log^2(x))dx, from x=0 to x=1
in the online calculator. I am not able to get easily the calculations for $$\int\frac{\log^2(x)}{1+x^3}dx$$ and after evaluate it as a definite integral. And you?
Question. This can be a good integral for this friday. Can you prove the closed-form for $$\int_0^1\frac{\log^2(x)}{1+x^3}dx?$$ Thanks in advance.