I was doing experiments with Wolfram Alpha online calculator, about similar integrals (simpler than the below one) and wondered about how get a closed-form for $$\int_0^1\frac{\log(1-x+x^2)\log(1+x-x^2)}{x}dx\tag{1}.$$
I've calculated the definite integral using the online calculator, but I believe that the output is an approximation, and since after of this, I've asked to Wolfram Alpha about the indefinite integral, I know that Wolfram Alpha can calculate it, but to me is impossible to evaluate the terms (are about two pages)
int log(1-x+x^2)log(1+x-x^2)/x dx
Question. Is there some way to evaluate this integral in $(1)$? This was just a curiosity, but I am asking here if you know such integral or do you know how get the evaluation of our integral. Thanks in advance.
Since Wolfram Alpha's answer seems to me difficult, I didn't any attempt (change or variable, integration by parts...).