What tools or ways would you propose for getting the closed form of this integral?
$$\int_0^{\pi/4}\frac{\log(1-x) \tan^2(x)}{1-x\tan^2(x)} \ dx$$
EDIT: It took a while since I made this post. I'll give a little bounty for the solver of the problem, 500 points bounty.
Supplementary question:
Calculate
$$\int_0^{\pi/4}\frac{\log(1-x)\log(x)\log(1+x) \tan^2(x)}{1-x\tan^2(x)} \ dx$$