I am having some difficulty with this exercise:
Calculate $$\int_{0}^{\infty}{\frac{\ln{x}}{x^{\frac{2}{3}}(1+x)}}\mathrm dx$$
By Wolfram Alpha, I know that the Answer is $\frac{-2}{3}\pi^2$. I tried to change variable $\arctan(\sqrt{x})=t$ cause I see (1+x), and it lead to $\int_{0}^{\pi/2}{\frac{\ln({\tan(t)})}{\tan(t)^{1/3}}}\mathrm dx$, and got stuck.
Can anyone help me solve this or give me some ideas.