I am looking for a closed form solution to the following, $$I_n=\int_0^{\pi/2}x\tan^{n}(x)\ dx,\quad 0\leq n<1.$$ I have tried integrating by parts to maybe introduce the Beta function, but the $\tan^n(x)$ term makes it difficult.
Substituting $x=\arctan(u)$, $$I_n=\int_0^{\infty}u^n\underbrace{\frac{\arctan(u)}{1+u^2}}_{f(u)}\ du.$$ If I can obtain a power series expansion for $f(u)$, I can find the Mellin transform of it by Ramanujan's master theorem and the integral will follow, but I haven't been successful with this either.
Any ideas? Thank you in advance.