$\int x \tan x \,dx=\frac{1}{2} i \text{Li}_2\left(-e^{2 i x}\right)+\frac{i x^2}{2}-x \log\left(1+e^{2 i x}\right)+C $

Where Li is the polylogarithm function

You can express the result in closed form also through polygamma function, Hurwitz zeta function or generalizer (to real orders) Bernoulli polynomials.

$\int x \tan x \,dx=\frac{1}{2} i \text{Li}_2\left(-e^{2 i x}\right)+\frac{i x^2}{2}-x \log\left(1+e^{2 i x}\right)+C $

Where $\text{Li}_2$ is the dilogarithm function