How to find a sum of the series? $$ S=\sum_{n=1}^\infty\left(\frac{\sin nx}{n}+\frac{\cos nx}{n^2}\right) $$
I tried to do something like that: $$ \begin{aligned} &z=\cos x+i\sin x\\ &S=\sum_{n=1}^\infty\text{Im}\left(\frac{z^n}{n}\right)+\sum_{n=1}^\infty\text{Re}\left(\frac{z^n}{n^2}\right) \end{aligned} $$ However, $\sum_{n=1}^\infty\frac{z^n}{n^2}=\text{Li}_2(z)$, and I have no idea what $\text{Re}(\text{Li}_2(z))$ is here.
So, I'd be very grateful if someone pointed out the right way of solving this problem.