Can you derive a closed form for this Fourier series: $\sum_{n=1}^{\infty} \frac{1}{n^2}\frac{a^n+b^n}{1-(ab)^n}\sin(n\phi)$

Question

During solving Laplace equation on an annular domain using separation of variables I encountered the following Fourier series: $$\sum_{n=1}^{\infty} \frac{1}{n^2}\frac{a^n+b^n}{1-(ab)^n}\sin(n\phi)$$ where $0<a<1$ and $0<b<1$. Since it is in a compact form I thought that it might have closed solution.