Is there a nice formula for this function of $|x|<1$? $$f(x)=\sum_{n=1}^\infty x^n\cos\left(\frac{3 \pi}{2\cdot5^n}\right)=\Re \sum_{n=1}^\infty x^ne^{\frac{3\pi i}{2}\cdot5^{-n}}$$
Ideally the result would be so pleasant that I can symbolically invert the function $f$ to compute, for example, $f^{-1}(\sin\frac{\pi}{5})$. And I'd like to generalize to other forms $\cos(a\cdot b^{-n})$ where $a$ is a rational multiple of $\pi$ and $b$ is an integer $\geq2$.
(I suspect that none of this is possible, but it doesn't hurt to ask!)