Fourier Series $\sin(\sin(x))$

Can anyone find the Fourier Series of $ \sin(\sin(x))$?

I have tried evaluating the integrals to determine the coefficients of each of the coefficients of the sine waves, but have no idea where to start computing the integrals.

First note that $f(x)$ is anti-symmetric so the series only has $\sin$'s, and that only odd $n$'s will appear since $f(x-\pi)=-f(x)$.

The integrals give combinations of the Bessel J functions, at $x=1$. The coefficient of $\sin nx$ is: \begin{align} b_n& = \frac{1}{\pi}\int_{-\pi}^{\pi} \sin(\sin(x))\sin(nx) \, dx \\ & = \frac{1}{\pi}\int_{-\pi}^{\pi} \cos(nx-\sin(x)) - \cos(nx)\cos(\sin(x))\, dx \\ & =2 J_n(1) \end{align} Since: $$J_n(1) = \frac{1}{\pi} \int_0^\pi \cos(nx - \sin x)\, dx\ ,$$ and the second integral becomes $0$ for odd $n$, as it is anti-symmetric around $\pm \pi/2$.

Thus we have: $$\large\color{blue}{\sin(\sin(x)) = 2\sum_{k=0}^\infty J_{2k+1}(1)\sin \left((2k+1)x\right)}$$

P.S: This gives an interesting identity for the sum of all odd $J_k(1)$: $$\sin(1)=2\sum_{k=0}^\infty (-1)^kJ_{2k+1}(1)$$