I am trying to find PDF of $Y=a\sin(X)$ when $X\sim \mathcal{N}(\mu,\,\sigma^2)$. I know it is equal to: $$f_Y(y) = \sum_{k=-\infty}^\infty \frac 1 {\sqrt{a^2-y^2}}\left( f_X(2(k+1)\pi-\arcsin(y/a) ) + f_X(2k\pi+\arcsin(y/a)) \right)$$
$$f_Y(y)=\frac{1}{\sqrt{2\pi(a^2-y^2)}} \sum_{k=-\infty}^\infty \begin{bmatrix} \exp\left( -\left( \frac{\arcsin(y/a)+2n\pi}{2\sigma^2} \right)^2 \right) + \exp\left( -\left( \frac{2(n+1)\pi-\arcsin(y/a)}{2\sigma^2} \right)^2 \right)\end{bmatrix}.$$
I want to find the closed form solution of this distribution that I can use for further manipulation.