How to integrate Elliptic function with gaussian kernel

Question

I have an integral problem as follows: $$ I(k,\sigma)=\int_{-\infty}^{\infty}\frac{\exp\left(-\frac{x^2}{2\sigma^2}\right)}{\sqrt{1+k^2\sin^2(x)}}dx, $$ where $k,\sigma>0$. The function to be integrated is an elliptic function multiplied by a Gaussian kernel. I searched from Google and have no solution. Maybe one possible approach is first to find the power series of the elliptic function: $$ \frac{1}{\sqrt{1+k^2\sin^2(x)}}=\sum_{n=0}^{\infty}a_n(k)x^n. $$ However, this is not easy too.