Expectation of $\cos(\|X\|)$ where $X \sim \mathcal{N}(\mu,\Sigma)$

Question

Do:

$$ \int_{-\infty}^\infty \int_{-\infty}^\infty \cos\left(\sqrt{x^2+y^2}\right) e^{-\frac{1}{2}\left[\frac{(x-\mu_x)^2}{\sigma_x^2} + \frac{(y-\mu_y)^2}{\sigma_y^2}\right]}\:\mathrm{d}x\:\mathrm{d}y $$

This problem arises from an expectation of $\cos(\|X\|)$ where $X$ is a Gauss distributed.

Thank you!