I am trying to derive the following result: $$\mathbb{E}\left[\exp\left(-aX^{2}\right)\right],$$ such that $$X\sim\mathcal{N}\left(\mu,\sigma^{2}\right),$$ $$a = a_{R} + ia_{I},$$ $$a_{R},a_{I} \in \mathbb{R},$$ $$i=\sqrt{-1}.$$
I'm aware that the solution to this is
$$\frac{\exp\left(-\frac{\mu^{2}a}{2a\sigma^2+1}\right)}{\sqrt{2a\sigma^2+1}},$$
but I am having a devil of a time arriving at this solution. It's clear to me that the expression will have to be manipulated to somehow such that one obtains the Laplace Transform of the Chi-Squared Distribution (one degree of freedom) PDF scaled by a constant, but it's really not clear to me how to get there.
Would anyone please be so kind to give me a hand here? Thanks in advance.