I have two random variables which do not follow $\mathcal{N}(0,1)$. The characteristic function of the product of two random normal variables is
$\frac{1}{\sqrt{\sigma^2 t^2 +1}}exp \left[ - \frac{\mu^2 t^2}{2(\sigma^2 t^2 + 1)} \right ]$ according to my calculations.
To find the pdf I tried computing this $\frac{1}{2 \pi} \int_{-\infty}^{\infty}e^{-itz}\frac{1}{\sqrt{\sigma^2 t^2 +1}}exp \left[ - \frac{\mu^2 t^2}{2(\sigma^2 t^2 + 1)} \right ] dt$
And the result I got is $\frac{1}{2 \pi} \int_{-\infty}^{\infty} \frac{cos(zt)}{\sqrt{\sigma^2 t^2 +1}}exp \left[ - \frac{\mu^2 t^2}{2(\sigma^2 t^2 + 1)} \right ] dt$.
But I can't proceed after this. What is the result of this integral??
Does this integral equate to any special functions like Bessel or modified vessel?
Any hint will be appreciated.