My statistical model involves the multiplication of a scalar random variable $X|X \geq 0 \sim 2\mathcal{N}(x;0,\sigma^2)\ \mathbb{I} \ [x \in \mathcal{R}_+]$, or a gaussian variable that must be strictly above zero, and and scalar exponentially distributed magnitude $M \sim \exp(m;\lambda)$.
I would like to find the distribution of $C =M\cdot X$ which is clearly still positive.
So far I have that:
$f_C(c) = \int_0^{\infty}f_{X|M}(\frac{c}{M}|M = m)f_M(m)\frac{dm}{m}$
$\ \ \ \ \ \ \ \ \ \ = \int_0^{\infty}f_{X}(\frac{c}{m})f_M(m)\frac{dm}{m}$
$\ \ \ \ \ \ \ \ \ \ = \sqrt{\frac{2}\pi} \frac{\lambda}{\sigma}\int_0^{\infty}\exp \left({-\frac{1}{2}\left(\frac{c}{\sigma m}\right)^2} \right) \cdot \exp \left(- \lambda m\right) \frac{dm}{m}$
I haven't really founding a promising direction to pursue to integrate this. Part of what makes it difficult is the integration of the form $\int\exp(-\frac{\alpha}{m^2} - \beta m)\frac{dm}{m}$ and everything I try seems fruitless. I've considered modifying the $M$ distribution, but the exponential seems reasonable for my application. It's modelling something physical (diffusion of a single particle), but the magnitude $M$ is not determined by some well-defined rigourous process so it's quite possible the $M$ distribution has nothing to do with physics inherently. But I thought I could change the magnitude to the form $\sim \frac{1}{Z}e^{(k_bT)^{-1}U(m)}$ or even $\sim \frac{1}{Z}e^{-(k_bT)^{-1}U(m,x)} $ for some $U(\cdots)$'s if necessary, but I'm not sure if either of those would be physically justifiable. Any ideas?
EDIT: for clarity.