Given the following p.d.f., which is the p.d.f. of the real and imaginary parts of a random variable that is the ratio between a complex Gaussian and a Chi-squared RVs:

\begin{equation*} f_U(u)=\exp\Big\{{-\frac{1}{4 u^2}}\Big\} \,\frac{\left(8 n u^2-1\right) I_n\left(\frac{1}{4 u^2}\right)+I_{n+1}\left(\frac{1}{4 u^2}\right)}{4 |u|^3}, \end{equation*} where $n$ is a constant integer and $I_n(z)$ is the modified Bessel function of the first kind.

I'd like to find its Moment-generating function or at least a closed form expression for its mean and variance.

**UPDATE 17/02/2019**
Further information on this p.d.f. and how it was obtained can be found at: Distribution of ratio between complex Gaussian and Chi-square R.V.s