I am trying to find out the expression of the CDF of the product of two independent non-central chi distributions (not chi-squared) each with $k=2$ degrees of freedom. As I understand, non-central chi distribution (with k=2) is square root of the sum of squares of two Guassian random variables of different means and variances.
If I assume that the two Guassian variables have same means and variances, then the non-central chi distribution reduces to standard Rayleigh. It turns out that a closed form of CDF of the product of two independent Rayleigh distributions does exist (e.g. p. 56 of this book).
However, I have been unable to find a closed form of CDF of the product of the two independent non-central chi distributions. Is there a good reference that I can look into? Any help would be greatly appreciated.