A Product of Gaussian and Rayleigh distributions gives what distribution?

Finding out the distribution of the product of a Gaussian and a Rayleigh distributed random variables.

I'm trying to find out the distribution function (cumulative or density) of the product of two independent random variables respectively following a non-zero-mean Gaussian and a Rayleigh distribution. The math is too intricate for me, I've found in the appendix of [Probability Distributions Involving Gaussian Random Variables - Simon, Marvin K.] the density for the product of a zero-mean Gaussian and a Rayleigh but this will not work for what I am trying to do.

Would anyone have a reference that could help me ?

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