I am looking for the value of the following integral:
$$\int_u^\infty \sqrt{(x^2-a)} \exp\left({-\left(bx^2+\frac{c}{x^2}\right)}\right)\text dx$$
I encountered this problem when trying to find the expected value of a normal PDF where the standard deviation is a function of a Rayleigh random variable. I have seen integrals like $\int e^{-\left(bx^2+\frac{c}{x^2}\right)}\text dx$, or $\int e^{-\left(bx+\frac{c}{x}\right)}\text dx$ where the solutions involve closed form exponentials or modified Bessel functions, but I can't figure it out for the integral explained above.