Integral with modified Bessel function of the first kind with hyperbolic function

Question

I try to compute the following integral, $$J = \int_{0}^{\pi/2}dy\,\cosh(2Ay)I_1(2B\cos y).$$ To be honest, I have no ideas how to perform the integration. My best idea was to use integral representaion for $I_1(2B\cos y)$ function but it was unsuccessful. Can any one comment it?

Answer 1

There is the closed form of the mentioned integral, it is given in Gradstein-Ryzhik book (6.681.3): $$\int_{0}^{\pi/2}dx\,I_{2\nu}(2a\cos x)\cos(2\mu x)=\frac{\pi}{2}I_{\nu-\mu}(a)I_{\nu+\mu}(a).$$

The mentioned integral can be rewritten in this form with replacement $\mu\rightarrow i\mu$ and setting $\mu=2A$ with $\nu=1/2$ and $a=2B$.