Integral involving bessel functions, exponential and ratio of polynomials

Question

I need to solve this integral:

$$\int_0^{+\infty}\quad\frac{k}{k^2-\alpha^2}\,J_1(2\pi R k)J_1(ak)\,\exp(-4\pi^2\omega^2k^2)\, dk$$

where $\omega,\,R,\,a>0$ and $J_1(a\alpha)=0$.

Thanks!