Integration of Bessel function of first kind multiplied with Gaussian

Question

I like to know how to proceed with the following two integrations

$$\int_{-\infty}^{\infty}J^{2}_{n\pm1}(x)e^{-(a+nb)^{2}}\,dn$$

$$\int_{-\infty}^{\infty}J_{n+1}(x)J_{n-1}(x)e^{-(a+nb)^{2}}\,dn$$

Note that $x$ is independent of $n$.