Can the integral
$$I(R,\sigma)\equiv \int_R^{\infty} x^{2\nu+1}J_{\nu}(x)\exp\{-\frac{x^2}{4\sigma^2}\}dx$$
be written in a closed form in terms of elementary functions? Here $\nu$ is a positive integer, $J_{\nu}(x)$ is a Bessel function of the first kind, $\sigma$ is positive, and $R$ is large and positive such that the asymptotic approximation $J_{\nu}(x) \approx \sqrt{\frac{2}{\pi x}}\cos(x-\frac{\pi}{2}\nu-\frac{\pi}{4})$ is a valid approximation in the integration region.