I enconter a Bessel transform when I deal with a 2D hydrodynamics problem as follows: $$H(k)=\int_{0}^{\infty}(\frac{1}{r(r+1)}+\frac{1}{r^3})e^{-a^2r^2}J_{0}(kr)rdr,$$ where $a$ is a real positive constant.
I found some results using Mathematica: $$\int_{0}^{\infty}\frac{1}{r}e^{-a^2r^2}J_{0}(kr)rdr=\frac{e^{-\frac{k^2}{8 a^2}} \sqrt{\pi} I_{0}(\frac{k^2}{8a^2})}{2a},$$ $$\int_{0}^{\infty}\frac{1}{r^3}e^{-a^2r^2}J_{0}(kr)rdr=-\frac{e^{-\frac{k^2}{8 a^2}} \sqrt{\pi}\left(\left(4a^2+k^2\right) I_{0}(\frac{k^2}{8a^2})+k^2 I_{1}(\frac{k^2}{8a^2})\right)}{4a}.$$ But I cannot find the integral $$H(k)=\int_{0}^{\infty}\frac{1}{r+1}e^{-a^2r^2}J_{0}(kr)rdr.$$ Any thoughts on how to solve this? Thanks!