Any suggestions how to solve this or how to find an approximate solution \begin{equation} \int_0^a\int_0^\infty J_0 (\lambda r)J_1(\lambda a)\frac{1}{\sqrt{n+\lambda^2 }}d\lambda dr \end{equation}
(J0,J1 Bessel function of first kind, 0th and 1st order, respectively) Thanks, Ehsan
For $n=0$ Mathematica seems to think the integral is equal to $$ I= \frac{a}{4\pi}\left(4 + 8G-i\pi^2 \right) $$ where $G$ is the Catalan constant. However, numerically for $a=1$ it seems there is a branch problem giving the $-i\pi^2$. So it seems more like $$ I= \frac{a}{\pi}\left(1 + 2G\right) $$ it solved the internal integral over $\lambda$ to be $$ \frac{2}{a \pi r}\left(E\left(\frac{a^2}{r^2}\right)+(a-r)(a+r)K\left(\frac{a^2}{r^2}\right) \right) $$ for complete elliptic $E$ and $K$ functions, beware that the Mathematica notation is with elliptic modulus $k^2=a^2/r^2$ here.