How to study large $r \gg 1$ asymptotics of $$I(r):=\int_0^{\infty} \frac{1-e^{-q}}{1+q} J_0(rq) \ dq,$$ where $J_0$ is the zeroth order Bessel function of the first kind.

I did some numerics and it seems to indicate that $I(r) \sim 1/r^3$ for large $r$, but this is not obvious to me where it comes from.