I am not working in physics field, but I still somehow encountered a problem involving integration of Bessel functions. Currently the problem can be reduced to finding an point-wise upper and lower bound of this following function:
$$f(x) = \log(x) + \frac{-x^2}{8}{}_2F_3\left([1,1];[2,2,2],\frac{-x^2}{4}\right)$$
for some sufficient large $x$ (say $x \geq 1$) this function comes from a result of integration of $\int{\frac{J_0(x)}{x} dx}$, and the reference can be found from Bessel-Integral function by Humbert (1933). A more explicit form of $f(x)$ could be expressed as
$$f(x) = \log(x) + \sum_{k=1}^\infty\frac{1}{2k(k!)^2}\left(\frac{-x^2}{4}\right)^k$$
From a Matlab plot, this function seems to be a sinusoidal function with decreasing amplitude when $x$ increases with offset value roughly 0.117, so I'm suspecting the envelope should be an rational polynomial as Bessel function $J_0(x)$