Difficult Bessel integral In my research involving quantum field theory and computing correlation functions in perturbation theory, I encountered the following integral that seems pretty hard to solve:
$$f(x)=\int_{0}^{\infty}\frac{dz}{z(z^2+1)}\text{arcsinh}^2(z)J_0(xz)$$
I wonder if anyone has any insights towards a solution. A series expansion and it's convergence properties would be fantastic as well.

Not elegant at all since using brute force.
Computing numerically the integrals, it seems that a good (totally empirical) approximation could be
$$\log(f(x))=a-b\, x^c$$ Based of $40$ equally spaced data points between $0.5$ and $20.0$, a quick and dirty nolinear regression gives
$$\begin{array}{clclclclc}
 \text{} & \text{Estimate} & \text{Standard Error} & \text{Confidence Interval} \\
 a & 0.17160 & 0.00218 & \{0.17603,0.16717\} \\
 b & 1.10970 & 0.00091 & \{1.10785,1.11155\} \\
 c & 0.97817 & 0.00025 & \{0.97765,0.97868\} \\
\end{array}$$
