How to integrate Bessel function $\int_0^{\infty } J_0(a x) \left(\frac{\sin (x)}{x^3}-\frac{\cos (x)}{x^2}\right) \, dx$

Question

How to integrate Bessel function $$\int_0^{\infty}J_0(ax)\left(\frac{\sin x}{x^3}-\frac{\cos x}{x^2}\right)dx.$$ Here $a>1$, $J_0(x)$ is Bessel function.

I try write $J_0(ax)=\sum_{k=0}^{\infty}\frac{(-1)^k}{(k!)^2}(\frac{ax}{2})^{2k}=\frac{1}{\pi}\int_{0}^{\pi}e^{iax\cos\theta}d\theta$, but I can not go on.