I have an integral involves products of Bessel functions and sine function,
$$\int_{0}^{\infty}\frac{\Big(k-\frac12\Big)^{u+j}}{\Big(k+\frac12\Big)^{u+j+2}}~\sqrt{k}~J^2_{\ell}(k)~\sin{\big(\tau~\sqrt{k}\big)}~dk$$
in which, $u$ and $j$ are natural numbers, $\ell$ is arbitrary integer and $\tau$ is a real number, large or small. $$J^2_{\ell}(k) =\frac{1}{\pi}\int_{0}^{\pi} J_{0}(2k\sin\theta)\cos{(2\ell\theta)} \mathrm{d}\theta$$ And initial integral turns out to be $$ \frac{1}{\pi} \sum_{n=0}^{u+j} \binom{u+j}{n}(-1)^n \int_{0}^{\pi} [\int_{0}^{\infty} \frac{\sqrt{k}}{(k+\frac{1}{2})^{n+2}} J_{0}(2k\sin\theta) \sin{(\tau\sqrt{k})}\mathrm{d}k] \cos{(2\ell\theta)} \mathrm{d}\theta $$
Any help on its computation, numerically or analytically, is much appreciated. Thanks in advance.