Calculate the integral $\int_{-\infty}^{+\infty}\sin(a\cosh(x)) \, dx$

Question

I got into trouble when I tried to calculate this integral while doing a quantum field theory ( QFT ) problem: $$ \int_{-\infty}^{+\infty} \sin\left(\,{a\cosh\left(\,{x}\,\right)}\right)\,{\rm d}x $$ The value is $\pi\operatorname{J}_{0}\left(\,{a}\,\right)$, where $\operatorname{J}_{n}\left(\,{x}\,\right)$ is the Bessel function of the first kind.

How can I prove it analytically?