integration - Integral of sine multiplied by Bessel function with complicated argument

I need a help with integral below,
$$\int_0^\infty \sin(ax)\ J_0\left(b\sqrt{1+x^2}\right)\ \mathrm{d}x,$$
where $a,b > 0$ and real, $J_0(x)$ is the zeroth-order of Bessel function of the first kind.

I found some integrals similar to the integral above, but I don't have any idea on how to apply it. Here are some integrals that might help.
$$\int_0^\infty \cos(ax)\ J_0\left(b\sqrt{1+x^2}\right)\ \mathrm{d}x = \frac{\cos\sqrt{b^2-a^2}}{\sqrt{b^2-a^2}}; \mathrm{~~for~0 < a < b}$$

$$\int_0^\infty \sin(ax)\ J_0(bx)\ \mathrm{d}x = \frac{1}{\sqrt{a^2-b^2}}; \mathrm{~~for~0 < b < a}$$

The proof of the first integral can be seen here.