Any ideas on how to tackle this integral ?

$\displaystyle \int_{-1}^{1} dt \cos(a t) \cos (b \sqrt{1-t^{2}})$

I know that the 'sine' version can be solved in terms of a Bessel function (Gradstein 3.711):

$\displaystyle \int_{-1}^{1} dt \cos(b t) \sin (a \sqrt{1-t^{2}}) = -\pi \frac{\partial}{\partial a} J_{0}(\sqrt{a^2 + b^2})$