I just ran into integrals of the Bessel type, but which are unfortunately indefinite integrals, such as $$ f(t)=\int \cos(\gamma\cos(\omega t))\cos(\omega t)\mathrm dt. $$ I'm conscious of the fact that in a sense this is game over - if these integrals were doable in terms of getting elementary expressions for $f$, then Bessel functions would also be elementary and they are not. However, that's not a reason why there might not be standard ways to deal with a function like this one out there. Is $f(t)$ known in terms of other special functions, for example?
This is quite hard to google as most searches will just return indefinite integrals that have $J_\nu(x)$s in the integrand itself. If it's any help, this came up naturally while studying the motion of charged particles in oscillating electric and magnetic fields.