I am trying to do the following integral:
$$ \int_0^{2\pi} e^{-A(b+c\cos\phi)^2} d\phi$$
where $A>0$, $b>0$, and $c$ real. I have tried trig substitutions, differentiating under the integral, Fourier transforming, without any luck. I know I can replace $\cos$ with $\sin$ without changing the value of the integral, because it is over the full period. I also know that in the case that $b=0$ the answer is simply a Bessel function, but indeed the presence of both $\cos$ and $\cos^2$ in the exponential is what makes it tricky. Any help or advice is appreciated!