The first part is just a Gaussian integral and the second is the modified Bessel function of the first kind for $n=0$, but I couldn't find any information and what to do with their summation. Any tips would be appreciated. Thank you!
$$\int_{0}^{2\pi} \exp\left(-\frac{(\theta-\frac{\pi}{2})^2}{c_2}+ c_1 \cos\left(\theta-\frac{\pi}{2}\right)\right) \,\, d\theta$$