I am trying to get a closed form for the following definite integral:

$$f(\theta)= \int_\frac{\pi}{2}^\pi \frac{1}{\sqrt{1-\alpha^2 \cos^2\theta}}\exp\left(C_2\cos\theta-C_1\sqrt{1-\alpha^2 \cos^2\theta}\right) \,d\theta$$

where $0<\alpha<1$ and $C_1$ and $C_2$ are constants .

Can this function be related to some special function? Can anyone give me some advice or recommend some references?

Thanks in advance.