In my research I have come across the following integral\begin{equation}f = \int_0^{2\pi} \text{d}\theta \exp\left\{\frac{3}{2}(h_1 \cos^2\theta + h_2 \sin^2\theta + 2 h_{12} \sin\theta \cos\theta)\right\}\end{equation}where $h_1$, $h_2$ and $h_{12}$ are constants of order unity, but which can be either positive or negative. Eventually, I will have to evaluate this integral numerically a large number of times. Hence, it would be quite interesting if it could be written in terms of special functions (if $h_2 = h_{12} = 0$, for instance, it can be written in terms of Bessel Functions). However, I was unable to figure out how. Any suggestions?More generally, do you have any tips on how to implement it numerically efficiently?Thank you very much in advance.Best regards, Gabriel 
Representing an integral as a special function
special functions;integral equations
Using the half angle formulas you can convert the exponent into the form$$a + b \cos 2\theta + c \sin 2\theta$$which then integrates nicely into Bessel functions.  Mathematica gives$$\int_0^{2\pi}e^{a + b \cos 2\theta + c \sin 2\theta} d\theta = e^a \pi I_0\left(\sqrt{b^2 + c^2}\right)$$