After recently seeing the definite integral:
$$ \int^{2 \pi}_0 d \phi \exp \left\{ a \cos \phi + b \sin \phi \right\} = 2 \pi I_0 (\sqrt{a^2 + b^2}) $$ where $I_0$ is the modified Bessel function of the first kind, on wikipedia https://en.wikipedia.org/wiki/Lists_of_integrals and then confirming it on DLMF http://dlmf.nist.gov/10.32 (Eq.(10.32.1)), I have recently become obsessed with wanting to know if a similar closed form solution exists if we add a linear term in the exponential, i.e.
$$ \int^{2 \pi}_0 d \phi \exp \left\{ a \cos \phi + b \sin \phi + c \phi \right\} . $$
There are some really suggestive expressions in the modified Bessel reference of DLMF, such as Eq.(10.32.12),
$$ I_{\nu}(z)=\frac{1}{2πi}\int^{∞+iπ}_{∞−iπ}\exp\{z \cosh \phi−\nu \phi\}d \phi $$
and the like, but alas, my knowledge of these integral techniques is lacking of late. If anyone has worked in this area and/or knows of a helpful reference I am thanking you in advance :)