I'm trying to integrate a formula but I can’t figure it out. Its calculation involves an error function. Here's the formula:

$f(a, b, c)=\int_{0}^{+\pi} d \theta \exp (a \cos \theta) \operatorname{erf}(b \cos \theta+c)$

where, a, b, c presents constant respectively, and $\operatorname{erf}$ presents the error function integral. The exact expression of $\operatorname{erf}$ is
$\operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} e^{-\eta^{2}} d \eta$

Thanks a lot.