Evaluate $ \int \frac{e^x}{\left({1+\cos(x)}\right)} dx$ Background: I was in the process of solving some interesting integrals from this site, only to find out I needed a lot more practice before becoming familiar with special functions. 
So while doing some problems, I encountered some difficulty with one particular integral; I happened to incorrectly copy it onto a notebook. But I'm curious to know as to how exactly I can evaluate this particular integral.
Essentially, I need help in evaluating the following integral :-

$$ \int \frac{e^x}{\left({1+\cos(x)}\right)} dx$$
Question: How exactly can I evaluate this integral? 
Both solutions as well as hints would be greatly appreciated.

Note: Original problem had $\cosh(x)$ instead of $\cos(x)$.

The integrand does not possess an elementary antiderivative. This can be shown using either Liouville's theorem or the Risch algorithm. However, doing so requires advanced knowledge  of abstract algebra. Alternately, expand $~\dfrac1{1+\cos x}~$ into its binomial series, then switch the  order of summation and integration to obtain an infinite series, which you might rewrite in  terms of hypergeometric functions.
