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 $cos(x)$.

Answer

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.

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