Integration Integrate $\int_{-\theta c}^{\theta c} e^{-K/\cos(\theta)} \, d\theta$ I'm trying to integrate $\displaystyle\int_{-\theta c}^{\theta c} e^{-K/\cos(\theta)} \, d\theta$
Numericaly the integrale look like clean, I try various method to have analytic form:


*

*Mathematica

*Taylor series


I wasn't able to find something.
For details:


*

*$0\lt\theta c\le\frac{\pi}{2}$

*$0\lt K\lt \infty$


Have you any guess about this form?
Regards

Given: 
$$I = \displaystyle\int_{-\theta c}^{\theta c} e^{-K/\cos(\theta)} \, d\theta$$
* $0\lt\theta c\le\frac{\pi}{2}$, $0\lt K\lt \infty$
Then:
\begin{align}
I &= 2 \, \int_{0}^{\theta_{c}} e^{-K/\cos(\theta)} \, d\theta \\
&= 2 \, \sum_{n=0}^{\infty} \frac{(-K)^{n}}{n!} \, \int_{0}^{\theta_{c}} sec^{n}\theta \, d\theta \\
&= 2 \, \sum_{n=0}^{\infty} \frac{(-K)^{n}}{n!} \, \sin\theta_{c} \, {}_{2}F_{1}\left( \frac{1}{2}, \frac{n+1}{2}; \frac{3}{2}; \sin^{2}\theta_{c} \right). \\
\end{align}
