calculus - Integration Integrate $int_{-theta c}^{theta c} e^{-K/cos(theta)} , dtheta$

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

Answer

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}