I was trying to analyze a large amplitude problem and I got stuck at an equation like this.
$$\int_0^T dt=\sqrt{\frac{l}{2g}}\int_{\theta_{\text{max}}}^{ \theta_0} \frac{d \theta }{ \sqrt{\cos\theta - \cos \theta_{\text{max}}}}$$ where everything is constant except the dummy integration variable $t$ and $\theta$.
I went online to find a solution but found none so far.
I thought about an infinite Taylor series: Let $y$ be the required function on the right hand side, then $\frac{dy}{d\theta}$ will be the known function itself. $$ y = y(0)+y'(0) + y''(0)\ldots$$ I was able to calculate $y(0)$ as an indefinite integral. But there are limits in my integral. I am not sure if this is how I approach the problem.
It seems (?) that I have to find an approximation to an elliptic integral.