The pdf $f(\theta)$ of the wrapped cauchy distribution is $$ f(\theta) = \frac{1}{2\pi}\frac{\sinh(\kappa)}{\cosh(\kappa) - \cos(\theta - \mu)} $$ where $\mu \in \mathcal{R}$ and $\kappa > 0$ are parameters, and $\theta \in [-\pi,\pi)$.
To find the cumulative density function $F(\theta)$ we need to evaluate the integral $$ F(\theta) = \frac{\sinh(\kappa)}{2\pi}\int_{-\pi}^{\theta}\frac{1}{\cosh(\kappa) - \cos(\theta' - \mu)}d\theta' $$
But this integral seems intractable to me.
The CDF, evaluated numerically, looks pretty much like a sigmoid function. Is there a good numerical approximation to that integral that lets me work with something "close" to the true CDF?