Typical Indefinite Integration!

Calculus Level 5

$\large \int \frac{\cos(2016x)}{\sin(x)} \, dx$

Let $x$ be a real number except at multiples of $\pi$ . Ignoring the arbitrary constant, if the indefinite integral is equals to

$\large I = \alpha \left( \sum_{n=1}^{1008} \frac{ \cos((2n-1)x)}{2n-1} \right) + \ln \left| \tan \left( \frac {2x}\beta \right) \right |$

For constants $\alpha$ and $\beta$ , evaluate $\alpha^\beta - \beta^\alpha$ .