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 \(\alpha^\beta - \beta^\alpha \).

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