"I am back" says Integration - Part 2

Calculus Level 5

\[ \large \int \frac{\sqrt{\cos(2x)}}{\sin(x)} \, dx \]

If the integral above can be expressed as \(\sqrt{a}\log(\sqrt{b}\cos{x} + \sqrt{\cos(cx)}) - \log(d\cot{x} + ((\cot{x})^e - g)^f)\), for constants \(a,b,c,d,e,g\) and \(f\), evaluate \(\frac{abc}{dg} + ef + 11\).

Note: \(f\) is not an integer. And all these constants may not be distinct.

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