"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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