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integratn of 1+cos4x/cotx-tanx

$Given \int \frac{1 + \cos 4 x}{\cot x - \tan x} ⅆ x \Rightarrow \int \frac{1 + \cos 4 x}{\frac{\cos x}{\sin x} - \frac{\sin x}{\cos x}} ⅆ x \Rightarrow \int \frac{(1 + \cos 4 x) \sin x \cos x}{\cos^{2} x - \sin^{2} x} ⅆ x \Rightarrow \int \frac{(1 + 2 \cos^{2} 2 x - 1) \sin x \cos x}{\cos 2 x} ⅆ x \Rightarrow \int \frac{2 \cos^{2} 2 x (2 \sin x \cos x)}{2 \cos 2 x} ⅆ x \Rightarrow \int \frac{2 \cos^{2} 2 x (\sin 2 x)}{2 \cos 2 x} ⅆ x \Rightarrow \frac{1}{2} \int 2 \sin 2 x \cos 2 x ⅆ x \Rightarrow \frac{1}{2} \int \sin 4 x ⅆ x \Rightarrow \frac{- 1}{8} \cos 4 x + c$

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