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$\int \frac{\cos 2 x . \sin 4 x}{\cos^{4} x (1 + \cos^{2} 2 x)} d x =$

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$\log \frac{{(1 + \cos 2 x)}^{2}}{(1 + \cos^{2} x)} + \sec x + C$

$\log \frac{1 + \cos^{2} 2 x}{{(1 + \cos 2 x)}^{2}} + \sec x + C$

$\log \frac{{(1 + \cos 2 x)}^{2}}{(1 + \cos^{2} 2 x)} + \sec^{2} x + C$

$\log (\frac{1 + \cos 2 x}{1 + \cos^{2} 2 x}) + \sec^{2} x + C$

answer is C.

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Detailed Solution

$\begin{aligned} \int \frac{\cos 2 x \times (2 \sin x \cos 2 x)}{{(\frac{1 + \cos 2 x}{2})}^{2} (1 + \cos 2 x)} d x \\ put t = \cos 2 x \Rightarrow dt = - 2 \sin 2 xdx \end{aligned}$

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