Find the antiderivative/primitive/integrals of the \( \mathrm{P} \)...

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Find the antiderivative/primitive/integrals of the
\( \mathrm{P} \) following by simple manipulation/simplifying.

W
\[
\int\left[\frac{\cot ^{2} 2 x-1}{2 \cot 2 x}-\cos 8 x \cot 4 x\right] d x
\]
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