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Answer
$$ \displaystyle\int \dfrac{\sin\left(2x\right)-\cos\left(2x\right)}{{\left(\sin\left(2x\right)+\cos\left(2x\right)\right)}^{2}}\, \mathrm d x = {{1}\over{2\,\left(\sin \left(2\,x\right)+\cos \left(2\,x\right) \right)}} $$
Explanation
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