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How to integrate this :

Problem :

$\int \frac{\sqrt{\cot x }-\sqrt{\tan x}}{\sqrt{2}(\cos x +\sin x)}dx$

My approach :

=$\int \frac{\cos x -\sin x}{\sqrt{2\sin x \cos x}(\cos x +\sin x)}dx$

Now what to do further as if we see derivative of $\cos x +\sin x$ is available in numerator, but $\sqrt{2\sin x\cos x}$ still left in denominator.

Please help. Thanks.

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Hint: $$\frac{\cos x -\sin x}{\sqrt{2\sin x \cos x}(\cos x +\sin x)}\times \frac{\cos x +\sin x}{\cos x +\sin x}=\frac{\cos2x}{\sqrt{\sin2x}(1+\sin2x)}$$ and let $\sin2x=u$

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