Evaluate: `intcos{2cot^(-1)sqrt((1+x)/(1-x))}`

Explanation

Let `I=int cos {2 cot ^{-1} sqrt{frac{1+x}{1-x}}} d x`

Let `x=cos 2 theta`

On differentiating both sides, we get

`d x=-2 sin 2 theta d theta`

`therefore I=-int cos {2 cot ^{-1} sqrt{frac{1+cos 2 theta}{1-cos 2 theta}}} 2 sin 2 theta d theta`


`=-2 int cos {2 cot ^{-1} sqrt{frac{2 cos ^{2} theta}{2 sin ^{2} theta}}} 2 sin ^{2} theta d theta`


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