Evaluate: `inttan^(-1)sqrt((1-x)/(1+x))\ dx`

Explanation

`I=tan ^{-1} sqrt{frac{1-x}{1+x}} d x`
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Let x=cos theta Rightarrow d x=-sin theta d theta`
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I=int tan ^{-1} sqrt{frac{1-cos theta}{1+cos theta}}(-sin theta d theta) `
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=-int tan ^{-1} sqrt{frac{2 sin ^{2} frac{theta}{2}}{2 cos ^{2} frac{theta}{2}}} sin theta d theta`
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=-int tan ^{-1} tan frac{theta}{2} cdot sin theta d theta`
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=-frac{1}{2} int theta cdot sin theta d theta`
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=-frac{1}{2}[theta cdot(-cos theta)-int 1 cdot(-cos theta) d theta] `
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=-frac{1}{2}[-theta cos theta+sin theta] `
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