Evaluate: `int1/(cotx+cos e c\ x)\ dx`

Explanation

`text { Let } I=int frac{1}{cos x+cosec x} d x `

`int frac{sin x}{1+cos x+sin x} d x `

`int frac{2 sin x}{2+2 cos x+sin x} d x`

`int frac{sin x+cos x+sin x-cos x}{2+2 cos x sin x} d x`

`int frac{sin x+cos x}{2+2 cos x sin x} d x+int frac{sin x-cos x}{2+2 cos x sin x} d x`

`int frac{sin x+cos x}{3-sin ^{2} x-cos ^{2} x+2 cos x sin x} d x+int frac{sin x-cos x}{1+sin ^{2} x+cos ^{2} x+2 cos x sin x} d x`

`int frac{sin x+cos x}{3-(sin x-cos x)^{2}} d x+int frac{sin x-cos x}{1+(sin x+cos x+)^{2}} d x`

where, `I_{1}=int frac{sin x+cos x}{3-(sin x-cos x)^{2}} d x` and `quad I_{2}=int frac{sin x-cos x}{1+(sin x+cos x+)^{2}} d x`

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