`int(1+sinx)/(sinx(1+cosx))dx`

Explanation

`int(1+sinx)/(sinx(1+cosx))dx`

`=int frac{1}{sinx(1+cosx)}dx+int frac{1}{1+2cos^2x-1}dx`

`sinx=frac{2 tan(x/2)}{1+tan^2(x/2)}` and `cosx=frac{1-tan^2(x/2)}{1+tan^2(x/2)}`

`I_1=int frac{1}{frac{2 tan(x/2)}{1+tan^2 (x/2)}(1+frac{1-tan^2(x/2)}{1+tan^2(x/2)})}dx`

`I_1=int frac{sec^2(x/2)}{2 tan (x/2)(1+tan^2(x/2)+1-tan^2(x/2))}dx`

`tan (x/2)=t=>1/2 sec^2(x/2)dx=dt`

`I_1=int frac{dt}{2t}`

`=1/2 log t=>1/2 log tan (x/2)`
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