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

Explanation

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

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

substitute the value of `sinx` and `cosx` in terms of `tan(x/2)`

`sinx=frac{2 tan x/2}{1+tan^2 x/2}`

`cosx=frac{1-frac{tan^2x}{2}}{1+frac{tan^2x}{2}}`

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

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

`I_1=log t=>log tan x/2`
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