How do you find the integral of int 1/(1 + csc(x))?

$x + \sec x - \tan x + c$
$\int \frac{1}{1 + \csc x} \mathrm{dx} = \int \sin \frac{x}{1 + \sin x} \mathrm{dx} = \int \frac{\sin x - {\sin}^{2} x}{{\cos}^{2} x} \mathrm{dx}$
=$\int \tan x \sec x \mathrm{dx} - \int {\tan}^{2} x \mathrm{dx}$
=$\int \tan x \sec x \mathrm{dx} - \int \left({\sec}^{2} x - 1\right) \mathrm{dx}$
=$\sec x - \left(\tan x - x\right) + c$
=$x + \sec x - \tan x + c$