Primitive of Reciprocal of 1 plus Sine of a x

Theorem

 * $\displaystyle \int \frac {\d x} {1 + \sin a x} = -\frac 1 a \tan \left({\frac \pi 4 - \frac {a x} 2}\right) + C$

Proof
Then:

Also see

 * Primitive of $\dfrac 1 {1 + \cos a x}$