Primitive of Reciprocal of Square of 1 plus Sine of a x

Theorem

 * $\displaystyle \int \frac {\mathrm d x} {\left({1 + \sin a x}\right)^2} = \frac {-1} {2a} \tan \left({\frac \pi 4 - \frac {a x} 2}\right) - \frac 1 {6 a} \tan^3 \left({\frac \pi 4 - \frac {a x} 2}\right) + C$

Proof
Let:

Thus:

Also see

 * Primitive of $\dfrac 1 {\left({1 + \cos a x}\right)^2}$