Primitive of Reciprocal of Cosine of a x by 1 minus Sine of a x

Theorem

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

Proof
Let:

Then:

Also see

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


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