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

Theorem

 * $\displaystyle \int \frac {\d x} {\cos a x \paren {1 - \sin a x} } = \frac 1 {2 a \paren {1 - \sin a x} } + \frac 1 {2 a} \ln \size {\map \tan {\frac {a x} 2 + \frac \pi 4} } + C$

Proof
Let:

Then:

Also see

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


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