Primitive of Cosine of a x over Sine of a x minus Cosine of a x

Theorem

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

Proof
Let $u = \tan \dfrac {a x} 2$

Then:

Also see

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