Primitive of x over 1 plus Cosine of a x

Theorem

 * $\ds \int \frac {x \rd x} {1 + \cos a x} = \frac x a \tan \frac {a x} 2 + \frac 2 {a^2} \ln \size {\cos \frac {a x} 2} + C$

Proof
With a view to expressing the primitive in the form:
 * $\ds \int u \frac {\d v} {\d x} \rd x = u v - \int v \frac {\d u} {\d x} \rd x$

let:

and let:

Then:

Also see

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