Originally Posted by

**Matt Westwood** THis gets me to:

$\displaystyle \frac {-2 \sin a x} {3 a \cos^3 a x} + \frac 1 {3 a} \tan a x + \frac 2 {3 a \cos^3 a x}$

from which I haven't got a clue as to how to get to:

$\displaystyle \frac 1 {2a} \tan \left({\frac \pi 4 + \frac {a x} 2}\right) + \frac 1 {6 a} \tan^3 \left({\frac \pi 4 + \frac {a x} 2}\right)$

Presumably there are half-angle substitutions to be used, but one supposes there's an easier way to get there without going through this admittedly perfectly legitimate solution.

However, the form of the solution suggests that there may be integrations of secants to be found, as:

$\displaystyle \int \sec a x = \tan \left({\frac \pi 4 + \frac {a x} 2}\right)$

but I haven't been able to find it.

The reason I am going through this particular exercise is specifically to validate the results in this source work. Some of them appear to be wrong, and I want to make sure I know precisely where the errors are before I can recommend it as a learning aid.