The rearrangement technique is best illustrated through examples.
We can try expanding by the half angle formula:
Observe how the rearrangement we used led to a simpler expression that was easily integrable.
The denominator is of the form , where and are variable; but notice an important fact: is a constant. This should give us the required hint:
What we had to do in this question was therefore to realise that since is a constant, an introduction of the term in the numerator would lead to cancellations and simple ‘cot’ and ‘tan’ terms which can easily be integrated.
The numerator has a degree higher then the denominator which hints that some reduction of this rational expression is possible. This reduction can be accomplished if we somehow rearrange the numerator in such a way that it leads to a cancellation of common factors with the denominator; since the denominator is , we try to rearrange the numerator in terms of :