4: If you decided to mimic the reduction formula deduction, you probably did this
to prepare the integral for integration by parts. This leads to an elementary integral, then back substitution, a little algebraic simplification and you got the answer.
If you decided to use the standard approach - the tangent substitution - you get
This integral is best approached via trig identities (if you did not remember the reduction formula for the integral we started with, you probably do not remember the reduction formula for a power of cosine). After than you do the back substitution.