Your difficulty is understandable.
And we have the dreaded secant-cubed integral . . .
The formula can be derived by integrating-by-parts twice.
Or we can simply memorize the formula:
So we have: .
And we have: .
What I would have done after reaching
Rewrite it as
Since that is now an odd power of , multiply both numerator and denominator by to get
and make the substitution :
a rational integral. It can be done by partial fractions:
Eventually giving the same result Soroban got.