Integral of secant cubed


The integral of secant cubed is a frequent and challenging indefinite integral of elementary calculus:
There are a number of reasons why this particular antiderivative is worthy of special attention:

Integration by parts

This antiderivative may be found by integration by parts, as follows:
where
Then
Next add to both sides of the equality just derived:
given that the integral of the secant function is
Finally, divide both sides by 2:
which was to be derived.

Reduction to an integral of a rational function

where, so that. This admits a decomposition by partial fractions:
Antidifferentiating term by-term, one gets

Hyperbolic functions

Integrals of the form: can be reduced using the Pythagorean identity if is even or and are both odd. If is odd and is even, hyperbolic substitutions can be used to replace the nested integration by parts with hyperbolic power reducing formulas.
Note that follows directly from this substitution.

Higher odd powers of secant

Just as the integration by parts above reduced the integral of secant cubed to the integral of secant to the first power, so a similar process reduces the integral of higher odd powers of secant to lower ones. This is the secant reduction formula, which follows the syntax:
Alternatively:
Even powers of tangents can be accommodated by using binomial expansion to form an odd polynomial of secant and using these formulae on the largest term and combining like terms.