The integral of secant cubed is one of the more challenging[1] indefinite integrals of elementary calculus:
There are a number of reasons why this particular antiderivative is worthy of special attention:
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This antiderivative may be found by integration by parts, as follows:
where
Then
Here we have assumed already known the integral of the secant function.
Next we add to both sides of the equality just derived:
Then divide both sides by 2:
Integrals of the form: can be reduced using the Pythagorean identity if n is even or n and m are both odd. If n is odd and m 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.
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.