The integral of secant cubed is a frequent and challenging[1] indefinite integral of elementary calculus: where
is the inverse Gudermannian function, the integral of the secant function.
There are a number of reasons why this particular antiderivative is worthy of special attention: This antiderivative may be found by integration by parts, as follows:[2] where Then Next add
to both sides:[a] using the integral of the secant function,
sec x + tan x
[2] Finally, divide both sides by 2: which was to be derived.
[2] A possible mnemonic is: "The integral of secant cubed is the average of the derivative and integral of secant".
This admits a decomposition by partial fractions: Antidifferentiating term-by-term, one gets Alternatively, one may use the tangent half-angle substitution for any rational function of trigonometric functions; for this particular integrand, that method leads to the integration of Integrals of the form:
can be reduced using the Pythagorean identity if
is even, hyperbolic substitutions can be used to replace the nested integration by parts with hyperbolic power-reducing formulas.
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: 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.