The integral of secant cubed is a frequent and challenging 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:

Derivations

Integration by parts

This antiderivative may be found by integration by parts, as follows: where Then Next add to both sides: using the integral of the secant function, Finally, divide both sides by 2: which was to be derived. A possible mnemonic is: "The integral of secant cubed is the average of the derivative and integral of secant".

Reduction to an integral of a rational function

where u = \sin x, so that. 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

Hyperbolic functions

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.

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: 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.