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: Derivations. Integration by parts. This antiderivative may be found by integration by parts, as follows: where Then Next add formula_7 to both sides of the equality just derived: given that the integral of the secant function is formula_9 Finally, divide both sides by 2: which was to be derived. Reduction to an integral of a rational function. where formula_12, so that formula_13. This admits a decomposition  by partial fractions: Antidifferentiating term by-term, one gets Hyperbolic functions. Integrals of the form: formula_16 can be reduced using the Pythagorean identity if formula_17 is even or formula_17 and formula_19 are both odd. If formula_17 is odd and formula_19 is even, hyperbolic substitutions can be used to replace the nested integration by parts with hyperbolic power reducing formulas. Note that formula_23 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.