Integral of secant cubed

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The integral of secant cubed is a frequent and challenging^[1] indefinite integral of elementary calculus:

${\textstyle {\begin{aligned}\int \sec ^{3}x\,dx&={\tfrac {1}{2}}\sec x\tan x+{\tfrac {1}{2}}\int \sec x\,dx+C\\[6mu]&={\tfrac {1}{2}}(\sec x\tan x+\ln \left|\sec x+\tan x\right|)+C\\[6mu]&={\tfrac {1}{2}}(\sec x\tan x+\operatorname {gd} ^{-1}x)+C,\qquad |x|<{\tfrac {1}{2}}\pi \end{aligned}}}$

where ${\textstyle \operatorname {gd} ^{-1}}$ 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:

• The technique used for reducing integrals of higher odd powers of secant to lower ones is fully present in this, the simplest case. The other cases are done in the same way.
• The utility of hyperbolic functions in integration can be demonstrated in cases of odd powers of secant (powers of tangent can also be included).
• This is one of several integrals usually done in a first-year calculus course in which the most natural way to proceed involves integrating by parts and returning to the same integral one started with (another is the integral of the product of an exponential function with a sine or cosine function; yet another the integral of a power of the sine or cosine function).
• This integral is used in evaluating any integral of the form

${\displaystyle \int {\sqrt {a^{2}+x^{2}}}\,dx,}$

where ${\displaystyle a}$ is a constant. In particular, it appears in the problems of:

• rectifying the parabola and the Archimedean spiral
• finding the surface area of the helicoid.