Integral of the secant function

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In calculus, the integral of the secant function can be evaluated using a variety of methods and there are multiple ways of expressing the antiderivative, all of which can be shown to be equivalent via trigonometric identities,

${\displaystyle \int \sec \theta \,d\theta ={\begin{cases}{\dfrac {1}{2}}\ln {\dfrac {1+\sin \theta }{1-\sin \theta }}+C\\[15mu]\ln {{\bigl |}\sec \theta +\tan \theta \,{\bigr |}}+C\\[15mu]\ln {\left|\,{\tan }{\biggl (}{\dfrac {\theta }{2}}+{\dfrac {\pi }{4}}{\biggr )}\right|}+C\end{cases}}}$

This formula is useful for evaluating various trigonometric integrals. In particular, it can be used to evaluate the integral of the secant cubed, which, though seemingly special, comes up rather frequently in applications.^[1]

The definite integral of the secant function starting from ${\displaystyle 0}$ is the inverse Gudermannian function, ${\textstyle \operatorname {gd} ^{-1}.}$ For numerical applications, all of the above expressions result in loss of significance for some arguments. An alternative expression in terms of the inverse hyperbolic sine arsinh is numerically well behaved for real arguments ${\textstyle |\phi |<{\tfrac {1}{2}}\pi }$:^[2]

${\displaystyle \operatorname {gd} ^{-1}\phi =\int _{0}^{\phi }\sec \theta \,d\theta =\operatorname {arsinh} (\tan \phi ).}$

The integral of the secant function was historically one of the first integrals of its type ever evaluated, before most of the development of integral calculus. It is important because it is the vertical coordinate of the Mercator projection, used for marine navigation with constant compass bearing.