# Primitive of Arcsecant of x over a

## Theorem

### Formulation 1

$\ds \int \arcsec \frac x a \rd x = \begin {cases} x \arcsec \dfrac x a - a \map \ln {x + \sqrt {x^2 - a^2} } + C & : 0 < \arcsec \dfrac x a < \dfrac \pi 2 \\ x \arcsec \dfrac x a + a \map \ln {x + \sqrt {x^2 - a^2} } + C & : \dfrac \pi 2 < \arcsec \dfrac x a < \pi \\ \end {cases}$

### Formulation 2

$\ds \int \arcsec \frac x a \rd x = x \arcsec \frac x a - a \ln \size {x + \sqrt {x^2 - a^2} } + C$

for $x^2 > 1$.

$\arcsec \dfrac x a$ is undefined on the real numbers for $x^2 < 1$.