Primitive of Reciprocal of a squared minus x squared/Inverse Hyperbolic Function Form

Theorem
Let $a \in \R_{>0}$ be a strictly positive real constant.
 * $\ds \int \dfrac {\d x} {a^2 - x^2} = \begin {cases} \dfrac 1 a \tanh^{-1} \dfrac x a + C & : \size x < a \\

& \\ \dfrac 1 a \coth^{-1} \dfrac x a + C & : \size x > a \\ & \\ \text {undefined} & : x = a \end {cases}$

Proof
First note that if $x = a$ then $a^2 - x^2 = 0$ and so $\dfrac 1 {a^2 - x^2}$ is undefined.

Proof

 * : Chapter $\text {III}$: Integration: Integration