Primitive of Reciprocal of x squared minus a squared/Logarithm Form 1/size of x less than a

Theorem
Let $a \in \R_{>0}$ be a strictly positive real constant. Let $\size x < a$.

Then:
 * $\ds \int \frac {\d x} {x^2 - a^2} = \dfrac 1 {2 a} \map \ln {\dfrac {a - x} {a + x} } + C$

Proof
Let $\size x < a$.

Then: