Primitive of Reciprocal of a squared minus x squared squared/Partial Fraction Expansion

Lemma for Primitive of Reciprocal of $\left({a^2 - x^2}\right)^2$

 * $\dfrac 1 {\left({a^2 - x^2}\right)^2} \equiv \dfrac 1 {4 a^3 \left({a + x}\right)} + \dfrac 1 {4 a^3 \left({a - x}\right)} + \dfrac 1 {4 a^2 \left({a + x}\right)^2} + \dfrac 1 {4 a^2 \left({a - x}\right)^2}$

Proof
Setting $x = a$ in $(1)$:

Setting $x = -a$ in $(1)$:

Equating coefficients of $x^3$ in $(1)$:

Equating coefficients of $x^2$ in $(1)$:

Summarising:

Hence the result.