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

Lemma for Primitive of Reciprocal of $\paren {x^2 - a^2}^2$

 * $\dfrac 1 {\paren {x^2 - a^2}^2} \equiv \dfrac 1 {4 a^3 \paren {x + a} } - \dfrac 1 {4 a^3 \paren {x - a} } + \dfrac 1 {4 a^2 \paren {x + a}^2} + \dfrac 1 {4 a^2 \paren {x - a}^2}$

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.