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

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

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

Proof
Setting $x = 0$ in $(1)$:

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

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

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

Equating coefficients of $x^4$ in $(1)$:

Summarising:

Hence the result.