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

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

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

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.