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

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

 * $\dfrac 1 {x^2 \paren {a^2 - x^2}^2} \equiv \dfrac 1 {a^4 x^2} + \dfrac 3 {4 a^5 \paren {a + x} } - \dfrac 3 {4 a^5 \paren {a - x} } + \dfrac 1 {4 a^4 \paren {a + x}^2} + \dfrac 1 {4 a^4 \paren {a - x}^2}$

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

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

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

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

Equating coefficients of $x^5$ in $(1)$:

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

Summarising:

Hence the result.