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

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

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

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

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

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

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

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

Summarising:

Thus:

Hence the result.