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

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

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

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

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

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

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

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

Summarising:

Hence the result.