Primitive of x squared over x fourth plus a fourth/Partial Fraction Expansion

Lemma for Primitive of $\dfrac {x^2} {x^4 + a^4}$

 * $\dfrac {x^2} {x^4 + a^4} = \dfrac x {2 a \sqrt 2 \paren {x^2 - a x \sqrt 2 + a^2} } - \dfrac x {2 a \sqrt 2 \paren {x^2 + a x \sqrt 2 + a^2} }$

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

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

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

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

Summarising:

Thus: