Primitive of Reciprocal of x fourth plus a fourth/Partial Fraction Expansion

Lemma for Primitive of Reciprocal of $x^4 + a^4$

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

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: