Primitive of Reciprocal of x cubed plus a cubed/Lemma

Lemma for Primitive of Reciprocal of $x^3 + a^3$

$\ds \int \frac {\d x} {x^2 - a x + a^2} = \frac 2 {a \sqrt 3} \map \arctan {\frac {2 x - a} {a \sqrt 3} }$

The discriminant of $x^2 - a x + a^2$ is:

$\ds \map {\mathrm {Disc} } {x^2 - a x + a^2}$

$=$

$\ds \paren {-a}^2 - 4 \times 1 \times a^2$

$\ds $

$=$

$\ds a^2 - 4 a^2$

$\ds $

$=$

$\ds -3 a^2$

$\ds $

$<$

$\ds 0$

Thus:

$\ds \int \frac {\d x} {x^2 - a x + a^2}$

$=$

$\ds \frac 2 {\sqrt {4 a^2 - \paren {-a}^2} } \map \arctan {\frac {2 x - a} {\sqrt {4 a^2 - \paren {-a}^2} } }$

$\ds $

$=$

$\ds \frac 2 {a \sqrt 3} \map \arctan {\frac {2 x - a} {a \sqrt 3} }$

simplifying

$\blacksquare$