$\cfrac{{{x^2}}}{{\sqrt {{x^6} + {a^2}} }}$

Let $I = \int {\cfrac{{{x^2}}}{{{x^6} + {a^6}}}dx = \int {\cfrac{{{x^2}dx}}{{\sqrt {{{\left( {{x^3}} \right)}^2} + {{\left( {{a^3}} \right)}^2}} }}} }$

Put ${x^3} = t$ $\Rightarrow$ $3{x^2}dx = dt$

$= \cfrac{1}{3}\int {\cfrac{{dt}}{{\sqrt {{t^2} + {{\left( {{a^3}} \right)}^2}} }} = \cfrac{1}{3}\log \left| {t + \sqrt {{t^2} + {a^6}} } \right| + C}$

$= \cfrac{1}{3}\log \left| {{x^3} + \sqrt {{x^6} + {a^6}} } \right| + C$