\[ \int \frac {x^2}{\left (1+x^4\right ) \sqrt [4]{-x^2+x^6}} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {1+\sqrt {2}}\, \arctan \left (\frac {\sqrt {2-\sqrt {2}}\, x}{-x \sqrt {2+\sqrt {2}}+2^{\frac {3}{4}} \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}\right )}{8}-\frac {\sqrt {1+\sqrt {2}}\, \arctan \left (\frac {\sqrt {2-\sqrt {2}}\, x}{x \sqrt {2+\sqrt {2}}+2^{\frac {3}{4}} \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}\right )}{8}+\frac {\sqrt {\sqrt {2}-1}\, \arctan \left (\frac {2^{\frac {3}{4}} \sqrt {2+\sqrt {2}}\, x \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}{-2 x^{2}+\sqrt {2}\, \sqrt {x^{6}-x^{2}}}\right )}{8}-\frac {\sqrt {1+\sqrt {2}}\, \arctanh \left (\frac {\frac {2^{\frac {1}{4}} x^{2}}{\sqrt {2-\sqrt {2}}}+\frac {\sqrt {x^{6}-x^{2}}\, 2^{\frac {3}{4}}}{2 \sqrt {2-\sqrt {2}}}}{x \left (x^{6}-x^{2}\right )^{\frac {1}{4}}}\right )}{8}-\frac {\sqrt {\sqrt {2}-1}\, \ln \left (-2 x^{2}+2^{\frac {3}{4}} \sqrt {2+\sqrt {2}}\, x \left (x^{6}-x^{2}\right )^{\frac {1}{4}}-\sqrt {2}\, \sqrt {x^{6}-x^{2}}\right )}{16}+\frac {\sqrt {\sqrt {2}-1}\, \ln \left (2 \sqrt {2-\sqrt {2}}\, x^{2}+2 \,2^{\frac {1}{4}} x \left (x^{6}-x^{2}\right )^{\frac {1}{4}}+\sqrt {4-2 \sqrt {2}}\, \sqrt {x^{6}-x^{2}}\right )}{16} \]
command
integrate(x^2/(x^4+1)/(x^6-x^2)^(1/4),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ \text {Timed out} \]