\[ \int \frac {x^{12}}{\left (1+x^4\right )^{3/2}} \, dx \]
Optimal antiderivative \[ -\frac {x^{9}}{2 \sqrt {x^{4}+1}}-\frac {15 x \sqrt {x^{4}+1}}{14}+\frac {9 x^{5} \sqrt {x^{4}+1}}{14}+\frac {15 \left (x^{2}+1\right ) \sqrt {\frac {\cos \left (4 \arctan \left (x \right )\right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sin \left (2 \arctan \left (x \right )\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {\frac {x^{4}+1}{\left (x^{2}+1\right )^{2}}}}{28 \cos \left (2 \arctan \left (x \right )\right ) \sqrt {x^{4}+1}} \]
command
integrate(x^12/(x^4+1)^(3/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ -\frac {15 \, \sqrt {i} {\left (-i \, x^{4} - i\right )} {\rm ellipticF}\left (\frac {\sqrt {i}}{x}, -1\right ) - {\left (2 \, x^{9} - 6 \, x^{5} - 15 \, x\right )} \sqrt {x^{4} + 1}}{14 \, {\left (x^{4} + 1\right )}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ {\rm integral}\left (\frac {\sqrt {x^{4} + 1} x^{12}}{x^{8} + 2 \, x^{4} + 1}, x\right ) \]