\[ \int \frac {\left (2+x^6\right ) \left (1-2 x^6+x^8+x^{12}\right )}{x^8 \sqrt [4]{-1+x^6} \left (-1+x^4+x^6\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \left (x^{6}-1\right )^{\frac {3}{4}} \left (3 x^{6}-7 x^{4}-3\right )}{21 x^{7}}-\sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, x \left (x^{6}-1\right )^{\frac {1}{4}}}{-x^{2}+\sqrt {x^{6}-1}}\right )-\sqrt {2}\, \arctanh \left (\frac {\sqrt {2}\, x \left (x^{6}-1\right )^{\frac {1}{4}}}{x^{2}+\sqrt {x^{6}-1}}\right ) \]
command
integrate((x^6+2)*(x^12+x^8-2*x^6+1)/x^8/(x^6-1)^(1/4)/(x^6+x^4-1),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \frac {84 \, \sqrt {2} x^{7} \arctan \left (-\frac {x^{12} + 2 \, x^{10} + x^{8} - 2 \, x^{6} - 2 \, x^{4} + 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{5} - x\right )} {\left (x^{6} - 1\right )}^{\frac {3}{4}} + 2 \, \sqrt {2} {\left (3 \, x^{9} - x^{7} - 3 \, x^{3}\right )} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{8} + x^{6} - x^{2}\right )} \sqrt {x^{6} - 1} - {\left (16 \, {\left (x^{6} - 1\right )}^{\frac {3}{4}} x^{5} + 2 \, \sqrt {2} {\left (x^{8} - 3 \, x^{6} - x^{2}\right )} \sqrt {x^{6} - 1} + \sqrt {2} {\left (x^{12} - 8 \, x^{10} - x^{8} - 2 \, x^{6} + 8 \, x^{4} + 1\right )} + 4 \, {\left (x^{9} + x^{7} - x^{3}\right )} {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{6} + x^{4} + 2 \, \sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} - 1} x^{2} + 2 \, \sqrt {2} {\left (x^{6} - 1\right )}^{\frac {3}{4}} x - 1}{x^{6} + x^{4} - 1}} + 1}{x^{12} - 14 \, x^{10} + x^{8} - 2 \, x^{6} + 14 \, x^{4} + 1}\right ) - 84 \, \sqrt {2} x^{7} \arctan \left (-\frac {x^{12} + 2 \, x^{10} + x^{8} - 2 \, x^{6} - 2 \, x^{4} - 2 \, \sqrt {2} {\left (x^{7} - 3 \, x^{5} - x\right )} {\left (x^{6} - 1\right )}^{\frac {3}{4}} - 2 \, \sqrt {2} {\left (3 \, x^{9} - x^{7} - 3 \, x^{3}\right )} {\left (x^{6} - 1\right )}^{\frac {1}{4}} + 4 \, {\left (x^{8} + x^{6} - x^{2}\right )} \sqrt {x^{6} - 1} - {\left (16 \, {\left (x^{6} - 1\right )}^{\frac {3}{4}} x^{5} - 2 \, \sqrt {2} {\left (x^{8} - 3 \, x^{6} - x^{2}\right )} \sqrt {x^{6} - 1} - \sqrt {2} {\left (x^{12} - 8 \, x^{10} - x^{8} - 2 \, x^{6} + 8 \, x^{4} + 1\right )} + 4 \, {\left (x^{9} + x^{7} - x^{3}\right )} {\left (x^{6} - 1\right )}^{\frac {1}{4}}\right )} \sqrt {\frac {x^{6} + x^{4} - 2 \, \sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} - 1} x^{2} - 2 \, \sqrt {2} {\left (x^{6} - 1\right )}^{\frac {3}{4}} x - 1}{x^{6} + x^{4} - 1}} + 1}{x^{12} - 14 \, x^{10} + x^{8} - 2 \, x^{6} + 14 \, x^{4} + 1}\right ) - 21 \, \sqrt {2} x^{7} \log \left (\frac {4 \, {\left (x^{6} + x^{4} + 2 \, \sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} - 1} x^{2} + 2 \, \sqrt {2} {\left (x^{6} - 1\right )}^{\frac {3}{4}} x - 1\right )}}{x^{6} + x^{4} - 1}\right ) + 21 \, \sqrt {2} x^{7} \log \left (\frac {4 \, {\left (x^{6} + x^{4} - 2 \, \sqrt {2} {\left (x^{6} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \, \sqrt {x^{6} - 1} x^{2} - 2 \, \sqrt {2} {\left (x^{6} - 1\right )}^{\frac {3}{4}} x - 1\right )}}{x^{6} + x^{4} - 1}\right ) + 8 \, {\left (3 \, x^{6} - 7 \, x^{4} - 3\right )} {\left (x^{6} - 1\right )}^{\frac {3}{4}}}{84 \, x^{7}} \]
Fricas 1.3.7 via sagemath 9.3 output
\[ \text {Timed out} \]