\[ \int \frac {\sqrt [4]{x^2+x^6} \left (1+x^8\right )}{x^4 \left (-1+x^4\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \left (x^{4}+1\right ) \left (x^{6}+x^{2}\right )^{\frac {1}{4}}}{5 x^{3}}+\frac {2^{\frac {1}{4}} \arctan \! \left (\frac {2^{\frac {1}{4}} x}{\left (x^{6}+x^{2}\right )^{\frac {1}{4}}}\right )}{2}+\frac {\arctan \! \left (\frac {2^{\frac {3}{4}} x \left (x^{6}+x^{2}\right )^{\frac {1}{4}}}{\sqrt {2}\, x^{2}-\sqrt {x^{6}+x^{2}}}\right ) 2^{\frac {3}{4}}}{4}-\frac {2^{\frac {1}{4}} \operatorname {arctanh}\! \left (\frac {2^{\frac {1}{4}} x}{\left (x^{6}+x^{2}\right )^{\frac {1}{4}}}\right )}{2}+\frac {\operatorname {arctanh}\! \left (\frac {\frac {x^{2} 2^{\frac {3}{4}}}{2}+\frac {\sqrt {x^{6}+x^{2}}\, 2^{\frac {1}{4}}}{2}}{x \left (x^{6}+x^{2}\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}}}{4} \]
command
Int[((x^2 + x^6)^(1/4)*(1 + x^8))/(x^4*(-1 + x^4)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \text {\$Aborted} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {4 \sqrt [4]{x^6+x^2} \operatorname {AppellF1}\left (-\frac {5}{8},1,-\frac {1}{4},\frac {3}{8},x^4,-x^4\right )}{5 \sqrt [4]{x^4+1} x^3}+\frac {8 \sqrt [4]{x^6+x^2} x \operatorname {Hypergeometric2F1}\left (\frac {3}{8},\frac {3}{4},\frac {11}{8},-x^4\right )}{15 \sqrt [4]{x^4+1}}+\frac {2}{5} \sqrt [4]{x^6+x^2} x-\frac {2 \sqrt [4]{x^6+x^2}}{5 x^3} \]