\[ \int \frac {\sqrt [4]{-1+2 x^4} \left (-2+x^8\right )}{x^6 \left (-1+x^4\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\left (2 x^{4}-1\right )^{\frac {1}{4}} \left (69 x^{8}-56 x^{4}-8\right )}{20 x^{5} \left (x^{4}-1\right )}+\frac {15 \arctan \! \left (\frac {x}{\left (2 x^{4}-1\right )^{\frac {1}{4}}}\right )}{8}-\frac {15 \,\operatorname {arctanh}\! \left (\frac {x}{\left (2 x^{4}-1\right )^{\frac {1}{4}}}\right )}{8} \]
command
Int[((-1 + 2*x^4)^(1/4)*(-2 + x^8))/(x^6*(-1 + x^4)^2),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \text {\$Aborted} \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ -\frac {4 \sqrt [4]{2 x^4-1} x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},2 x^4,x^4\right )}{3 \sqrt [4]{1-2 x^4}}+2 \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4-1}}\right )-2 \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{2 x^4-1}}\right )-\frac {\sqrt [4]{2 x^4-1} x^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {3}{4},\frac {7}{4},\frac {x^4}{1-x^4}\right )}{3 \sqrt [4]{1-2 x^4} \left (1-x^4\right )^{3/4}}+\frac {4 \sqrt [4]{2 x^4-1}}{x}-\frac {2 \left (2 x^4-1\right )^{5/4}}{5 x^5} \]