\[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^2+x^6}}{x^2 \left (1+x^2\right )} \, dx \]
Optimal antiderivative \[ \frac {2 \left (x^{6}+x^{2}\right )^{\frac {1}{4}}}{x}-\frac {\arctan \! \left (\frac {2^{\frac {3}{4}} x \left (x^{6}+x^{2}\right )^{\frac {1}{4}}}{\sqrt {2}\, x^{2}-\sqrt {x^{6}+x^{2}}}\right ) 2^{\frac {3}{4}}}{2}-\frac {\operatorname {arctanh}\! \left (\frac {\frac {x^{2} 2^{\frac {3}{4}}}{2}+\frac {\sqrt {x^{6}+x^{2}}\, 2^{\frac {1}{4}}}{2}}{x \left (x^{6}+x^{2}\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}}}{2} \]
command
Int[((-1 + x^2)*(x^2 + x^6)^(1/4))/(x^2*(1 + x^2)),x]
Rubi 4.17.3 under Mathematica 13.3.1 output
\[ \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^2+x^6}}{x^2 \left (1+x^2\right )} \, dx \]
Rubi 4.16.1 under Mathematica 13.3.1 output
\[ \frac {8 \sqrt [4]{x^6+x^2} x \operatorname {AppellF1}\left (\frac {3}{8},1,\frac {3}{4},\frac {11}{8},x^4,-x^4\right )}{3 \sqrt [4]{x^4+1}}-\frac {8 \sqrt [4]{x^6+x^2} x^3 \operatorname {AppellF1}\left (\frac {7}{8},1,\frac {3}{4},\frac {15}{8},x^4,-x^4\right )}{7 \sqrt [4]{x^4+1}}-\frac {4 \sqrt [4]{x^6+x^2} x \operatorname {Hypergeometric2F1}\left (\frac {3}{8},\frac {3}{4},\frac {11}{8},-x^4\right )}{3 \sqrt [4]{x^4+1}}+\frac {2 \sqrt [4]{x^6+x^2}}{x} \]