\[ \int \frac {\left (1+2 x^8\right ) \sqrt [4]{-1-2 x^4+2 x^8} \left (1-3 x^8+4 x^{16}\right )}{x^{10} \left (-1+2 x^8\right )} \, dx \]
Optimal antiderivative \[ \frac {\left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}} \left (20 x^{16}-4 x^{12}+9 x^{8}+2 x^{4}+5\right )}{45 x^{9}}-\frac {\arctan \left (\frac {2^{\frac {3}{4}} x \left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}}}{x^{2} \sqrt {2}-\sqrt {2 x^{8}-2 x^{4}-1}}\right ) 2^{\frac {3}{4}}}{4}-\frac {\arctanh \left (\frac {2 \,2^{\frac {1}{4}} x \left (2 x^{8}-2 x^{4}-1\right )^{\frac {1}{4}}}{2 x^{2}+\sqrt {2}\, \sqrt {2 x^{8}-2 x^{4}-1}}\right ) 2^{\frac {3}{4}}}{4} \]
command
Integrate[((1 + 2*x^8)*(-1 - 2*x^4 + 2*x^8)^(1/4)*(1 - 3*x^8 + 4*x^16))/(x^10*(-1 + 2*x^8)),x]
Mathematica 13.1 output
\[ \frac {1}{180} \left (\frac {4 \sqrt [4]{-1-2 x^4+2 x^8} \left (5+2 x^4+9 x^8-4 x^{12}+20 x^{16}\right )}{x^9}-45\ 2^{3/4} \text {ArcTan}\left (\frac {2^{3/4} x \sqrt [4]{-1-2 x^4+2 x^8}}{\sqrt {2} x^2-\sqrt {-1-2 x^4+2 x^8}}\right )-45\ 2^{3/4} \tanh ^{-1}\left (\frac {2 x \sqrt [4]{-2-4 x^4+4 x^8}}{2 x^2+\sqrt {-2-4 x^4+4 x^8}}\right )\right ) \]
Mathematica 12.3 output
\[ \int \frac {\left (1+2 x^8\right ) \sqrt [4]{-1-2 x^4+2 x^8} \left (1-3 x^8+4 x^{16}\right )}{x^{10} \left (-1+2 x^8\right )} \, dx \]________________________________________________________________________________________