\[ \int \frac {1-x^3+x^6}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx \]
Optimal antiderivative \[ -\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{-x +2 \left (x^{4}+x^{2}\right )^{\frac {1}{3}}}\right )}{6}-\frac {\arctan \left (\frac {\sqrt {3}\, x}{x +2 \left (x^{4}+x^{2}\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{-x +2^{\frac {2}{3}} \left (x^{4}+x^{2}\right )^{\frac {1}{3}}}\right ) 2^{\frac {2}{3}}}{8}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, x}{x +2^{\frac {2}{3}} \left (x^{4}+x^{2}\right )^{\frac {1}{3}}}\right ) 2^{\frac {2}{3}}}{24}+\frac {\ln \left (-x +\left (x^{4}+x^{2}\right )^{\frac {1}{3}}\right )}{2}-\frac {\ln \left (x +\left (x^{4}+x^{2}\right )^{\frac {1}{3}}\right )}{6}+\frac {\ln \left (-2 x +2^{\frac {2}{3}} \left (x^{4}+x^{2}\right )^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{24}-\frac {\ln \left (2 x +2^{\frac {2}{3}} \left (x^{4}+x^{2}\right )^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{8}+\frac {\ln \left (x^{2}-x \left (x^{4}+x^{2}\right )^{\frac {1}{3}}+\left (x^{4}+x^{2}\right )^{\frac {2}{3}}\right )}{12}-\frac {\ln \left (x^{2}+x \left (x^{4}+x^{2}\right )^{\frac {1}{3}}+\left (x^{4}+x^{2}\right )^{\frac {2}{3}}\right )}{4}+\frac {\ln \left (-2 x^{2}+2^{\frac {2}{3}} x \left (x^{4}+x^{2}\right )^{\frac {1}{3}}-2^{\frac {1}{3}} \left (x^{4}+x^{2}\right )^{\frac {2}{3}}\right ) 2^{\frac {2}{3}}}{16}-\frac {\ln \left (2 x^{2}+2^{\frac {2}{3}} x \left (x^{4}+x^{2}\right )^{\frac {1}{3}}+2^{\frac {1}{3}} \left (x^{4}+x^{2}\right )^{\frac {2}{3}}\right ) 2^{\frac {2}{3}}}{48} \]
command
int((x^6-x^3+1)/(x^4+x^2)^(1/3)/(x^6-1),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(15564\) |
Maple 2021.1 output
\[\int \frac {x^{6}-x^{3}+1}{\left (x^{4}+x^{2}\right )^{\frac {1}{3}} \left (x^{6}-1\right )}\, dx\]________________________________________________________________________________________