\[ \int \frac {(1-2 x) \left (1-x^3\right )^{2/3}}{\left (1-x+x^2\right )^2} \, dx \]
Optimal antiderivative \[ \frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}-x +1}+\frac {\ln \left (2^{\frac {1}{3}}-\left (-x^{3}+1\right )^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{2}-\frac {\ln \left (-2^{\frac {1}{3}} x -\left (-x^{3}+1\right )^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{2}+\ln \left (x +\left (-x^{3}+1\right )^{\frac {1}{3}}\right )-\frac {2 \arctan \left (\frac {\left (1-\frac {2 x}{\left (-x^{3}+1\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}+\frac {\arctan \left (\frac {\left (1-\frac {2 \,2^{\frac {1}{3}} x}{\left (-x^{3}+1\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) 2^{\frac {2}{3}} \sqrt {3}}{3}+\frac {\arctan \left (\frac {\left (1+2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right ) 2^{\frac {2}{3}} \sqrt {3}}{3} \]
command
int((1-2*x)*(-x^3+1)^(2/3)/(x^2-x+1)^2,x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| trager | \(\frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}-x +1}+\frac {\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{6}+432\right )^{4} x^{2}+\RootOf \left (\textit {\_Z}^{6}+432\right )^{4} x -\RootOf \left (\textit {\_Z}^{6}+432\right )^{4}+12 \RootOf \left (\textit {\_Z}^{6}+432\right )^{2} \left (-x^{3}+1\right )^{\frac {1}{3}} x +72 \left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}-x +1}\right ) \RootOf \left (\textit {\_Z}^{6}+432\right )^{4}}{72}+\frac {\ln \left (-\frac {\RootOf \left (\textit {\_Z}^{6}+432\right )^{4} x^{2}+\RootOf \left (\textit {\_Z}^{6}+432\right )^{4} x -\RootOf \left (\textit {\_Z}^{6}+432\right )^{4}+12 \RootOf \left (\textit {\_Z}^{6}+432\right )^{2} \left (-x^{3}+1\right )^{\frac {1}{3}} x +72 \left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}-x +1}\right ) \RootOf \left (\textit {\_Z}^{6}+432\right )}{6}+\frac {\RootOf \left (\textit {\_Z}^{6}+432\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{6}+432\right )^{5} \left (-x^{3}+1\right )^{\frac {1}{3}} x -\RootOf \left (\textit {\_Z}^{6}+432\right )^{4} x^{2}-\RootOf \left (\textit {\_Z}^{6}+432\right )^{4} x +\RootOf \left (\textit {\_Z}^{6}+432\right )^{4}-12 \RootOf \left (\textit {\_Z}^{6}+432\right )^{2} \left (-x^{3}+1\right )^{\frac {1}{3}} x +36 \RootOf \left (\textit {\_Z}^{6}+432\right ) x^{2}+36 \RootOf \left (\textit {\_Z}^{6}+432\right ) x +144 \left (-x^{3}+1\right )^{\frac {2}{3}}-36 \RootOf \left (\textit {\_Z}^{6}+432\right )}{x^{2}-x +1}\right )}{3}-\frac {\ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}-x \left (-x^{3}+1\right )^{\frac {1}{3}}+x^{2}\right ) \RootOf \left (\textit {\_Z}^{6}+432\right )^{3}}{12}-\ln \left (\left (-x^{3}+1\right )^{\frac {2}{3}}-x \left (-x^{3}+1\right )^{\frac {1}{3}}+x^{2}\right )+\frac {\RootOf \left (\textit {\_Z}^{6}+432\right )^{3} \ln \left (-\RootOf \left (\textit {\_Z}^{6}+432\right )^{6} x^{3}+24 \RootOf \left (\textit {\_Z}^{6}+432\right )^{3}+1728 x \left (-x^{3}+1\right )^{\frac {2}{3}}-1728 x^{2} \left (-x^{3}+1\right )^{\frac {1}{3}}+1296 x^{3}-864\right )}{18}\) | \(457\) |
| risch | \(\text {Expression too large to display}\) | \(1033\) |
Maple 2021.1 output
\[ \int \frac {\left (-2 x +1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{\left (x^{2}-x +1\right )^{2}}\, dx \]________________________________________________________________________________________