\[ \int \frac {x}{(1+x) \sqrt [3]{1-x^3}} \, dx \]
Optimal antiderivative \[ \frac {\ln \left (\left (1-x \right ) \left (1+x \right )^{2}\right ) 2^{\frac {2}{3}}}{8}+\frac {\ln \left (x +\left (-x^{3}+1\right )^{\frac {1}{3}}\right )}{2}-\frac {3 \ln \left (-1+x +2^{\frac {2}{3}} \left (-x^{3}+1\right )^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{8}-\frac {\arctan \left (\frac {\left (1-\frac {2 x}{\left (-x^{3}+1\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}}{3}+\frac {\arctan \left (\frac {\left (1+\frac {2^{\frac {1}{3}} \left (1-x \right )}{\left (-x^{3}+1\right )^{\frac {1}{3}}}\right ) \sqrt {3}}{3}\right ) \sqrt {3}\, 2^{\frac {2}{3}}}{4} \]
command
int(x/(1+x)/(-x^3+1)^(1/3),x,method=_RETURNVERBOSE)
Maple 2022.1 output
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(1790\) |
Maple 2021.1 output
\[ \int \frac {x}{\left (x +1\right ) \left (-x^{3}+1\right )^{\frac {1}{3}}}\, dx \]________________________________________________________________________________________