\[ \int \frac {x^3 \left (3+x^2\right )}{\left (1+x^2\right ) \sqrt [3]{1+x^2-x^3} \left (1+x^2+x^3\right )} \, dx \]
Optimal antiderivative \[ \sqrt {3}\, \arctan \! \left (\frac {\sqrt {3}\, x}{-x +2 \left (-x^{3}+x^{2}+1\right )^{\frac {1}{3}}}\right )-\frac {\sqrt {3}\, \arctan \! \left (\frac {\sqrt {3}\, x}{-x +2^{\frac {2}{3}} \left (-x^{3}+x^{2}+1\right )^{\frac {1}{3}}}\right ) 2^{\frac {2}{3}}}{2}+\ln \! \left (x +\left (-x^{3}+x^{2}+1\right )^{\frac {1}{3}}\right )-\frac {\ln \! \left (2 x +2^{\frac {2}{3}} \left (-x^{3}+x^{2}+1\right )^{\frac {1}{3}}\right ) 2^{\frac {2}{3}}}{2}-\frac {\ln \! \left (x^{2}-x \left (-x^{3}+x^{2}+1\right )^{\frac {1}{3}}+\left (-x^{3}+x^{2}+1\right )^{\frac {2}{3}}\right )}{2}+\frac {\ln \! \left (-2 x^{2}+2^{\frac {2}{3}} x \left (-x^{3}+x^{2}+1\right )^{\frac {1}{3}}-2^{\frac {1}{3}} \left (-x^{3}+x^{2}+1\right )^{\frac {2}{3}}\right ) 2^{\frac {2}{3}}}{4} \]
command
int(x^3*(x^2+3)/(x^2+1)/(-x^3+x^2+1)^(1/3)/(x^3+x^2+1),x)
Maple 2022.1 output
\[\int \frac {x^{3} \left (x^{2}+3\right )}{\left (x^{2}+1\right ) \left (-x^{3}+x^{2}+1\right )^{\frac {1}{3}} \left (x^{3}+x^{2}+1\right )}\, dx\]
Maple 2021.1 output
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(1845\) |