Integrand size = 13, antiderivative size = 107 \[ \int x^6 \left (-1+x^3\right )^{2/3} \, dx=\frac {1}{81} \left (-1+x^3\right )^{2/3} \left (-4 x-3 x^4+9 x^7\right )-\frac {4 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{81 \sqrt {3}}+\frac {4}{243} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {2}{243} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.89, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {285, 327, 245} \[ \int x^6 \left (-1+x^3\right )^{2/3} \, dx=-\frac {4 \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{81 \sqrt {3}}-\frac {4}{81} \left (x^3-1\right )^{2/3} x+\frac {2}{81} \log \left (\sqrt [3]{x^3-1}-x\right )+\frac {1}{9} \left (x^3-1\right )^{2/3} x^7-\frac {1}{27} \left (x^3-1\right )^{2/3} x^4 \]
[In]
[Out]
Rule 245
Rule 285
Rule 327
Rubi steps \begin{align*} \text {integral}& = \frac {1}{9} x^7 \left (-1+x^3\right )^{2/3}-\frac {2}{9} \int \frac {x^6}{\sqrt [3]{-1+x^3}} \, dx \\ & = -\frac {1}{27} x^4 \left (-1+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (-1+x^3\right )^{2/3}-\frac {4}{27} \int \frac {x^3}{\sqrt [3]{-1+x^3}} \, dx \\ & = -\frac {4}{81} x \left (-1+x^3\right )^{2/3}-\frac {1}{27} x^4 \left (-1+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (-1+x^3\right )^{2/3}-\frac {4}{81} \int \frac {1}{\sqrt [3]{-1+x^3}} \, dx \\ & = -\frac {4}{81} x \left (-1+x^3\right )^{2/3}-\frac {1}{27} x^4 \left (-1+x^3\right )^{2/3}+\frac {1}{9} x^7 \left (-1+x^3\right )^{2/3}-\frac {4 \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {2}{81} \log \left (-x+\sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.95 \[ \int x^6 \left (-1+x^3\right )^{2/3} \, dx=\frac {1}{243} \left (3 x \left (-1+x^3\right )^{2/3} \left (-4-3 x^3+9 x^6\right )-4 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )+4 \log \left (-x+\sqrt [3]{-1+x^3}\right )-2 \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.31
method | result | size |
meijerg | \(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} x^{7} \operatorname {hypergeom}\left (\left [-\frac {2}{3}, \frac {7}{3}\right ], \left [\frac {10}{3}\right ], x^{3}\right )}{7 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}}}\) | \(33\) |
risch | \(\frac {x \left (9 x^{6}-3 x^{3}-4\right ) \left (x^{3}-1\right )^{\frac {2}{3}}}{81}-\frac {4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right )}{81 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}}}\) | \(54\) |
pseudoelliptic | \(\frac {-2 \ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )+4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right )+4 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )+\left (27 x^{7}-9 x^{4}-12 x \right ) \left (x^{3}-1\right )^{\frac {2}{3}}}{243 {\left (\left (x^{3}-1\right )^{\frac {2}{3}}+x \left (x +\left (x^{3}-1\right )^{\frac {1}{3}}\right )\right )}^{3} {\left (x -\left (x^{3}-1\right )^{\frac {1}{3}}\right )}^{3}}\) | \(130\) |
trager | \(\frac {x \left (9 x^{6}-3 x^{3}-4\right ) \left (x^{3}-1\right )^{\frac {2}{3}}}{81}+\frac {4 \ln \left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}-3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}-2 x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1\right )}{243}+\frac {4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}-x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1\right )}{243}\) | \(189\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.93 \[ \int x^6 \left (-1+x^3\right )^{2/3} \, dx=\frac {1}{81} \, {\left (9 \, x^{7} - 3 \, x^{4} - 4 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + \frac {4}{243} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {4}{243} \, \log \left (-\frac {x - {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {2}{243} \, \log \left (\frac {x^{2} + {\left (x^{3} - 1\right )}^{\frac {1}{3}} x + {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 3.58 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.32 \[ \int x^6 \left (-1+x^3\right )^{2/3} \, dx=- \frac {x^{7} e^{- \frac {i \pi }{3}} \Gamma \left (\frac {7}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {7}{3} \\ \frac {10}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {10}{3}\right )} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.36 \[ \int x^6 \left (-1+x^3\right )^{2/3} \, dx=\frac {4}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + \frac {11 \, {\left (x^{3} - 1\right )}^{\frac {5}{3}}}{x^{5}} - \frac {4 \, {\left (x^{3} - 1\right )}^{\frac {8}{3}}}{x^{8}}}{81 \, {\left (\frac {3 \, {\left (x^{3} - 1\right )}}{x^{3}} - \frac {3 \, {\left (x^{3} - 1\right )}^{2}}{x^{6}} + \frac {{\left (x^{3} - 1\right )}^{3}}{x^{9}} - 1\right )}} - \frac {2}{243} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {4}{243} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]
[In]
[Out]
\[ \int x^6 \left (-1+x^3\right )^{2/3} \, dx=\int { {\left (x^{3} - 1\right )}^{\frac {2}{3}} x^{6} \,d x } \]
[In]
[Out]
Timed out. \[ \int x^6 \left (-1+x^3\right )^{2/3} \, dx=\int x^6\,{\left (x^3-1\right )}^{2/3} \,d x \]
[In]
[Out]