Integrand size = 11, antiderivative size = 104 \[ \int x \left (-1+x^6\right )^{2/3} \, dx=\frac {1}{6} x^2 \left (-1+x^6\right )^{2/3}-\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{3 \sqrt {3}}+\frac {1}{9} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )-\frac {1}{18} \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.66, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {281, 201, 245} \[ \int x \left (-1+x^6\right )^{2/3} \, dx=-\frac {\arctan \left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{6} \left (x^6-1\right )^{2/3} x^2+\frac {1}{6} \log \left (x^2-\sqrt [3]{x^6-1}\right ) \]
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Rule 201
Rule 245
Rule 281
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \left (-1+x^3\right )^{2/3} \, dx,x,x^2\right ) \\ & = \frac {1}{6} x^2 \left (-1+x^6\right )^{2/3}-\frac {1}{3} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,x^2\right ) \\ & = \frac {1}{6} x^2 \left (-1+x^6\right )^{2/3}-\frac {\arctan \left (\frac {1+\frac {2 x^2}{\sqrt [3]{-1+x^6}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{6} \log \left (x^2-\sqrt [3]{-1+x^6}\right ) \\ \end{align*}
Time = 0.47 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.96 \[ \int x \left (-1+x^6\right )^{2/3} \, dx=\frac {1}{18} \left (3 x^2 \left (-1+x^6\right )^{2/3}-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )+2 \log \left (-x^2+\sqrt [3]{-1+x^6}\right )-\log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.13 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.32
method | result | size |
meijerg | \(\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {2}{3}} x^{2} \operatorname {hypergeom}\left (\left [-\frac {2}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{6}\right )}{2 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {2}{3}}}\) | \(33\) |
risch | \(\frac {x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}}{6}-\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{6}\right )}{3 \operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\) | \(46\) |
pseudoelliptic | \(\frac {-3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}-2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{2}+2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x^{2}}\right )+\ln \left (\frac {x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{4}}\right )-2 \ln \left (\frac {-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}}{x^{2}}\right )}{18 \left (x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}\right ) \left (-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}\right )}\) | \(129\) |
trager | \(\frac {x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}}{6}+\frac {\ln \left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}-2 x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}+3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1\right )}{9}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}-2 x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+2\right )}{9}\) | \(200\) |
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Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int x \left (-1+x^6\right )^{2/3} \, dx=\frac {1}{6} \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} x^{2} + \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) + \frac {1}{9} \, \log \left (-\frac {x^{2} - {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) - \frac {1}{18} \, \log \left (\frac {x^{4} + {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}}\right ) \]
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Result contains complex when optimal does not.
Time = 0.62 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.33 \[ \int x \left (-1+x^6\right )^{2/3} \, dx=- \frac {x^{2} e^{- \frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {4}{3}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int x \left (-1+x^6\right )^{2/3} \, dx=\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) - \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{6 \, x^{4} {\left (\frac {x^{6} - 1}{x^{6}} - 1\right )}} - \frac {1}{18} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) + \frac {1}{9} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \]
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\[ \int x \left (-1+x^6\right )^{2/3} \, dx=\int { {\left (x^{6} - 1\right )}^{\frac {2}{3}} x \,d x } \]
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Timed out. \[ \int x \left (-1+x^6\right )^{2/3} \, dx=\int x\,{\left (x^6-1\right )}^{2/3} \,d x \]
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