Integrand size = 13, antiderivative size = 114 \[ \int x^{10} \sqrt [3]{-1+x^3} \, dx=\frac {1}{972} \sqrt [3]{-1+x^3} \left (-20 x^2-12 x^5-9 x^8+81 x^{11}\right )+\frac {10 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{243 \sqrt {3}}+\frac {10}{729} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {5}{729} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {285, 327, 337} \[ \int x^{10} \sqrt [3]{-1+x^3} \, dx=\frac {10 \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{243 \sqrt {3}}+\frac {5}{243} \log \left (x-\sqrt [3]{x^3-1}\right )+\frac {1}{12} \sqrt [3]{x^3-1} x^{11}-\frac {1}{108} \sqrt [3]{x^3-1} x^8-\frac {1}{81} \sqrt [3]{x^3-1} x^5-\frac {5}{243} \sqrt [3]{x^3-1} x^2 \]
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Rule 285
Rule 327
Rule 337
Rubi steps \begin{align*} \text {integral}& = \frac {1}{12} x^{11} \sqrt [3]{-1+x^3}-\frac {1}{12} \int \frac {x^{10}}{\left (-1+x^3\right )^{2/3}} \, dx \\ & = -\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}-\frac {2}{27} \int \frac {x^7}{\left (-1+x^3\right )^{2/3}} \, dx \\ & = -\frac {1}{81} x^5 \sqrt [3]{-1+x^3}-\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}-\frac {5}{81} \int \frac {x^4}{\left (-1+x^3\right )^{2/3}} \, dx \\ & = -\frac {5}{243} x^2 \sqrt [3]{-1+x^3}-\frac {1}{81} x^5 \sqrt [3]{-1+x^3}-\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}-\frac {10}{243} \int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx \\ & = -\frac {5}{243} x^2 \sqrt [3]{-1+x^3}-\frac {1}{81} x^5 \sqrt [3]{-1+x^3}-\frac {1}{108} x^8 \sqrt [3]{-1+x^3}+\frac {1}{12} x^{11} \sqrt [3]{-1+x^3}+\frac {10 \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{243 \sqrt {3}}+\frac {5}{243} \log \left (x-\sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96 \[ \int x^{10} \sqrt [3]{-1+x^3} \, dx=\frac {3 x^2 \sqrt [3]{-1+x^3} \left (-20-12 x^3-9 x^6+81 x^9\right )+40 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )+40 \log \left (-x+\sqrt [3]{-1+x^3}\right )-20 \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )}{2916} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.06 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.29
method | result | size |
meijerg | \(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} x^{11} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {11}{3}\right ], \left [\frac {14}{3}\right ], x^{3}\right )}{11 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}\) | \(33\) |
risch | \(\frac {x^{2} \left (81 x^{9}-9 x^{6}-12 x^{3}-20\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{972}-\frac {5 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{243 \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(63\) |
pseudoelliptic | \(\frac {-20 \ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )-40 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right )+40 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )+\left (243 x^{11}-27 x^{8}-36 x^{5}-60 x^{2}\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{2916 \left (x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )^{4} {\left (x -\left (x^{3}-1\right )^{\frac {1}{3}}\right )}^{4}}\) | \(138\) |
trager | \(\frac {x^{2} \left (81 x^{9}-9 x^{6}-12 x^{3}-20\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{972}+\frac {10 \ln \left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right )}{729}+\frac {10 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}-2 x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1\right )}{729}\) | \(198\) |
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Time = 0.25 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.93 \[ \int x^{10} \sqrt [3]{-1+x^3} \, dx=-\frac {10}{729} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{972} \, {\left (81 \, x^{11} - 9 \, x^{8} - 12 \, x^{5} - 20 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} + \frac {10}{729} \, \log \left (-\frac {x - {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {5}{729} \, \log \left (\frac {x^{2} + {\left (x^{3} - 1\right )}^{\frac {1}{3}} x + {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
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Result contains complex when optimal does not.
Time = 18.66 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.28 \[ \int x^{10} \sqrt [3]{-1+x^3} \, dx=\frac {x^{11} e^{\frac {i \pi }{3}} \Gamma \left (\frac {11}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {11}{3} \\ \frac {14}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {14}{3}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.49 \[ \int x^{10} \sqrt [3]{-1+x^3} \, dx=-\frac {10}{729} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {\frac {40 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {93 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}}}{x^{4}} - \frac {72 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}}}{x^{7}} + \frac {20 \, {\left (x^{3} - 1\right )}^{\frac {10}{3}}}{x^{10}}}{972 \, {\left (\frac {4 \, {\left (x^{3} - 1\right )}}{x^{3}} - \frac {6 \, {\left (x^{3} - 1\right )}^{2}}{x^{6}} + \frac {4 \, {\left (x^{3} - 1\right )}^{3}}{x^{9}} - \frac {{\left (x^{3} - 1\right )}^{4}}{x^{12}} - 1\right )}} - \frac {5}{729} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {10}{729} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]
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\[ \int x^{10} \sqrt [3]{-1+x^3} \, dx=\int { {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{10} \,d x } \]
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Timed out. \[ \int x^{10} \sqrt [3]{-1+x^3} \, dx=\int x^{10}\,{\left (x^3-1\right )}^{1/3} \,d x \]
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