Integrand size = 13, antiderivative size = 109 \[ \int x^7 \sqrt [3]{-1+x^3} \, dx=\frac {1}{162} \sqrt [3]{-1+x^3} \left (-5 x^2-3 x^5+18 x^8\right )+\frac {5 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{81 \sqrt {3}}+\frac {5}{243} \log \left (-x+\sqrt [3]{-1+x^3}\right )-\frac {5}{486} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 97, 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, 337} \[ \int x^7 \sqrt [3]{-1+x^3} \, dx=\frac {5 \arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {5}{162} \log \left (x-\sqrt [3]{x^3-1}\right )+\frac {1}{9} \sqrt [3]{x^3-1} x^8-\frac {1}{54} \sqrt [3]{x^3-1} x^5-\frac {5}{162} \sqrt [3]{x^3-1} x^2 \]
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Rule 285
Rule 327
Rule 337
Rubi steps \begin{align*} \text {integral}& = \frac {1}{9} x^8 \sqrt [3]{-1+x^3}-\frac {1}{9} \int \frac {x^7}{\left (-1+x^3\right )^{2/3}} \, dx \\ & = -\frac {1}{54} x^5 \sqrt [3]{-1+x^3}+\frac {1}{9} x^8 \sqrt [3]{-1+x^3}-\frac {5}{54} \int \frac {x^4}{\left (-1+x^3\right )^{2/3}} \, dx \\ & = -\frac {5}{162} x^2 \sqrt [3]{-1+x^3}-\frac {1}{54} x^5 \sqrt [3]{-1+x^3}+\frac {1}{9} x^8 \sqrt [3]{-1+x^3}-\frac {5}{81} \int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx \\ & = -\frac {5}{162} x^2 \sqrt [3]{-1+x^3}-\frac {1}{54} x^5 \sqrt [3]{-1+x^3}+\frac {1}{9} x^8 \sqrt [3]{-1+x^3}+\frac {5 \arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {5}{162} \log \left (x-\sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95 \[ \int x^7 \sqrt [3]{-1+x^3} \, dx=\frac {1}{486} \left (3 x^2 \sqrt [3]{-1+x^3} \left (-5-3 x^3+18 x^6\right )+10 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )+10 \log \left (-x+\sqrt [3]{-1+x^3}\right )-5 \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.98 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.30
method | result | size |
meijerg | \(\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} x^{8} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {8}{3}\right ], \left [\frac {11}{3}\right ], x^{3}\right )}{8 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}}}\) | \(33\) |
risch | \(\frac {x^{2} \left (18 x^{6}-3 x^{3}-5\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{162}-\frac {5 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{162 \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(58\) |
pseudoelliptic | \(\frac {-5 \ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right )-10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right )+10 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right )+\left (54 x^{8}-9 x^{5}-15 x^{2}\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{486 \left (x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}\right )^{3} {\left (x -\left (x^{3}-1\right )^{\frac {1}{3}}\right )}^{3}}\) | \(133\) |
trager | \(\frac {x^{2} \left (18 x^{6}-3 x^{3}-5\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{162}+\frac {5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1\right )}{243}-\frac {5 \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+4 x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )}{243}-\frac {5 \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}-1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+4 x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2\right )}{243}\) | \(320\) |
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Time = 0.25 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.93 \[ \int x^7 \sqrt [3]{-1+x^3} \, dx=-\frac {5}{243} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{162} \, {\left (18 \, x^{8} - 3 \, x^{5} - 5 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} + \frac {5}{243} \, \log \left (-\frac {x - {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {5}{486} \, \log \left (\frac {x^{2} + {\left (x^{3} - 1\right )}^{\frac {1}{3}} x + {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
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Result contains complex when optimal does not.
Time = 3.56 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.33 \[ \int x^7 \sqrt [3]{-1+x^3} \, dx=- \frac {x^{8} e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 \Gamma \left (\frac {11}{3}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.33 \[ \int x^7 \sqrt [3]{-1+x^3} \, dx=-\frac {5}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {\frac {10 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {13 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}}}{x^{4}} - \frac {5 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}}}{x^{7}}}{162 \, {\left (\frac {3 \, {\left (x^{3} - 1\right )}}{x^{3}} - \frac {3 \, {\left (x^{3} - 1\right )}^{2}}{x^{6}} + \frac {{\left (x^{3} - 1\right )}^{3}}{x^{9}} - 1\right )}} - \frac {5}{486} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {5}{243} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]
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\[ \int x^7 \sqrt [3]{-1+x^3} \, dx=\int { {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{7} \,d x } \]
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Timed out. \[ \int x^7 \sqrt [3]{-1+x^3} \, dx=\int x^7\,{\left (x^3-1\right )}^{1/3} \,d x \]
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