Integrand size = 13, antiderivative size = 117 \[ \int \frac {x^{19}}{\sqrt [3]{-1+x^6}} \, dx=\frac {1}{324} \left (-1+x^6\right )^{2/3} \left (28 x^2+21 x^8+18 x^{14}\right )+\frac {7 \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{81 \sqrt {3}}-\frac {7}{243} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+\frac {7}{486} \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {281, 327, 245} \[ \int \frac {x^{19}}{\sqrt [3]{-1+x^6}} \, dx=\frac {7 \arctan \left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {1}{18} \left (x^6-1\right )^{2/3} x^{14}+\frac {7}{108} \left (x^6-1\right )^{2/3} x^8+\frac {7}{81} \left (x^6-1\right )^{2/3} x^2-\frac {7}{162} \log \left (x^2-\sqrt [3]{x^6-1}\right ) \]
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Rule 245
Rule 281
Rule 327
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {x^9}{\sqrt [3]{-1+x^3}} \, dx,x,x^2\right ) \\ & = \frac {1}{18} x^{14} \left (-1+x^6\right )^{2/3}+\frac {7}{18} \text {Subst}\left (\int \frac {x^6}{\sqrt [3]{-1+x^3}} \, dx,x,x^2\right ) \\ & = \frac {7}{108} x^8 \left (-1+x^6\right )^{2/3}+\frac {1}{18} x^{14} \left (-1+x^6\right )^{2/3}+\frac {7}{27} \text {Subst}\left (\int \frac {x^3}{\sqrt [3]{-1+x^3}} \, dx,x,x^2\right ) \\ & = \frac {7}{81} x^2 \left (-1+x^6\right )^{2/3}+\frac {7}{108} x^8 \left (-1+x^6\right )^{2/3}+\frac {1}{18} x^{14} \left (-1+x^6\right )^{2/3}+\frac {7}{81} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,x^2\right ) \\ & = \frac {7}{81} x^2 \left (-1+x^6\right )^{2/3}+\frac {7}{108} x^8 \left (-1+x^6\right )^{2/3}+\frac {1}{18} x^{14} \left (-1+x^6\right )^{2/3}+\frac {7 \arctan \left (\frac {1+\frac {2 x^2}{\sqrt [3]{-1+x^6}}}{\sqrt {3}}\right )}{81 \sqrt {3}}-\frac {7}{162} \log \left (x^2-\sqrt [3]{-1+x^6}\right ) \\ \end{align*}
Time = 4.49 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.96 \[ \int \frac {x^{19}}{\sqrt [3]{-1+x^6}} \, dx=\frac {1}{972} \left (3 x^2 \left (-1+x^6\right )^{2/3} \left (28+21 x^6+18 x^{12}\right )+28 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )-28 \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+14 \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 2.22 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28
method | result | size |
meijerg | \(\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{20} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {10}{3}\right ], \left [\frac {13}{3}\right ], x^{6}\right )}{20 \operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\) | \(33\) |
risch | \(\frac {x^{2} \left (18 x^{12}+21 x^{6}+28\right ) \left (x^{6}-1\right )^{\frac {2}{3}}}{324}+\frac {7 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{6}\right )}{81 \operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\) | \(58\) |
pseudoelliptic | \(\frac {-54 \left (x^{6}-1\right )^{\frac {2}{3}} x^{14}-63 x^{8} \left (x^{6}-1\right )^{\frac {2}{3}}-84 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}+28 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{2}+2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x^{2}}\right )-14 \ln \left (\frac {x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{4}}\right )+28 \ln \left (\frac {-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}}{x^{2}}\right )}{972 \left (x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}\right )^{3} \left (-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}\right )^{3}}\) | \(155\) |
trager | \(\frac {x^{2} \left (18 x^{12}+21 x^{6}+28\right ) \left (x^{6}-1\right )^{\frac {2}{3}}}{324}+\frac {7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+4 x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}+3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-2\right )}{243}-\frac {7 \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1\right ) \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{243}+\frac {7 \ln \left (\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+x^{6}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-1\right )}{243}\) | \(338\) |
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Time = 0.27 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.91 \[ \int \frac {x^{19}}{\sqrt [3]{-1+x^6}} \, dx=\frac {1}{324} \, {\left (18 \, x^{14} + 21 \, x^{8} + 28 \, x^{2}\right )} {\left (x^{6} - 1\right )}^{\frac {2}{3}} - \frac {7}{243} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) - \frac {7}{243} \, \log \left (-\frac {x^{2} - {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) + \frac {7}{486} \, \log \left (\frac {x^{4} + {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}}\right ) \]
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Result contains complex when optimal does not.
Time = 6.63 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.29 \[ \int \frac {x^{19}}{\sqrt [3]{-1+x^6}} \, dx=- \frac {x^{20} e^{\frac {2 i \pi }{3}} \Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {13}{3}\right )} \]
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Time = 0.30 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.24 \[ \int \frac {x^{19}}{\sqrt [3]{-1+x^6}} \, dx=-\frac {7}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) - \frac {\frac {67 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} - \frac {77 \, {\left (x^{6} - 1\right )}^{\frac {5}{3}}}{x^{10}} + \frac {28 \, {\left (x^{6} - 1\right )}^{\frac {8}{3}}}{x^{16}}}{324 \, {\left (\frac {3 \, {\left (x^{6} - 1\right )}}{x^{6}} - \frac {3 \, {\left (x^{6} - 1\right )}^{2}}{x^{12}} + \frac {{\left (x^{6} - 1\right )}^{3}}{x^{18}} - 1\right )}} + \frac {7}{486} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) - \frac {7}{243} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \]
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\[ \int \frac {x^{19}}{\sqrt [3]{-1+x^6}} \, dx=\int { \frac {x^{19}}{{\left (x^{6} - 1\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {x^{19}}{\sqrt [3]{-1+x^6}} \, dx=\int \frac {x^{19}}{{\left (x^6-1\right )}^{1/3}} \,d x \]
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