Integrand size = 11, antiderivative size = 88 \[ \int \frac {x}{\sqrt [3]{-1+x^6}} \, dx=\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+\frac {1}{12} \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.60, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {281, 245} \[ \int \frac {x}{\sqrt [3]{-1+x^6}} \, dx=\frac {\arctan \left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} \log \left (x^2-\sqrt [3]{x^6-1}\right ) \]
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Rule 245
Rule 281
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^3}} \, dx,x,x^2\right ) \\ & = \frac {\arctan \left (\frac {1+\frac {2 x^2}{\sqrt [3]{-1+x^6}}}{\sqrt {3}}\right )}{2 \sqrt {3}}-\frac {1}{4} \log \left (x^2-\sqrt [3]{-1+x^6}\right ) \\ \end{align*}
Time = 0.43 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.95 \[ \int \frac {x}{\sqrt [3]{-1+x^6}} \, dx=\frac {1}{12} \left (2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )-2 \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+\log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 3.97 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.38
method | result | size |
meijerg | \(\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{6}\right )}{2 \operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}}}\) | \(33\) |
pseudoelliptic | \(-\frac {\ln \left (\frac {-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}}{x^{2}}\right )}{6}+\frac {\ln \left (\frac {x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{4}}\right )}{12}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{2}+2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x^{2}}\right )}{6}\) | \(78\) |
trager | \(-\frac {\ln \left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}-2 x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-3 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}+3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1\right )}{6}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{6}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{6}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}-x^{6}+3 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1\right )}{6}\) | \(190\) |
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Time = 0.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.93 \[ \int \frac {x}{\sqrt [3]{-1+x^6}} \, dx=-\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2} + 2 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{3 \, x^{2}}\right ) - \frac {1}{6} \, \log \left (-\frac {x^{2} - {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}}\right ) + \frac {1}{12} \, \log \left (\frac {x^{4} + {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{2} + {\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}}\right ) \]
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Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.35 \[ \int \frac {x}{\sqrt [3]{-1+x^6}} \, dx=\frac {x^{2} e^{- \frac {i \pi }{3}} \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {4}{3}\right )} \]
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Time = 0.31 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.78 \[ \int \frac {x}{\sqrt [3]{-1+x^6}} \, dx=-\frac {1}{6} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) + \frac {1}{12} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) - \frac {1}{6} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \]
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\[ \int \frac {x}{\sqrt [3]{-1+x^6}} \, dx=\int { \frac {x}{{\left (x^{6} - 1\right )}^{\frac {1}{3}}} \,d x } \]
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Timed out. \[ \int \frac {x}{\sqrt [3]{-1+x^6}} \, dx=\int \frac {x}{{\left (x^6-1\right )}^{1/3}} \,d x \]
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