Integrand size = 18, antiderivative size = 109 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^3} \, dx=\frac {\left (-3+x^6\right ) \sqrt [3]{-1+x^6}}{6 x^2}-\frac {\arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )}{3 \sqrt {3}}-\frac {1}{9} \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+\frac {1}{18} \log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.78, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {464, 281, 285, 337} \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^3} \, dx=-\frac {\arctan \left (\frac {\frac {2 x^2}{\sqrt [3]{x^6-1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{3} \sqrt [3]{x^6-1} x^4+\frac {\left (x^6-1\right )^{4/3}}{2 x^2}-\frac {1}{6} \log \left (x^2-\sqrt [3]{x^6-1}\right ) \]
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Rule 281
Rule 285
Rule 337
Rule 464
Rubi steps \begin{align*} \text {integral}& = \frac {\left (-1+x^6\right )^{4/3}}{2 x^2}-2 \int x^3 \sqrt [3]{-1+x^6} \, dx \\ & = \frac {\left (-1+x^6\right )^{4/3}}{2 x^2}-\text {Subst}\left (\int x \sqrt [3]{-1+x^3} \, dx,x,x^2\right ) \\ & = -\frac {1}{3} x^4 \sqrt [3]{-1+x^6}+\frac {\left (-1+x^6\right )^{4/3}}{2 x^2}+\frac {1}{3} \text {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^2\right ) \\ & = -\frac {1}{3} x^4 \sqrt [3]{-1+x^6}+\frac {\left (-1+x^6\right )^{4/3}}{2 x^2}-\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 = 0.40 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^3} \, dx=\frac {1}{18} \left (\frac {3 \left (-3+x^6\right ) \sqrt [3]{-1+x^6}}{x^2}-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^2}{x^2+2 \sqrt [3]{-1+x^6}}\right )-2 \log \left (-x^2+\sqrt [3]{-1+x^6}\right )+\log \left (x^4+x^2 \sqrt [3]{-1+x^6}+\left (-1+x^6\right )^{2/3}\right )\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.10 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.51
method | result | size |
risch | \(\frac {x^{12}-4 x^{6}+3}{6 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}}+\frac {{\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {2}{3}} x^{4} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{6}\right )}{6 \operatorname {signum}\left (x^{6}-1\right )^{\frac {2}{3}}}\) | \(56\) |
meijerg | \(\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} x^{4} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{6}\right )}{4 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}}}-\frac {\operatorname {signum}\left (x^{6}-1\right )^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], x^{6}\right )}{2 {\left (-\operatorname {signum}\left (x^{6}-1\right )\right )}^{\frac {1}{3}} x^{2}}\) | \(66\) |
pseudoelliptic | \(\frac {3 \left (x^{6}-1\right )^{\frac {1}{3}} x^{6}+2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x^{2}+2 \left (x^{6}-1\right )^{\frac {1}{3}}\right )}{3 x^{2}}\right ) x^{2}+\ln \left (\frac {x^{4}+x^{2} \left (x^{6}-1\right )^{\frac {1}{3}}+\left (x^{6}-1\right )^{\frac {2}{3}}}{x^{4}}\right ) x^{2}-2 \ln \left (\frac {-x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}}{x^{2}}\right ) x^{2}-9 \left (x^{6}-1\right )^{\frac {1}{3}}}{18 \left (x^{2}-\left (x^{6}-1\right )^{\frac {1}{3}}\right ) x^{2} \left (\left (x^{6}-1\right )^{\frac {2}{3}}+x^{2} \left (x^{2}+\left (x^{6}-1\right )^{\frac {1}{3}}\right )\right )}\) | \(152\) |
trager | \(\frac {\left (x^{6}-3\right ) \left (x^{6}-1\right )^{\frac {1}{3}}}{6 x^{2}}-\frac {\ln \left (-23398198288 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x^{6}-1099291994096 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x^{6}+2309110220532 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+575815167740 x^{6}-1203968676864 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}-300992169216 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-276285385917 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}+1497484690432 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2}-407313550340 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )-190928187198\right )}{9}+\frac {\ln \left (48489698336 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x^{6}-2321232645116 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x^{6}+1105141543668 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+279315992063 x^{6}+1203968676864 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+300992169216 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-577277555133 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}-3103340693504 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2}+1913919094776 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )-185723198911\right )}{9}-\frac {4 \ln \left (48489698336 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2} x^{6}-2321232645116 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) x^{6}+1105141543668 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {1}{3}} x^{4}+279315992063 x^{6}+1203968676864 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right ) \left (x^{6}-1\right )^{\frac {2}{3}} x^{2}+300992169216 x^{4} \left (x^{6}-1\right )^{\frac {1}{3}}-577277555133 x^{2} \left (x^{6}-1\right )^{\frac {2}{3}}-3103340693504 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )^{2}+1913919094776 \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )-185723198911\right ) \operatorname {RootOf}\left (16 \textit {\_Z}^{2}-4 \textit {\_Z} +1\right )}{9}\) | \(456\) |
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Time = 0.44 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^3} \, dx=-\frac {2 \, \sqrt {3} x^{2} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{4} - 13720 \, \sqrt {3} {\left (x^{6} - 1\right )}^{\frac {2}{3}} x^{2} + \sqrt {3} {\left (5831 \, x^{6} - 7200\right )}}{58653 \, x^{6} - 8000}\right ) + x^{2} \log \left (-3 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} x^{4} + 3 \, {\left (x^{6} - 1\right )}^{\frac {2}{3}} x^{2} + 1\right ) - 3 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} {\left (x^{6} - 3\right )}}{18 \, x^{2}} \]
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Result contains complex when optimal does not.
Time = 1.87 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.65 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^3} \, dx=- \frac {x^{4} e^{- \frac {2 i \pi }{3}} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 \Gamma \left (\frac {5}{3}\right )} + \frac {e^{\frac {i \pi }{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {x^{6}} \right )}}{6 x^{2} \Gamma \left (\frac {2}{3}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^3} \, dx=\frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + 1\right )}\right ) - \frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{2 \, x^{2}} - \frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{6 \, x^{2} {\left (\frac {x^{6} - 1}{x^{6}} - 1\right )}} + \frac {1}{18} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} + \frac {{\left (x^{6} - 1\right )}^{\frac {2}{3}}}{x^{4}} + 1\right ) - \frac {1}{9} \, \log \left (\frac {{\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2}} - 1\right ) \]
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\[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^3} \, dx=\int { \frac {{\left (x^{6} + 1\right )} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt [3]{-1+x^6} \left (1+x^6\right )}{x^3} \, dx=\int \frac {{\left (x^6-1\right )}^{1/3}\,\left (x^6+1\right )}{x^3} \,d x \]
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