Integrand size = 18, antiderivative size = 101 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^5} \, dx=\frac {\left (-1-3 x^3\right ) \sqrt [3]{-1+x^3}}{4 x^4}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
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Time = 0.01 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.76, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {462, 283, 337} \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^5} \, dx=-\frac {\arctan \left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\sqrt [3]{x^3-1}}{x}-\frac {1}{2} \log \left (x-\sqrt [3]{x^3-1}\right )+\frac {\left (x^3-1\right )^{4/3}}{4 x^4} \]
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Rule 283
Rule 337
Rule 462
Rubi steps \begin{align*} \text {integral}& = \frac {\left (-1+x^3\right )^{4/3}}{4 x^4}+\int \frac {\sqrt [3]{-1+x^3}}{x^2} \, dx \\ & = -\frac {\sqrt [3]{-1+x^3}}{x}+\frac {\left (-1+x^3\right )^{4/3}}{4 x^4}+\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx \\ & = -\frac {\sqrt [3]{-1+x^3}}{x}+\frac {\left (-1+x^3\right )^{4/3}}{4 x^4}-\frac {\arctan \left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (x-\sqrt [3]{-1+x^3}\right ) \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^5} \, dx=\frac {\left (-1-3 x^3\right ) \sqrt [3]{-1+x^3}}{4 x^4}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{-1+x^3}\right )+\frac {1}{6} \log \left (x^2+x \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.58 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.57
method | result | size |
risch | \(-\frac {3 x^{6}-2 x^{3}-1}{4 x^{4} \left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{2} \operatorname {hypergeom}\left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{2 \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(58\) |
meijerg | \(-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], x^{3}\right )}{{\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x}-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {4}{3}}}{4 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{3}} x^{4}}\) | \(66\) |
pseudoelliptic | \(\frac {4 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{4}+2 \ln \left (\frac {x^{2}+x \left (x^{3}-1\right )^{\frac {1}{3}}+\left (x^{3}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{4}-4 \ln \left (\frac {-x +\left (x^{3}-1\right )^{\frac {1}{3}}}{x}\right ) x^{4}-9 \left (x^{3}-1\right )^{\frac {1}{3}} x^{3}-3 \left (x^{3}-1\right )^{\frac {1}{3}}}{12 x^{4}}\) | \(107\) |
trager | \(-\frac {\left (3 x^{3}+1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{4 x^{4}}-\frac {\ln \left (-17421502720 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2} x^{3}+3103405824 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +21033432144 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) x^{2}-23047994048 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) x^{3}-1508552373 x \left (x^{3}-1\right )^{\frac {2}{3}}+193962864 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+1246536764 x^{3}+139372021760 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2}+369329904 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )-396625334\right )}{3}+\frac {\ln \left (14505155072 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2} x^{3}-3103405824 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +24136837968 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) x^{2}-21940004336 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) x^{3}-1314589509 x \left (x^{3}-1\right )^{\frac {2}{3}}-193962864 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+1565213135 x^{3}-116041240576 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2}+22309334240 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )-1020791175\right )}{3}-\frac {16 \ln \left (14505155072 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2} x^{3}-3103405824 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +24136837968 \left (x^{3}-1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) x^{2}-21940004336 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) x^{3}-1314589509 x \left (x^{3}-1\right )^{\frac {2}{3}}-193962864 x^{2} \left (x^{3}-1\right )^{\frac {1}{3}}+1565213135 x^{3}-116041240576 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2}+22309334240 \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )-1020791175\right ) \operatorname {RootOf}\left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )}{3}\) | \(446\) |
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Time = 0.40 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.11 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^5} \, dx=-\frac {4 \, \sqrt {3} x^{4} \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} - 7200\right )}}{58653 \, x^{3} - 8000}\right ) + 2 \, x^{4} \log \left (-3 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x + 1\right ) + 3 \, {\left (3 \, x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{12 \, x^{4}} \]
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Result contains complex when optimal does not.
Time = 1.37 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.63 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^5} \, dx=\begin {cases} \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} - \frac {\sqrt [3]{-1 + \frac {1}{x^{3}}} e^{- \frac {2 i \pi }{3}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 \Gamma \left (- \frac {1}{3}\right )} + \frac {\sqrt [3]{1 - \frac {1}{x^{3}}} \Gamma \left (- \frac {4}{3}\right )}{3 x^{3} \Gamma \left (- \frac {1}{3}\right )} & \text {otherwise} \end {cases} + \frac {e^{\frac {i \pi }{3}} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {x^{3}} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^5} \, dx=\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) - \frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {4}{3}}}{4 \, x^{4}} + \frac {1}{6} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) - \frac {1}{3} \, \log \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \]
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\[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^5} \, dx=\int { \frac {{\left (x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{5}} \,d x } \]
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Time = 6.47 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.54 \[ \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^5} \, dx=-\frac {{\left (x^3-1\right )}^{1/3}-x^3\,{\left (x^3-1\right )}^{1/3}}{4\,x^4}-\frac {{\left (x^3-1\right )}^{1/3}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},-\frac {1}{3};\ \frac {2}{3};\ x^3\right )}{x\,{\left (1-x^3\right )}^{1/3}} \]
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