Optimal. Leaf size=101 \[ -\frac {1}{3} \log \left (\sqrt [3]{x^3-1}-x\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3-1}+x}\right )}{\sqrt {3}}+\frac {\sqrt [3]{x^3-1} \left (-3 x^3-1\right )}{4 x^4}+\frac {1}{6} \log \left (\sqrt [3]{x^3-1} x+\left (x^3-1\right )^{2/3}+x^2\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 109, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {451, 277, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} -\frac {\sqrt [3]{x^3-1}}{x}-\frac {1}{3} \log \left (1-\frac {x}{\sqrt [3]{x^3-1}}\right )-\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\left (x^3-1\right )^{4/3}}{4 x^4}+\frac {1}{6} \log \left (\frac {x}{\sqrt [3]{x^3-1}}+\frac {x^2}{\left (x^3-1\right )^{2/3}}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 277
Rule 292
Rule 331
Rule 451
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-1+x^3} \left (1+x^3\right )}{x^5} \, dx &=\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}+\operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\sqrt [3]{-1+x^3}}{x}+\frac {\left (-1+x^3\right )^{4/3}}{4 x^4}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\sqrt [3]{-1+x^3}}{x}+\frac {\left (-1+x^3\right )^{4/3}}{4 x^4}-\frac {1}{3} \log \left (1-\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\sqrt [3]{-1+x^3}}{x}+\frac {\left (-1+x^3\right )^{4/3}}{4 x^4}-\frac {1}{3} \log \left (1-\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {1}{6} \log \left (1+\frac {x^2}{\left (-1+x^3\right )^{2/3}}+\frac {x}{\sqrt [3]{-1+x^3}}\right )+\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{-1+x^3}}\right )\\ &=-\frac {\sqrt [3]{-1+x^3}}{x}+\frac {\left (-1+x^3\right )^{4/3}}{4 x^4}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{-1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (1-\frac {x}{\sqrt [3]{-1+x^3}}\right )+\frac {1}{6} \log \left (1+\frac {x^2}{\left (-1+x^3\right )^{2/3}}+\frac {x}{\sqrt [3]{-1+x^3}}\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 50, normalized size = 0.50 \begin {gather*} \frac {\sqrt [3]{x^3-1} \left (-\frac {4 x^3 \, _2F_1\left (-\frac {1}{3},-\frac {1}{3};\frac {2}{3};x^3\right )}{\sqrt [3]{1-x^3}}+x^3-1\right )}{4 x^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 101, normalized size = 1.00 \begin {gather*} \frac {\left (-1-3 x^3\right ) \sqrt [3]{-1+x^3}}{4 x^4}-\frac {\tan ^{-1}\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 ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.78, size = 112, normalized size = 1.11 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{3} + 1\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{5}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.60, size = 58, normalized size = 0.57
method | result | size |
risch | \(-\frac {3 x^{6}-2 x^{3}-1}{4 x^{4} \left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {\left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} x^{2} \hypergeom \left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], x^{3}\right )}{2 \mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(58\) |
meijerg | \(-\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \hypergeom \left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], x^{3}\right )}{\left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}} x}-\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-x^{3}+1\right )^{\frac {4}{3}}}{4 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}} x^{4}}\) | \(66\) |
trager | \(-\frac {\left (3 x^{3}+1\right ) \left (x^{3}-1\right )^{\frac {1}{3}}}{4 x^{4}}-\frac {\ln \left (-17421502720 \RootOf \left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2} x^{3}+3103405824 \RootOf \left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +21033432144 \RootOf \left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-23047994048 \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 \RootOf \left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2}+369329904 \RootOf \left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )-396625334\right )}{3}+\frac {\ln \left (14505155072 \RootOf \left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2} x^{3}-3103405824 \RootOf \left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +24136837968 \RootOf \left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-21940004336 \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 \RootOf \left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2}+22309334240 \RootOf \left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )-1020791175\right )}{3}-\frac {16 \ln \left (14505155072 \RootOf \left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2} x^{3}-3103405824 \RootOf \left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}} x +24136837968 \RootOf \left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {1}{3}} x^{2}-21940004336 \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 \RootOf \left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )^{2}+22309334240 \RootOf \left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )-1020791175\right ) \RootOf \left (256 \textit {\_Z}^{2}-16 \textit {\_Z} +1\right )}{3}\) | \(446\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 93, normalized size = 0.92 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.17, size = 55, normalized size = 0.54 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.13, size = 165, normalized size = 1.63 \begin {gather*} \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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