Optimal. Leaf size=104 \[ \frac {5}{243} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {5}{486} \log \left (\left (x^3-1\right )^{2/3}-\sqrt [3]{x^3-1}+1\right )-\frac {5 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {\sqrt [3]{x^3-1} \left (5 x^6+3 x^3-18\right )}{162 x^9} \]
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Rubi [A] time = 0.11, antiderivative size = 100, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {266, 47, 51, 58, 618, 204, 31} \begin {gather*} \frac {5 \sqrt [3]{x^3-1}}{162 x^3}+\frac {5}{162} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {5 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{81 \sqrt {3}}-\frac {\sqrt [3]{x^3-1}}{9 x^9}+\frac {\sqrt [3]{x^3-1}}{54 x^6}-\frac {5 \log (x)}{162} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 47
Rule 51
Rule 58
Rule 204
Rule 266
Rule 618
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-1+x^3}}{x^{10}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x}}{x^4} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{-1+x^3}}{9 x^9}+\frac {1}{27} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x^3} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{-1+x^3}}{9 x^9}+\frac {\sqrt [3]{-1+x^3}}{54 x^6}+\frac {5}{162} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x^2} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{-1+x^3}}{9 x^9}+\frac {\sqrt [3]{-1+x^3}}{54 x^6}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^3}+\frac {5}{243} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x} \, dx,x,x^3\right )\\ &=-\frac {\sqrt [3]{-1+x^3}}{9 x^9}+\frac {\sqrt [3]{-1+x^3}}{54 x^6}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^3}-\frac {5 \log (x)}{162}+\frac {5}{162} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^3}\right )+\frac {5}{162} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^3}\right )\\ &=-\frac {\sqrt [3]{-1+x^3}}{9 x^9}+\frac {\sqrt [3]{-1+x^3}}{54 x^6}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^3}-\frac {5 \log (x)}{162}+\frac {5}{162} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {5}{81} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^3}\right )\\ &=-\frac {\sqrt [3]{-1+x^3}}{9 x^9}+\frac {\sqrt [3]{-1+x^3}}{54 x^6}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^3}-\frac {5 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{81 \sqrt {3}}-\frac {5 \log (x)}{162}+\frac {5}{162} \log \left (1+\sqrt [3]{-1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 28, normalized size = 0.27 \begin {gather*} \frac {1}{4} \left (x^3-1\right )^{4/3} \, _2F_1\left (\frac {4}{3},4;\frac {7}{3};1-x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.17, size = 104, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{-1+x^3} \left (-18+3 x^3+5 x^6\right )}{162 x^9}-\frac {5 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {5}{243} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {5}{486} \log \left (1-\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 = 1.07, size = 93, normalized size = 0.89 \begin {gather*} \frac {10 \, \sqrt {3} x^{9} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 5 \, x^{9} \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 10 \, x^{9} \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 3 \, {\left (5 \, x^{6} + 3 \, x^{3} - 18\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{486 \, x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 90, normalized size = 0.87 \begin {gather*} \frac {5}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {5 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + 13 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 10 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{162 \, x^{9}} - \frac {5}{486} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {5}{243} \, \log \left ({\left | {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.76, size = 89, normalized size = 0.86
method | result | size |
meijerg | \(\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {10 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {11}{3}\right ], \left [2, 5\right ], x^{3}\right )}{81}-\frac {5 \left (\frac {4}{15}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{27}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{9}}+\frac {\Gamma \left (\frac {2}{3}\right )}{2 x^{6}}+\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}\) | \(89\) |
risch | \(\frac {5 x^{9}-2 x^{6}-21 x^{3}+18}{162 x^{9} \left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {5 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} \left (\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )\right )}{243 \Gamma \left (\frac {2}{3}\right ) \mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(96\) |
trager | \(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (5 x^{6}+3 x^{3}-18\right )}{162 x^{9}}-\frac {5 \ln \left (\frac {-376963072 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2} x^{3}-5045760 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) x^{3}+2817024 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+1477 x^{3}-14247 \left (x^{3}-1\right )^{\frac {2}{3}}-2817024 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )+3015704576 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2}+14247 \left (x^{3}-1\right )^{\frac {1}{3}}+8707072 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )-2743}{x^{3}}\right )}{243}-\frac {2560 \ln \left (\frac {-376963072 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2} x^{3}-5045760 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) x^{3}+2817024 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}+1477 x^{3}-14247 \left (x^{3}-1\right )^{\frac {2}{3}}-2817024 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )+3015704576 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2}+14247 \left (x^{3}-1\right )^{\frac {1}{3}}+8707072 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )-2743}{x^{3}}\right ) \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )}{243}+\frac {2560 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \ln \left (-\frac {376963072 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2} x^{3}-3573248 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) x^{3}+2817024 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-9894 x^{3}+19749 \left (x^{3}-1\right )^{\frac {2}{3}}-2817024 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )-3015704576 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2}-19749 \left (x^{3}-1\right )^{\frac {1}{3}}-3073024 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )+8245}{x^{3}}\right )}{243}\) | \(451\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 111, normalized size = 1.07 \begin {gather*} \frac {5}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {5 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + 13 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 10 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{162 \, {\left ({\left (x^{3} - 1\right )}^{3} + 3 \, x^{3} + 3 \, {\left (x^{3} - 1\right )}^{2} - 2\right )}} - \frac {5}{486} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {5}{243} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.99, size = 124, normalized size = 1.19 \begin {gather*} \frac {5\,\ln \left (\frac {25\,{\left (x^3-1\right )}^{1/3}}{6561}+\frac {25}{6561}\right )}{243}+\frac {\frac {13\,{\left (x^3-1\right )}^{4/3}}{162}-\frac {5\,{\left (x^3-1\right )}^{1/3}}{81}+\frac {5\,{\left (x^3-1\right )}^{7/3}}{162}}{3\,{\left (x^3-1\right )}^2+{\left (x^3-1\right )}^3+3\,x^3-2}-\ln \left (\frac {5}{54}-\frac {5\,{\left (x^3-1\right )}^{1/3}}{27}+\frac {\sqrt {3}\,5{}\mathrm {i}}{54}\right )\,\left (\frac {5}{486}+\frac {\sqrt {3}\,5{}\mathrm {i}}{486}\right )+\ln \left (\frac {5\,{\left (x^3-1\right )}^{1/3}}{27}-\frac {5}{54}+\frac {\sqrt {3}\,5{}\mathrm {i}}{54}\right )\,\left (-\frac {5}{486}+\frac {\sqrt {3}\,5{}\mathrm {i}}{486}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.42, size = 34, normalized size = 0.33 \begin {gather*} - \frac {\Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{3}}} \right )}}{3 x^{8} \Gamma \left (\frac {11}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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