Optimal. Leaf size=104 \[ \frac {13}{243} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {13}{486} \log \left (\left (x^3-1\right )^{2/3}-\sqrt [3]{x^3-1}+1\right )-\frac {13 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {\sqrt [3]{x^3-1} \left (13 x^6-57 x^3+18\right )}{162 x^9} \]
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Rubi [A] time = 0.08, antiderivative size = 100, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {446, 78, 47, 51, 58, 618, 204, 31} \begin {gather*} \frac {13 \sqrt [3]{x^3-1}}{162 x^3}+\frac {13}{162} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {13 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{81 \sqrt {3}}-\frac {\left (x^3-1\right )^{4/3}}{9 x^9}-\frac {13 \sqrt [3]{x^3-1}}{54 x^6}-\frac {13 \log (x)}{162} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 47
Rule 51
Rule 58
Rule 78
Rule 204
Rule 446
Rule 618
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{-1+x^3} \left (-1+2 x^3\right )}{x^{10}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x} (-1+2 x)}{x^4} \, dx,x,x^3\right )\\ &=-\frac {\left (-1+x^3\right )^{4/3}}{9 x^9}+\frac {13}{27} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x}}{x^3} \, dx,x,x^3\right )\\ &=-\frac {13 \sqrt [3]{-1+x^3}}{54 x^6}-\frac {\left (-1+x^3\right )^{4/3}}{9 x^9}+\frac {13}{162} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x^2} \, dx,x,x^3\right )\\ &=-\frac {13 \sqrt [3]{-1+x^3}}{54 x^6}+\frac {13 \sqrt [3]{-1+x^3}}{162 x^3}-\frac {\left (-1+x^3\right )^{4/3}}{9 x^9}+\frac {13}{243} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x} \, dx,x,x^3\right )\\ &=-\frac {13 \sqrt [3]{-1+x^3}}{54 x^6}+\frac {13 \sqrt [3]{-1+x^3}}{162 x^3}-\frac {\left (-1+x^3\right )^{4/3}}{9 x^9}-\frac {13 \log (x)}{162}+\frac {13}{162} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^3}\right )+\frac {13}{162} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^3}\right )\\ &=-\frac {13 \sqrt [3]{-1+x^3}}{54 x^6}+\frac {13 \sqrt [3]{-1+x^3}}{162 x^3}-\frac {\left (-1+x^3\right )^{4/3}}{9 x^9}-\frac {13 \log (x)}{162}+\frac {13}{162} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {13}{81} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^3}\right )\\ &=-\frac {13 \sqrt [3]{-1+x^3}}{54 x^6}+\frac {13 \sqrt [3]{-1+x^3}}{162 x^3}-\frac {\left (-1+x^3\right )^{4/3}}{9 x^9}-\frac {13 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{81 \sqrt {3}}-\frac {13 \log (x)}{162}+\frac {13}{162} \log \left (1+\sqrt [3]{-1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 38, normalized size = 0.37 \begin {gather*} \frac {\left (x^3-1\right )^{4/3} \left (13 x^9 \, _2F_1\left (\frac {4}{3},3;\frac {7}{3};1-x^3\right )-4\right )}{36 x^9} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 104, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{-1+x^3} \left (18-57 x^3+13 x^6\right )}{162 x^9}-\frac {13 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {13}{243} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {13}{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 = 0.54, size = 93, normalized size = 0.89 \begin {gather*} \frac {26 \, \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 ) - 13 \, x^{9} \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 26 \, x^{9} \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 3 \, {\left (13 \, x^{6} - 57 \, 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 = 0.18, size = 90, normalized size = 0.87 \begin {gather*} \frac {13}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {13 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} - 31 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 26 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{162 \, x^{9}} - \frac {13}{486} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {13}{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 = 2.66, size = 96, normalized size = 0.92
method | result | size |
risch | \(\frac {13 x^{9}-70 x^{6}+75 x^{3}-18}{162 x^{9} \left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {13 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} \left (\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )\right )}{243 \Gamma \left (\frac {2}{3}\right ) \mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(96\) |
meijerg | \(-\frac {2 \mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (\frac {5 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {8}{3}\right ], \left [2, 4\right ], x^{3}\right )}{27}+\frac {\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{3}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{2 x^{6}}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}-\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {10 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {11}{3}\right ], \left [2, 5\right ], x^{3}\right )}{81}-\frac {5 \left (\frac {4}{15}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{27}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{9}}+\frac {\Gamma \left (\frac {2}{3}\right )}{2 x^{6}}+\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}\) | \(170\) |
trager | \(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (13 x^{6}-57 x^{3}+18\right )}{162 x^{9}}+\frac {6656 \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}-\frac {13 \ln \left (-\frac {163840 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2} x^{3}+1728 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) x^{3}+2496 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-7 x^{3}-1310720 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2}-2496 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )+18 \left (x^{3}-1\right )^{\frac {2}{3}}-64 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )-18 \left (x^{3}-1\right )^{\frac {1}{3}}+13}{x^{3}}\right )}{243}-\frac {6656 \ln \left (-\frac {163840 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2} x^{3}+1728 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) x^{3}+2496 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-7 x^{3}-1310720 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )^{2}-2496 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )+18 \left (x^{3}-1\right )^{\frac {2}{3}}-64 \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )-18 \left (x^{3}-1\right )^{\frac {1}{3}}+13}{x^{3}}\right ) \RootOf \left (262144 \textit {\_Z}^{2}+512 \textit {\_Z} +1\right )}{243}\) | \(453\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 146, normalized size = 1.40 \begin {gather*} \frac {13}{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 {{\left (x^{3} - 1\right )}^{\frac {4}{3}} - 2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{9 \, {\left (2 \, x^{3} + {\left (x^{3} - 1\right )}^{2} - 1\right )}} - \frac {13}{486} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {13}{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 = 1.38, size = 231, normalized size = 2.22 \begin {gather*} \frac {2\,\ln \left (\frac {4\,{\left (x^3-1\right )}^{1/3}}{81}+\frac {4}{81}\right )}{27}-\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}-\frac {\frac {2\,{\left (x^3-1\right )}^{1/3}}{9}-\frac {{\left (x^3-1\right )}^{4/3}}{9}}{{\left (x^3-1\right )}^2+2\,x^3-1}-\ln \left (\frac {1}{3}-\frac {2\,{\left (x^3-1\right )}^{1/3}}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )\,\left (\frac {1}{27}+\frac {\sqrt {3}\,1{}\mathrm {i}}{27}\right )+\ln \left (\frac {2\,{\left (x^3-1\right )}^{1/3}}{3}-\frac {1}{3}+\frac {\sqrt {3}\,1{}\mathrm {i}}{3}\right )\,\left (-\frac {1}{27}+\frac {\sqrt {3}\,1{}\mathrm {i}}{27}\right )+\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 [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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