Optimal. Leaf size=109 \[ \frac {25}{729} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {25 \log \left (\left (x^3-1\right )^{2/3}-\sqrt [3]{x^3-1}+1\right )}{1458}-\frac {25 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{243 \sqrt {3}}+\frac {\sqrt [3]{x^3-1} \left (50 x^9+30 x^6-99 x^3-81\right )}{972 x^{12}} \]
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Rubi [A] time = 0.07, antiderivative size = 116, normalized size of antiderivative = 1.06, number of steps used = 9, number of rules used = 8, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {446, 78, 47, 51, 58, 618, 204, 31} \begin {gather*} \frac {25 \sqrt [3]{x^3-1}}{486 x^3}+\frac {25}{486} \log \left (\sqrt [3]{x^3-1}+1\right )-\frac {25 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{243 \sqrt {3}}+\frac {\left (x^3-1\right )^{4/3}}{12 x^{12}}-\frac {5 \sqrt [3]{x^3-1}}{27 x^9}+\frac {5 \sqrt [3]{x^3-1}}{162 x^6}-\frac {25 \log (x)}{486} \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+x^3\right )}{x^{13}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x} (1+x)}{x^5} \, dx,x,x^3\right )\\ &=\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}+\frac {5}{9} \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x}}{x^4} \, dx,x,x^3\right )\\ &=-\frac {5 \sqrt [3]{-1+x^3}}{27 x^9}+\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}+\frac {5}{81} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x^3} \, dx,x,x^3\right )\\ &=-\frac {5 \sqrt [3]{-1+x^3}}{27 x^9}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^6}+\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}+\frac {25}{486} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x^2} \, dx,x,x^3\right )\\ &=-\frac {5 \sqrt [3]{-1+x^3}}{27 x^9}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^6}+\frac {25 \sqrt [3]{-1+x^3}}{486 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}+\frac {25}{729} \operatorname {Subst}\left (\int \frac {1}{(-1+x)^{2/3} x} \, dx,x,x^3\right )\\ &=-\frac {5 \sqrt [3]{-1+x^3}}{27 x^9}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^6}+\frac {25 \sqrt [3]{-1+x^3}}{486 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}-\frac {25 \log (x)}{486}+\frac {25}{486} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\sqrt [3]{-1+x^3}\right )+\frac {25}{486} \operatorname {Subst}\left (\int \frac {1}{1-x+x^2} \, dx,x,\sqrt [3]{-1+x^3}\right )\\ &=-\frac {5 \sqrt [3]{-1+x^3}}{27 x^9}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^6}+\frac {25 \sqrt [3]{-1+x^3}}{486 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}-\frac {25 \log (x)}{486}+\frac {25}{486} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {25}{243} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,-1+2 \sqrt [3]{-1+x^3}\right )\\ &=-\frac {5 \sqrt [3]{-1+x^3}}{27 x^9}+\frac {5 \sqrt [3]{-1+x^3}}{162 x^6}+\frac {25 \sqrt [3]{-1+x^3}}{486 x^3}+\frac {\left (-1+x^3\right )^{4/3}}{12 x^{12}}-\frac {25 \tan ^{-1}\left (\frac {1-2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{243 \sqrt {3}}-\frac {25 \log (x)}{486}+\frac {25}{486} \log \left (1+\sqrt [3]{-1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 38, normalized size = 0.35 \begin {gather*} \frac {\left (x^3-1\right )^{4/3} \left (5 x^{12} \, _2F_1\left (\frac {4}{3},4;\frac {7}{3};1-x^3\right )+1\right )}{12 x^{12}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 109, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{-1+x^3} \left (-81-99 x^3+30 x^6+50 x^9\right )}{972 x^{12}}-\frac {25 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^3}}{\sqrt {3}}\right )}{243 \sqrt {3}}+\frac {25}{729} \log \left (1+\sqrt [3]{-1+x^3}\right )-\frac {25 \log \left (1-\sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right )}{1458} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 98, normalized size = 0.90 \begin {gather*} \frac {100 \, \sqrt {3} x^{12} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 50 \, x^{12} \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 100 \, x^{12} \log \left ({\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + 3 \, {\left (50 \, x^{9} + 30 \, x^{6} - 99 \, x^{3} - 81\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{2916 \, x^{12}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 99, normalized size = 0.91 \begin {gather*} \frac {25}{729} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {50 \, {\left (x^{3} - 1\right )}^{\frac {10}{3}} + 180 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + 111 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 100 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{972 \, x^{12}} - \frac {25}{1458} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {25}{729} \, \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 = 3.04, size = 101, normalized size = 0.93
method | result | size |
risch | \(\frac {50 x^{12}-20 x^{9}-129 x^{6}+18 x^{3}+81}{972 x^{12} \left (x^{3}-1\right )^{\frac {2}{3}}}+\frac {25 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {2}{3}} \left (\frac {2 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {5}{3}\right ], \left [2, 2\right ], x^{3}\right )}{3}+\left (\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )\right )}{729 \Gamma \left (\frac {2}{3}\right ) \mathrm {signum}\left (x^{3}-1\right )^{\frac {2}{3}}}\) | \(101\) |
meijerg | \(\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (-\frac {10 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {11}{3}\right ], \left [2, 5\right ], x^{3}\right )}{81}-\frac {5 \left (\frac {4}{15}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{27}-\frac {\Gamma \left (\frac {2}{3}\right )}{x^{9}}+\frac {\Gamma \left (\frac {2}{3}\right )}{2 x^{6}}+\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}-\frac {\mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{3}} \left (\frac {22 \Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [1, 1, \frac {14}{3}\right ], \left [2, 6\right ], x^{3}\right )}{243}+\frac {10 \left (\frac {47}{120}+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )+i \pi \right ) \Gamma \left (\frac {2}{3}\right )}{81}+\frac {3 \Gamma \left (\frac {2}{3}\right )}{4 x^{12}}-\frac {\Gamma \left (\frac {2}{3}\right )}{3 x^{9}}-\frac {\Gamma \left (\frac {2}{3}\right )}{6 x^{6}}-\frac {5 \Gamma \left (\frac {2}{3}\right )}{27 x^{3}}\right )}{9 \Gamma \left (\frac {2}{3}\right ) \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{3}}}\) | \(185\) |
trager | \(\frac {\left (x^{3}-1\right )^{\frac {1}{3}} \left (50 x^{9}+30 x^{6}-99 x^{3}-81\right )}{972 x^{12}}+\frac {102400 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) \ln \left (-\frac {24125636608 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2} x^{3}-28585984 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) x^{3}+22536192 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-9894 x^{3}-193005092864 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2}-22536192 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )+19749 \left (x^{3}-1\right )^{\frac {2}{3}}-24584192 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )-19749 \left (x^{3}-1\right )^{\frac {1}{3}}+8245}{x^{3}}\right )}{729}-\frac {25 \ln \left (-\frac {24125636608 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2} x^{3}+40366080 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) x^{3}-22536192 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-1477 x^{3}-193005092864 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2}+22536192 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )+14247 \left (x^{3}-1\right )^{\frac {2}{3}}-69656576 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )-14247 \left (x^{3}-1\right )^{\frac {1}{3}}+2743}{x^{3}}\right )}{729}-\frac {102400 \ln \left (-\frac {24125636608 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2} x^{3}+40366080 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) x^{3}-22536192 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right ) \left (x^{3}-1\right )^{\frac {2}{3}}-1477 x^{3}-193005092864 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )^{2}+22536192 \left (x^{3}-1\right )^{\frac {1}{3}} \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )+14247 \left (x^{3}-1\right )^{\frac {2}{3}}-69656576 \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )-14247 \left (x^{3}-1\right )^{\frac {1}{3}}+2743}{x^{3}}\right ) \RootOf \left (16777216 \textit {\_Z}^{2}+4096 \textit {\_Z} +1\right )}{729}\) | \(458\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 184, normalized size = 1.69 \begin {gather*} \frac {25}{729} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {20 \, {\left (x^{3} - 1\right )}^{\frac {10}{3}} + 72 \, {\left (x^{3} - 1\right )}^{\frac {7}{3}} + 93 \, {\left (x^{3} - 1\right )}^{\frac {4}{3}} - 40 \, {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{972 \, {\left ({\left (x^{3} - 1\right )}^{4} + 4 \, {\left (x^{3} - 1\right )}^{3} + 4 \, x^{3} + 6 \, {\left (x^{3} - 1\right )}^{2} - 3\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 {25}{1458} \, \log \left ({\left (x^{3} - 1\right )}^{\frac {2}{3}} - {\left (x^{3} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {25}{729} \, \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 = 1.61, size = 265, normalized size = 2.43 \begin {gather*} \frac {5\,\ln \left (\frac {25\,{\left (x^3-1\right )}^{1/3}}{6561}+\frac {25}{6561}\right )}{243}+\frac {10\,\ln \left (\frac {100\,{\left (x^3-1\right )}^{1/3}}{59049}+\frac {100}{59049}\right )}{729}+\frac {\frac {31\,{\left (x^3-1\right )}^{4/3}}{324}-\frac {10\,{\left (x^3-1\right )}^{1/3}}{243}+\frac {2\,{\left (x^3-1\right )}^{7/3}}{27}+\frac {5\,{\left (x^3-1\right )}^{10/3}}{243}}{6\,{\left (x^3-1\right )}^2+4\,{\left (x^3-1\right )}^3+{\left (x^3-1\right )}^4+4\,x^3-3}+\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 )-\ln \left (\frac {5}{81}-\frac {10\,{\left (x^3-1\right )}^{1/3}}{81}+\frac {\sqrt {3}\,5{}\mathrm {i}}{81}\right )\,\left (\frac {5}{729}+\frac {\sqrt {3}\,5{}\mathrm {i}}{729}\right )+\ln \left (\frac {10\,{\left (x^3-1\right )}^{1/3}}{81}-\frac {5}{81}+\frac {\sqrt {3}\,5{}\mathrm {i}}{81}\right )\,\left (-\frac {5}{729}+\frac {\sqrt {3}\,5{}\mathrm {i}}{729}\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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