Optimal. Leaf size=115 \[ \frac {14}{243} \log \left (\sqrt [3]{x^3+1}-1\right )-\frac {7}{243} \log \left (\left (x^3+1\right )^{2/3}+\sqrt [3]{x^3+1}+1\right )+\frac {14 \tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {\left (x^3+1\right )^{2/3} \left (-729 x^7+560 x^6+486 x^4-420 x^3-405 x+360\right )}{3240 x^9} \]
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Rubi [A] time = 0.10, antiderivative size = 150, normalized size of antiderivative = 1.30, number of steps used = 13, number of rules used = 9, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {1844, 266, 51, 55, 618, 204, 31, 271, 264} \begin {gather*} \frac {14 \left (x^3+1\right )^{2/3}}{81 x^3}+\frac {7}{81} \log \left (1-\sqrt [3]{x^3+1}\right )+\frac {14 \tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {\left (x^3+1\right )^{2/3}}{9 x^9}-\frac {\left (x^3+1\right )^{2/3}}{8 x^8}-\frac {7 \left (x^3+1\right )^{2/3}}{54 x^6}+\frac {3 \left (x^3+1\right )^{2/3}}{20 x^5}-\frac {9 \left (x^3+1\right )^{2/3}}{40 x^2}-\frac {7 \log (x)}{81} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 51
Rule 55
Rule 204
Rule 264
Rule 266
Rule 271
Rule 618
Rule 1844
Rubi steps
\begin {align*} \int \frac {-1+x}{x^{10} \sqrt [3]{1+x^3}} \, dx &=\int \left (-\frac {1}{x^{10} \sqrt [3]{1+x^3}}+\frac {1}{x^9 \sqrt [3]{1+x^3}}\right ) \, dx\\ &=-\int \frac {1}{x^{10} \sqrt [3]{1+x^3}} \, dx+\int \frac {1}{x^9 \sqrt [3]{1+x^3}} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{8 x^8}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^4 \sqrt [3]{1+x}} \, dx,x,x^3\right )-\frac {3}{4} \int \frac {1}{x^6 \sqrt [3]{1+x^3}} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{9 x^9}-\frac {\left (1+x^3\right )^{2/3}}{8 x^8}+\frac {3 \left (1+x^3\right )^{2/3}}{20 x^5}+\frac {7}{27} \operatorname {Subst}\left (\int \frac {1}{x^3 \sqrt [3]{1+x}} \, dx,x,x^3\right )+\frac {9}{20} \int \frac {1}{x^3 \sqrt [3]{1+x^3}} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{9 x^9}-\frac {\left (1+x^3\right )^{2/3}}{8 x^8}-\frac {7 \left (1+x^3\right )^{2/3}}{54 x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{20 x^5}-\frac {9 \left (1+x^3\right )^{2/3}}{40 x^2}-\frac {14}{81} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt [3]{1+x}} \, dx,x,x^3\right )\\ &=\frac {\left (1+x^3\right )^{2/3}}{9 x^9}-\frac {\left (1+x^3\right )^{2/3}}{8 x^8}-\frac {7 \left (1+x^3\right )^{2/3}}{54 x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{20 x^5}+\frac {14 \left (1+x^3\right )^{2/3}}{81 x^3}-\frac {9 \left (1+x^3\right )^{2/3}}{40 x^2}+\frac {14}{243} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{1+x}} \, dx,x,x^3\right )\\ &=\frac {\left (1+x^3\right )^{2/3}}{9 x^9}-\frac {\left (1+x^3\right )^{2/3}}{8 x^8}-\frac {7 \left (1+x^3\right )^{2/3}}{54 x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{20 x^5}+\frac {14 \left (1+x^3\right )^{2/3}}{81 x^3}-\frac {9 \left (1+x^3\right )^{2/3}}{40 x^2}-\frac {7 \log (x)}{81}-\frac {7}{81} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^3}\right )+\frac {7}{81} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^3}\right )\\ &=\frac {\left (1+x^3\right )^{2/3}}{9 x^9}-\frac {\left (1+x^3\right )^{2/3}}{8 x^8}-\frac {7 \left (1+x^3\right )^{2/3}}{54 x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{20 x^5}+\frac {14 \left (1+x^3\right )^{2/3}}{81 x^3}-\frac {9 \left (1+x^3\right )^{2/3}}{40 x^2}-\frac {7 \log (x)}{81}+\frac {7}{81} \log \left (1-\sqrt [3]{1+x^3}\right )-\frac {14}{81} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^3}\right )\\ &=\frac {\left (1+x^3\right )^{2/3}}{9 x^9}-\frac {\left (1+x^3\right )^{2/3}}{8 x^8}-\frac {7 \left (1+x^3\right )^{2/3}}{54 x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{20 x^5}+\frac {14 \left (1+x^3\right )^{2/3}}{81 x^3}-\frac {9 \left (1+x^3\right )^{2/3}}{40 x^2}+\frac {14 \tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{81 \sqrt {3}}-\frac {7 \log (x)}{81}+\frac {7}{81} \log \left (1-\sqrt [3]{1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.03, size = 46, normalized size = 0.40 \begin {gather*} -\frac {\left (x^3+1\right )^{2/3} \left (20 x^8 \, _2F_1\left (\frac {2}{3},4;\frac {5}{3};x^3+1\right )+9 x^6-6 x^3+5\right )}{40 x^8} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 17.29, size = 115, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^3\right )^{2/3} \left (360-405 x-420 x^3+486 x^4+560 x^6-729 x^7\right )}{3240 x^9}+\frac {14 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{81 \sqrt {3}}+\frac {14}{243} \log \left (-1+\sqrt [3]{1+x^3}\right )-\frac {7}{243} \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 = 0.86, size = 124, normalized size = 1.08 \begin {gather*} -\frac {560 \, \sqrt {3} x^{9} \arctan \left (-\frac {\sqrt {3} {\left (x^{3} + 1\right )} - 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 4 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{3} + 9}\right ) - 280 \, x^{9} \log \left (\frac {x^{3} - 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{3}}\right ) + 3 \, {\left (729 \, x^{7} - 560 \, x^{6} - 486 \, x^{4} + 420 \, x^{3} + 405 \, x - 360\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{9720 \, x^{9}} \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 {x - 1}{{\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{10}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.65, size = 111, normalized size = 0.97
method | result | size |
risch | \(-\frac {729 x^{10}-560 x^{9}+243 x^{7}-140 x^{6}-81 x^{4}+60 x^{3}+405 x -360}{3240 x^{9} \left (x^{3}+1\right )^{\frac {1}{3}}}+\frac {7 \sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{3} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], -x^{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{243 \pi }\) | \(111\) |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {70 \pi \sqrt {3}\, x^{3} \hypergeom \left (\left [1, 1, \frac {13}{3}\right ], \left [2, 5\right ], -x^{3}\right )}{729 \Gamma \left (\frac {2}{3}\right )}-\frac {28 \left (\frac {197}{84}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )\right ) \pi \sqrt {3}}{243 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right ) x^{9}}+\frac {\pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right ) x^{6}}-\frac {4 \pi \sqrt {3}}{27 \Gamma \left (\frac {2}{3}\right ) x^{3}}\right )}{6 \pi }-\frac {\left (\frac {9}{5} x^{6}-\frac {6}{5} x^{3}+1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{8 x^{8}}\) | \(128\) |
trager | \(-\frac {\left (729 x^{7}-560 x^{6}-486 x^{4}+420 x^{3}+405 x -360\right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{3240 x^{9}}+\frac {28 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (-\frac {16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}-18 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}-30 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{3}-16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}-30 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+9 \left (x^{3}+1\right )^{\frac {2}{3}}-38 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+9 \left (x^{3}+1\right )^{\frac {1}{3}}+5}{x^{3}}\right )}{243}-\frac {14 \ln \left (-\frac {16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}+34 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}+30 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{3}-16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+30 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+24 \left (x^{3}+1\right )^{\frac {2}{3}}+22 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+24 \left (x^{3}+1\right )^{\frac {1}{3}}+20}{x^{3}}\right )}{243}-\frac {28 \ln \left (-\frac {16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}+34 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}+30 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+15 x^{3}-16 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2}+30 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+24 \left (x^{3}+1\right )^{\frac {2}{3}}+22 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )+24 \left (x^{3}+1\right )^{\frac {1}{3}}+20}{x^{3}}\right ) \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )}{243}\) | \(466\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 145, normalized size = 1.26 \begin {gather*} \frac {14}{243} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {28 \, {\left (x^{3} + 1\right )}^{\frac {8}{3}} - 77 \, {\left (x^{3} + 1\right )}^{\frac {5}{3}} + 67 \, {\left (x^{3} + 1\right )}^{\frac {2}{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 {2}{3}}}{2 \, x^{2}} + \frac {2 \, {\left (x^{3} + 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} - \frac {{\left (x^{3} + 1\right )}^{\frac {8}{3}}}{8 \, x^{8}} - \frac {7}{243} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {14}{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 = 1.29, size = 158, normalized size = 1.37 \begin {gather*} \frac {14\,\ln \left (\frac {196\,{\left (x^3+1\right )}^{1/3}}{6561}-\frac {196}{6561}\right )}{243}+\ln \left (\frac {196\,{\left (x^3+1\right )}^{1/3}}{6561}-9\,{\left (-\frac {7}{243}+\frac {\sqrt {3}\,7{}\mathrm {i}}{243}\right )}^2\right )\,\left (-\frac {7}{243}+\frac {\sqrt {3}\,7{}\mathrm {i}}{243}\right )-\ln \left (\frac {196\,{\left (x^3+1\right )}^{1/3}}{6561}-9\,{\left (\frac {7}{243}+\frac {\sqrt {3}\,7{}\mathrm {i}}{243}\right )}^2\right )\,\left (\frac {7}{243}+\frac {\sqrt {3}\,7{}\mathrm {i}}{243}\right )+\frac {\frac {67\,{\left (x^3+1\right )}^{2/3}}{162}-\frac {77\,{\left (x^3+1\right )}^{5/3}}{162}+\frac {14\,{\left (x^3+1\right )}^{8/3}}{81}}{{\left (x^3+1\right )}^3-3\,{\left (x^3+1\right )}^2+3\,x^3+2}-\frac {{\left (x^3+1\right )}^{2/3}\,\left (9\,x^6-6\,x^3+5\right )}{40\,x^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.85, size = 112, normalized size = 0.97 \begin {gather*} \frac {2 \left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{3 \Gamma \left (\frac {1}{3}\right )} - \frac {4 \left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{3} \Gamma \left (\frac {1}{3}\right )} + \frac {10 \left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{27 x^{6} \Gamma \left (\frac {1}{3}\right )} + \frac {\Gamma \left (\frac {10}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {10}{3} \\ \frac {13}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 x^{10} \Gamma \left (\frac {13}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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