Optimal. Leaf size=95 \[ \frac {\left (x^3+1\right )^{2/3} (2-3 x)}{6 x^3}+\frac {1}{9} \log \left (\sqrt [3]{x^3+1}-1\right )-\frac {1}{18} \log \left (\left (x^3+1\right )^{2/3}+\sqrt [3]{x^3+1}+1\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.08, antiderivative size = 86, normalized size of antiderivative = 0.91, number of steps used = 9, number of rules used = 8, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {1844, 266, 51, 55, 618, 204, 31, 264} \begin {gather*} \frac {\left (x^3+1\right )^{2/3}}{3 x^3}+\frac {1}{6} \log \left (1-\sqrt [3]{x^3+1}\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\left (x^3+1\right )^{2/3}}{2 x^2}-\frac {\log (x)}{6} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 51
Rule 55
Rule 204
Rule 264
Rule 266
Rule 618
Rule 1844
Rubi steps
\begin {align*} \int \frac {-1+x}{x^4 \sqrt [3]{1+x^3}} \, dx &=\int \left (-\frac {1}{x^4 \sqrt [3]{1+x^3}}+\frac {1}{x^3 \sqrt [3]{1+x^3}}\right ) \, dx\\ &=-\int \frac {1}{x^4 \sqrt [3]{1+x^3}} \, dx+\int \frac {1}{x^3 \sqrt [3]{1+x^3}} \, dx\\ &=-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {1}{3} \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}}{3 x^3}-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{1+x}} \, dx,x,x^3\right )\\ &=\frac {\left (1+x^3\right )^{2/3}}{3 x^3}-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {\log (x)}{6}-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^3}\right )+\frac {1}{6} \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}}{3 x^3}-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {\log (x)}{6}+\frac {1}{6} \log \left (1-\sqrt [3]{1+x^3}\right )-\frac {1}{3} \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}}{3 x^3}-\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\log (x)}{6}+\frac {1}{6} \log \left (1-\sqrt [3]{1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.02, size = 35, normalized size = 0.37 \begin {gather*} -\frac {\left (x^3+1\right )^{2/3} \left (x^2 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};x^3+1\right )+1\right )}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 8.02, size = 95, normalized size = 1.00 \begin {gather*} \frac {(2-3 x) \left (1+x^3\right )^{2/3}}{6 x^3}+\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {1}{9} \log \left (-1+\sqrt [3]{1+x^3}\right )-\frac {1}{18} \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 = 0.73, size = 104, normalized size = 1.09 \begin {gather*} -\frac {2 \, \sqrt {3} x^{3} \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 ) - x^{3} \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 (x^{3} + 1\right )}^{\frac {2}{3}} {\left (3 \, x - 2\right )}}{18 \, x^{3}} \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^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 3.35, size = 90, normalized size = 0.95
method | result | size |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (\frac {4 \pi \sqrt {3}\, x^{3} \hypergeom \left (\left [1, 1, \frac {7}{3}\right ], \left [2, 3\right ], -x^{3}\right )}{27 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \left (2-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )\right ) \pi \sqrt {3}}{9 \Gamma \left (\frac {2}{3}\right )}-\frac {2 \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right ) x^{3}}\right )}{6 \pi }-\frac {\left (x^{3}+1\right )^{\frac {2}{3}}}{2 x^{2}}\) | \(90\) |
risch | \(-\frac {3 x^{4}-2 x^{3}+3 x -2}{6 x^{3} \left (x^{3}+1\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \left (-\frac {2 \pi \sqrt {3}\, x^{3} \hypergeom \left (\left [1, 1, \frac {4}{3}\right ], \left [2, 2\right ], -x^{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {2 \left (-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )\right ) \pi \sqrt {3}}{3 \Gamma \left (\frac {2}{3}\right )}\right )}{18 \pi }\) | \(91\) |
trager | \(-\frac {\left (-2+3 x \right ) \left (x^{3}+1\right )^{\frac {2}{3}}}{6 x^{3}}-\frac {\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 )}{9}-\frac {2 \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 )}{9}+\frac {2 \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 )}{9}\) | \(446\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 78, normalized size = 0.82 \begin {gather*} \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} + \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{3 \, x^{3}} - \frac {1}{18} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{9} \, \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 = 1.34, size = 104, normalized size = 1.09 \begin {gather*} \frac {\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{9}-\frac {1}{9}\right )}{9}+\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{9}-9\,{\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )}^2\right )\,\left (-\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )-\ln \left (\frac {{\left (x^3+1\right )}^{1/3}}{9}-9\,{\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )}^2\right )\,\left (\frac {1}{18}+\frac {\sqrt {3}\,1{}\mathrm {i}}{18}\right )-\frac {{\left (x^3+1\right )}^{2/3}}{2\,x^2}+\frac {{\left (x^3+1\right )}^{2/3}}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.81, size = 53, normalized size = 0.56 \begin {gather*} \frac {\left (1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {2}{3}\right )}{3 \Gamma \left (\frac {1}{3}\right )} + \frac {\Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {e^{i \pi }}{x^{3}}} \right )}}{3 x^{4} \Gamma \left (\frac {7}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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