Optimal. Leaf size=101 \[ \frac {\sqrt [3]{x^3+1} \left (x^3+3\right )}{3 x}+\frac {2}{9} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}+x}\right )}{3 \sqrt {3}}-\frac {1}{9} \log \left (\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 110, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {453, 279, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} \frac {\left (x^3+1\right )^{4/3}}{x}+\frac {2}{9} \log \left (1-\frac {x}{\sqrt [3]{x^3+1}}\right )+\frac {2 \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2}{3} \sqrt [3]{x^3+1} x^2-\frac {1}{9} \log \left (\frac {x}{\sqrt [3]{x^3+1}}+\frac {x^2}{\left (x^3+1\right )^{2/3}}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 204
Rule 279
Rule 292
Rule 331
Rule 453
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x^2} \, dx &=\frac {\left (1+x^3\right )^{4/3}}{x}-2 \int x \sqrt [3]{1+x^3} \, dx\\ &=-\frac {2}{3} x^2 \sqrt [3]{1+x^3}+\frac {\left (1+x^3\right )^{4/3}}{x}-\frac {2}{3} \int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx\\ &=-\frac {2}{3} x^2 \sqrt [3]{1+x^3}+\frac {\left (1+x^3\right )^{4/3}}{x}-\frac {2}{3} \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2}{3} x^2 \sqrt [3]{1+x^3}+\frac {\left (1+x^3\right )^{4/3}}{x}-\frac {2}{9} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {2}{9} \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2}{3} x^2 \sqrt [3]{1+x^3}+\frac {\left (1+x^3\right )^{4/3}}{x}+\frac {2}{9} \log \left (1-\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {1}{9} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2}{3} x^2 \sqrt [3]{1+x^3}+\frac {\left (1+x^3\right )^{4/3}}{x}+\frac {2}{9} \log \left (1-\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {1}{9} \log \left (1+\frac {x^2}{\left (1+x^3\right )^{2/3}}+\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {2}{3} x^2 \sqrt [3]{1+x^3}+\frac {\left (1+x^3\right )^{4/3}}{x}+\frac {2 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {2}{9} \log \left (1-\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {1}{9} \log \left (1+\frac {x^2}{\left (1+x^3\right )^{2/3}}+\frac {x}{\sqrt [3]{1+x^3}}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 34, normalized size = 0.34 \begin {gather*} \frac {\left (x^3+1\right )^{4/3}}{x}-x^2 \, _2F_1\left (-\frac {1}{3},\frac {2}{3};\frac {5}{3};-x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.18, size = 101, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{1+x^3} \left (3+x^3\right )}{3 x}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{3 \sqrt {3}}+\frac {2}{9} \log \left (-x+\sqrt [3]{1+x^3}\right )-\frac {1}{9} \log \left (x^2+x \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 = 0.78, size = 105, normalized size = 1.04 \begin {gather*} \frac {2 \, \sqrt {3} x \arctan \left (-\frac {25382 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 13720 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (5831 \, x^{3} + 7200\right )}}{58653 \, x^{3} + 8000}\right ) + x \log \left (3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1\right ) + 3 \, {\left (x^{3} + 3\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{9 \, x} \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 {{\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{3} - 1\right )}}{x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.17, size = 33, normalized size = 0.33
method | result | size |
meijerg | \(\frac {\hypergeom \left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], -x^{3}\right )}{x}+\frac {x^{2} \hypergeom \left (\left [-\frac {1}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )}{2}\) | \(33\) |
risch | \(\frac {x^{6}+4 x^{3}+3}{3 x \left (x^{3}+1\right )^{\frac {2}{3}}}-\frac {x^{2} \hypergeom \left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )}{3}\) | \(40\) |
trager | \(\frac {\left (x^{3}+1\right )^{\frac {1}{3}} \left (x^{3}+3\right )}{3 x}+\frac {2 \ln \left (317 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-555 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +2358 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}-1486 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-2358 x \left (x^{3}+1\right )^{\frac {2}{3}}+1803 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+872 x^{3}-317 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-733 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+654\right )}{9}+\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-535 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+555 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +1803 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {1}{3}} x^{2}-2893 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-1803 x \left (x^{3}+1\right )^{\frac {2}{3}}+2358 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-1090 x^{3}+535 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-852 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-436\right )}{9}\) | \(254\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 105, normalized size = 1.04 \begin {gather*} -\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) + \frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x {\left (\frac {x^{3} + 1}{x^{3}} - 1\right )}} - \frac {1}{9} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {2}{9} \, \log \left (\frac {{\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^3-1\right )\,{\left (x^3+1\right )}^{1/3}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.16, size = 65, normalized size = 0.64 \begin {gather*} \frac {x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {2}{3} \\ \frac {5}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} - \frac {\Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, - \frac {1}{3} \\ \frac {2}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 x \Gamma \left (\frac {2}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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