Optimal. Leaf size=92 \[ -\frac {\sqrt [3]{x^3+1}}{x}-\frac {1}{3} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}+x}\right )}{\sqrt {3}}+\frac {1}{6} \log \left (\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 93, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {277, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} -\frac {\sqrt [3]{x^3+1}}{x}-\frac {1}{3} \log \left (1-\frac {x}{\sqrt [3]{x^3+1}}\right )-\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{6} \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 277
Rule 292
Rule 331
Rule 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {\sqrt [3]{1+x^3}}{x^2} \, dx &=-\frac {\sqrt [3]{1+x^3}}{x}+\int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx\\ &=-\frac {\sqrt [3]{1+x^3}}{x}+\operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {\sqrt [3]{1+x^3}}{x}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {\sqrt [3]{1+x^3}}{x}-\frac {1}{3} \log \left (1-\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {\sqrt [3]{1+x^3}}{x}-\frac {1}{3} \log \left (1-\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{6} \log \left (1+\frac {x^2}{\left (1+x^3\right )^{2/3}}+\frac {x}{\sqrt [3]{1+x^3}}\right )+\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{1+x^3}}\right )\\ &=-\frac {\sqrt [3]{1+x^3}}{x}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (1-\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{6} \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 = 0.00, size = 20, normalized size = 0.22 \begin {gather*} -\frac {\, _2F_1\left (-\frac {1}{3},-\frac {1}{3};\frac {2}{3};-x^3\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.13, size = 92, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [3]{1+x^3}}{x}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{6} \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 = 0.79, size = 100, normalized size = 1.09 \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 ) + 6 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{6 \, 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}}}{x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.25, size = 17, normalized size = 0.18
method | result | size |
meijerg | \(-\frac {\hypergeom \left (\left [-\frac {1}{3}, -\frac {1}{3}\right ], \left [\frac {2}{3}\right ], -x^{3}\right )}{x}\) | \(17\) |
risch | \(-\frac {\left (x^{3}+1\right )^{\frac {1}{3}}}{x}+\frac {x^{2} \hypergeom \left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )}{2}\) | \(30\) |
trager | \(-\frac {\left (x^{3}+1\right )^{\frac {1}{3}}}{x}+\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -3 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-5 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-54 x \left (x^{3}+1\right )^{\frac {2}{3}}-54 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-52 x^{3}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+19 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-39\right )}{3}-\frac {\ln \left (2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-57 x \left (x^{3}+1\right )^{\frac {2}{3}}-57 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-55 x^{3}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-15 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-22\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{3}+\frac {\ln \left (2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}-57 x \left (x^{3}+1\right )^{\frac {2}{3}}-57 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}-55 x^{3}-2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-15 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-22\right )}{3}\) | \(408\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.02, size = 81, normalized size = 0.88 \begin {gather*} \frac {1}{3} \, \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 {1}{6} \, \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 {1}{3} \, \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 [B] time = 0.90, size = 15, normalized size = 0.16 \begin {gather*} -\frac {{{}}_2{\mathrm {F}}_1\left (-\frac {1}{3},-\frac {1}{3};\ \frac {2}{3};\ -x^3\right )}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.82, size = 32, normalized size = 0.35 \begin {gather*} \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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