Optimal. Leaf size=89 \[ \frac {1}{4} \sqrt [3]{x^3+1} \left (x^3-3\right )-\frac {1}{3} \log \left (\sqrt [3]{x^3+1}-1\right )+\frac {1}{6} \log \left (\left (x^3+1\right )^{2/3}+\sqrt [3]{x^3+1}+1\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [A] time = 0.05, antiderivative size = 75, normalized size of antiderivative = 0.84, number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {446, 80, 50, 57, 618, 204, 31} \begin {gather*} \frac {1}{4} \left (x^3+1\right )^{4/3}-\sqrt [3]{x^3+1}-\frac {1}{2} \log \left (1-\sqrt [3]{x^3+1}\right )+\frac {\tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log (x)}{2} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 50
Rule 57
Rule 80
Rule 204
Rule 446
Rule 618
Rubi steps
\begin {align*} \int \frac {\left (-1+x^3\right ) \sqrt [3]{1+x^3}}{x} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {(-1+x) \sqrt [3]{1+x}}{x} \, dx,x,x^3\right )\\ &=\frac {1}{4} \left (1+x^3\right )^{4/3}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {\sqrt [3]{1+x}}{x} \, dx,x,x^3\right )\\ &=-\sqrt [3]{1+x^3}+\frac {1}{4} \left (1+x^3\right )^{4/3}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x (1+x)^{2/3}} \, dx,x,x^3\right )\\ &=-\sqrt [3]{1+x^3}+\frac {1}{4} \left (1+x^3\right )^{4/3}+\frac {\log (x)}{2}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\sqrt [3]{1+x^3}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\sqrt [3]{1+x^3}\right )\\ &=-\sqrt [3]{1+x^3}+\frac {1}{4} \left (1+x^3\right )^{4/3}+\frac {\log (x)}{2}-\frac {1}{2} \log \left (1-\sqrt [3]{1+x^3}\right )-\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+2 \sqrt [3]{1+x^3}\right )\\ &=-\sqrt [3]{1+x^3}+\frac {1}{4} \left (1+x^3\right )^{4/3}+\frac {\tan ^{-1}\left (\frac {1+2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\log (x)}{2}-\frac {1}{2} \log \left (1-\sqrt [3]{1+x^3}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 96, normalized size = 1.08 \begin {gather*} \frac {1}{12} \left (3 \sqrt [3]{x^3+1} x^3-9 \sqrt [3]{x^3+1}-4 \log \left (1-\sqrt [3]{x^3+1}\right )+2 \log \left (\left (x^3+1\right )^{2/3}+\sqrt [3]{x^3+1}+1\right )+4 \sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{x^3+1}+1}{\sqrt {3}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 89, normalized size = 1.00 \begin {gather*} \frac {1}{4} \left (-3+x^3\right ) \sqrt [3]{1+x^3}+\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 \sqrt [3]{1+x^3}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{3} \log \left (-1+\sqrt [3]{1+x^3}\right )+\frac {1}{6} \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 = 0.45, size = 70, normalized size = 0.79 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{4} \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} {\left (x^{3} - 3\right )} + \frac {1}{6} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \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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giac [A] time = 0.29, size = 73, normalized size = 0.82 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{4} \, {\left (x^{3} + 1\right )}^{\frac {4}{3}} - {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{6} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \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 = 2.40, size = 65, normalized size = 0.73
method | result | size |
meijerg | \(\frac {-\Gamma \left (\frac {2}{3}\right ) x^{3} \hypergeom \left (\left [\frac {2}{3}, 1, 1\right ], \left [2, 2\right ], -x^{3}\right )-3 \left (3+\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \relax (3)}{2}+3 \ln \relax (x )\right ) \Gamma \left (\frac {2}{3}\right )}{9 \Gamma \left (\frac {2}{3}\right )}+\frac {x^{3} \hypergeom \left (\left [-\frac {1}{3}, 1\right ], \relax [2], -x^{3}\right )}{3}\) | \(65\) |
trager | \(\left (\frac {x^{3}}{4}-\frac {3}{4}\right ) \left (x^{3}+1\right )^{\frac {1}{3}}+\frac {\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\frac {-4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+17 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+15 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}-15 x^{3}-24 \left (x^{3}+1\right )^{\frac {2}{3}}+15 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}-24 \left (x^{3}+1\right )^{\frac {1}{3}}+11 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-20}{x^{3}}\right )}{3}-\frac {\ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+9 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+15 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{3}+9 \left (x^{3}+1\right )^{\frac {2}{3}}+15 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+9 \left (x^{3}+1\right )^{\frac {1}{3}}+19 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+5}{x^{3}}\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{3}+\frac {\ln \left (-\frac {4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+9 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+15 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}}+2 x^{3}+9 \left (x^{3}+1\right )^{\frac {2}{3}}+15 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )-4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2}+9 \left (x^{3}+1\right )^{\frac {1}{3}}+19 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+5}{x^{3}}\right )}{3}\) | \(403\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 72, normalized size = 0.81 \begin {gather*} \frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) + \frac {1}{4} \, {\left (x^{3} + 1\right )}^{\frac {4}{3}} - {\left (x^{3} + 1\right )}^{\frac {1}{3}} + \frac {1}{6} \, \log \left ({\left (x^{3} + 1\right )}^{\frac {2}{3}} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} + 1\right ) - \frac {1}{3} \, \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 = 0.95, size = 86, normalized size = 0.97 \begin {gather*} \frac {{\left (x^3+1\right )}^{4/3}}{4}-{\left (x^3+1\right )}^{1/3}-\frac {\ln \left ({\left (x^3+1\right )}^{1/3}-1\right )}{3}-\ln \left (3\,{\left (x^3+1\right )}^{1/3}+\frac {3}{2}-\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )+\ln \left (3\,{\left (x^3+1\right )}^{1/3}+\frac {3}{2}+\frac {\sqrt {3}\,3{}\mathrm {i}}{2}\right )\,\left (\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 26.65, size = 82, normalized size = 0.92 \begin {gather*} \frac {\left (x^{3} + 1\right )^{\frac {4}{3}}}{4} - \sqrt [3]{x^{3} + 1} - \frac {\log {\left (\sqrt [3]{x^{3} + 1} - 1 \right )}}{3} + \frac {\log {\left (\left (x^{3} + 1\right )^{\frac {2}{3}} + \sqrt [3]{x^{3} + 1} + 1 \right )}}{6} + \frac {\sqrt {3} \operatorname {atan}{\left (\frac {2 \sqrt {3} \left (\sqrt [3]{x^{3} + 1} + \frac {1}{2}\right )}{3} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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