Optimal. Leaf size=96 \[ -\frac {1}{9} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}+x}\right )}{3 \sqrt {3}}+\frac {1}{3} \sqrt [3]{x^3+1} x^2+\frac {1}{18} \log \left (\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 97, normalized size of antiderivative = 1.01, number of steps used = 8, number of rules used = 8, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {279, 331, 292, 31, 634, 618, 204, 628} \begin {gather*} -\frac {1}{9} \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 )}{3 \sqrt {3}}+\frac {1}{3} \sqrt [3]{x^3+1} x^2+\frac {1}{18} \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 618
Rule 628
Rule 634
Rubi steps
\begin {align*} \int x \sqrt [3]{1+x^3} \, dx &=\frac {1}{3} x^2 \sqrt [3]{1+x^3}+\frac {1}{3} \int \frac {x}{\left (1+x^3\right )^{2/3}} \, dx\\ &=\frac {1}{3} x^2 \sqrt [3]{1+x^3}+\frac {1}{3} \operatorname {Subst}\left (\int \frac {x}{1-x^3} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=\frac {1}{3} x^2 \sqrt [3]{1+x^3}+\frac {1}{9} \operatorname {Subst}\left (\int \frac {1}{1-x} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {1}{9} \operatorname {Subst}\left (\int \frac {1-x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=\frac {1}{3} x^2 \sqrt [3]{1+x^3}-\frac {1}{9} \log \left (1-\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{18} \operatorname {Subst}\left (\int \frac {1+2 x}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x+x^2} \, dx,x,\frac {x}{\sqrt [3]{1+x^3}}\right )\\ &=\frac {1}{3} x^2 \sqrt [3]{1+x^3}-\frac {1}{9} \log \left (1-\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{18} \log \left (1+\frac {x^2}{\left (1+x^3\right )^{2/3}}+\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 x}{\sqrt [3]{1+x^3}}\right )\\ &=\frac {1}{3} x^2 \sqrt [3]{1+x^3}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{9} \log \left (1-\frac {x}{\sqrt [3]{1+x^3}}\right )+\frac {1}{18} \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 = 22, normalized size = 0.23 \begin {gather*} \frac {1}{2} 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 = 0.16, size = 96, normalized size = 1.00 \begin {gather*} -\frac {1}{9} \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 )}{3 \sqrt {3}}+\frac {1}{3} \sqrt [3]{x^3+1} x^2+\frac {1}{18} \log \left (\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 88, normalized size = 0.92 \begin {gather*} \frac {1}{3} \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + \frac {1}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \frac {1}{9} \, \log \left (-\frac {x - {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{18} \, \log \left (\frac {x^{2} + {\left (x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \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 {\left (x^{3} + 1\right )}^{\frac {1}{3}} x\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.34, size = 30, normalized size = 0.31 \begin {gather*} \frac {x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}}{3}+\frac {x^{2} \hypergeom \left (\left [\frac {2}{3}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )}{6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 94, normalized size = 0.98 \begin {gather*} \frac {1}{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}}}{3 \, x {\left (\frac {x^{3} + 1}{x^{3}} - 1\right )}} + \frac {1}{18} \, \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}{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 x\,{\left (x^3+1\right )}^{1/3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.83, size = 31, normalized size = 0.32 \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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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