Optimal. Leaf size=94 \[ \frac {1}{3} \left (x^3+1\right )^{2/3} x-\frac {2}{9} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {2 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}+x}\right )}{3 \sqrt {3}}+\frac {1}{9} \log \left (\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 63, normalized size of antiderivative = 0.67, number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {195, 239} \begin {gather*} \frac {1}{3} \left (x^3+1\right )^{2/3} x-\frac {1}{3} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {2 \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 195
Rule 239
Rubi steps
\begin {align*} \int \left (1+x^3\right )^{2/3} \, dx &=\frac {1}{3} x \left (1+x^3\right )^{2/3}+\frac {2}{3} \int \frac {1}{\sqrt [3]{1+x^3}} \, dx\\ &=\frac {1}{3} x \left (1+x^3\right )^{2/3}+\frac {2 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {1}{3} \log \left (-x+\sqrt [3]{1+x^3}\right )\\ \end {align*}
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Mathematica [C] time = 0.06, size = 99, normalized size = 1.05 \begin {gather*} \frac {3 (x+1) \left (x^3+1\right )^{2/3} F_1\left (\frac {5}{3};-\frac {2}{3},-\frac {2}{3};\frac {8}{3};-\frac {x+1}{-1+(-1)^{2/3}},-\frac {x+1}{-1-\sqrt [3]{-1}}\right )}{5 \left (\frac {x+1}{-1-\sqrt [3]{-1}}+1\right )^{2/3} \left (\frac {x+1}{(-1)^{2/3}-1}+1\right )^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 0.18, size = 94, normalized size = 1.00 \begin {gather*} \frac {1}{3} x \left (1+x^3\right )^{2/3}+\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 = 0.44, size = 86, normalized size = 0.91 \begin {gather*} \frac {1}{3} \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x - \frac {2}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) - \frac {2}{9} \, \log \left (-\frac {x - {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{9} \, \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 {2}{3}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.64, size = 14, normalized size = 0.15
method | result | size |
meijerg | \(x \hypergeom \left (\left [-\frac {2}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{3}\right )\) | \(14\) |
risch | \(\frac {x \left (x^{3}+1\right )^{\frac {2}{3}}}{3}+\frac {2 x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -x^{3}\right )}{3}\) | \(26\) |
trager | \(\frac {x \left (x^{3}+1\right )^{\frac {2}{3}}}{3}+\frac {2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}-3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x -3 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}-4 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}+1\right )^{\frac {2}{3}}+3 x^{2} \left (x^{3}+1\right )^{\frac {1}{3}}+4 x^{3}-\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+2\right )}{9}-\frac {2 \ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1\right ) \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )}{9}+\frac {2 \ln \left (\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )^{2} x^{3}+3 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) \left (x^{3}+1\right )^{\frac {2}{3}} x +3 \left (x^{3}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{2}+2 \RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right ) x^{3}+x^{3}+\RootOf \left (\textit {\_Z}^{2}-\textit {\_Z} +1\right )+1\right )}{9}\) | \(314\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.76, size = 94, normalized size = 1.00 \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 {2}{3}}}{3 \, x^{2} {\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 [B] time = 0.77, size = 12, normalized size = 0.13 \begin {gather*} x\,{{}}_2{\mathrm {F}}_1\left (-\frac {2}{3},\frac {1}{3};\ \frac {4}{3};\ -x^3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 0.81, size = 29, normalized size = 0.31 \begin {gather*} \frac {x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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