Optimal. Leaf size=63 \[ \frac {1}{3} \left (x^3+2\right )^{2/3} x+\frac {5}{6} \log \left (\sqrt [3]{x^3+2}-x\right )-\frac {5 \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.01, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {388, 239} \[ \frac {1}{3} \left (x^3+2\right )^{2/3} x+\frac {5}{6} \log \left (\sqrt [3]{x^3+2}-x\right )-\frac {5 \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 239
Rule 388
Rubi steps
\begin {align*} \int \frac {-1+x^3}{\sqrt [3]{2+x^3}} \, dx &=\frac {1}{3} x \left (2+x^3\right )^{2/3}-\frac {5}{3} \int \frac {1}{\sqrt [3]{2+x^3}} \, dx\\ &=\frac {1}{3} x \left (2+x^3\right )^{2/3}-\frac {5 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{2+x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {5}{6} \log \left (-x+\sqrt [3]{2+x^3}\right )\\ \end {align*}
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Mathematica [A] time = 0.03, size = 91, normalized size = 1.44 \[ \frac {1}{18} \left (6 \left (x^3+2\right )^{2/3} x+10 \log \left (1-\frac {x}{\sqrt [3]{x^3+2}}\right )-10 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt {3}}\right )-5 \log \left (\frac {x}{\sqrt [3]{x^3+2}}+\frac {x^2}{\left (x^3+2\right )^{2/3}}+1\right )\right ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.15, size = 94, normalized size = 1.49 \[ \frac {1}{3} \left (x^3+2\right )^{2/3} x+\frac {5}{9} \log \left (\sqrt [3]{x^3+2}-x\right )-\frac {5 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+2}+x}\right )}{3 \sqrt {3}}-\frac {5}{18} \log \left (\sqrt [3]{x^3+2} x+\left (x^3+2\right )^{2/3}+x^2\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 86, normalized size = 1.37 \[ \frac {1}{3} \, {\left (x^{3} + 2\right )}^{\frac {2}{3}} x + \frac {5}{9} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (x^{3} + 2\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {5}{9} \, \log \left (-\frac {x - {\left (x^{3} + 2\right )}^{\frac {1}{3}}}{x}\right ) - \frac {5}{18} \, \log \left (\frac {x^{2} + {\left (x^{3} + 2\right )}^{\frac {1}{3}} x + {\left (x^{3} + 2\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
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 \[ \int \frac {x^{3} - 1}{{\left (x^{3} + 2\right )}^{\frac {1}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.63, size = 29, normalized size = 0.46
method | result | size |
risch | \(\frac {x \left (x^{3}+2\right )^{\frac {2}{3}}}{3}-\frac {5 \,2^{\frac {2}{3}} x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -\frac {x^{3}}{2}\right )}{6}\) | \(29\) |
meijerg | \(-\frac {2^{\frac {2}{3}} x \hypergeom \left (\left [\frac {1}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], -\frac {x^{3}}{2}\right )}{2}+\frac {2^{\frac {2}{3}} x^{4} \hypergeom \left (\left [\frac {1}{3}, \frac {4}{3}\right ], \left [\frac {7}{3}\right ], -\frac {x^{3}}{2}\right )}{8}\) | \(38\) |
trager | \(\frac {x \left (x^{3}+2\right )^{\frac {2}{3}}}{3}+\frac {5 \ln \left (-4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}+6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-8 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}+3 x \left (x^{3}+2\right )^{\frac {2}{3}}-4 x^{3}-4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-4\right )}{9}+\frac {10 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \ln \left (4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )^{2} x^{3}-6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {2}{3}} x +6 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}+2 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right ) x^{3}+3 \left (x^{3}+2\right )^{\frac {1}{3}} x^{2}-2 x^{3}-4 \RootOf \left (4 \textit {\_Z}^{2}+2 \textit {\_Z} +1\right )-4\right )}{9}\) | \(226\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.21, size = 94, normalized size = 1.49 \[ \frac {5}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (x^{3} + 2\right )}^{\frac {1}{3}}}{x} + 1\right )}\right ) + \frac {2 \, {\left (x^{3} + 2\right )}^{\frac {2}{3}}}{3 \, x^{2} {\left (\frac {x^{3} + 2}{x^{3}} - 1\right )}} - \frac {5}{18} \, \log \left (\frac {{\left (x^{3} + 2\right )}^{\frac {1}{3}}}{x} + \frac {{\left (x^{3} + 2\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) + \frac {5}{9} \, \log \left (\frac {{\left (x^{3} + 2\right )}^{\frac {1}{3}}}{x} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^3-1}{{\left (x^3+2\right )}^{1/3}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.00, size = 71, normalized size = 1.13 \[ \frac {2^{\frac {2}{3}} x^{4} \Gamma \left (\frac {4}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {4}{3} \\ \frac {7}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{2}} \right )}}{6 \Gamma \left (\frac {7}{3}\right )} - \frac {2^{\frac {2}{3}} x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {\frac {x^{3} e^{i \pi }}{2}} \right )}}{6 \Gamma \left (\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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