Optimal. Leaf size=67 \[ \frac {1}{3} \left (1-x^3\right )^{2/3} x+\frac {1}{3} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac {2 \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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Rubi [A] time = 0.01, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {195, 239} \[ \frac {1}{3} \left (1-x^3\right )^{2/3} x+\frac {1}{3} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac {2 \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}} \]
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 = 0.12, size = 101, normalized size = 1.51 \[ \frac {3 (x-1) \left (1-x^3\right )^{2/3} F_1\left (\frac {5}{3};-\frac {2}{3},-\frac {2}{3};\frac {8}{3};-\frac {x-1}{1-(-1)^{2/3}},-\frac {x-1}{1+\sqrt [3]{-1}}\right )}{5 \left (\frac {x-1}{1+\sqrt [3]{-1}}+1\right )^{2/3} \left (\frac {x-1}{1-(-1)^{2/3}}+1\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.62, size = 94, normalized size = 1.40 \[ \frac {1}{3} \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x - \frac {2}{9} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {2}{9} \, \log \left (\frac {x + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{9} \, \log \left (\frac {x^{2} - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
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 \[ \int {\left (-x^{3} + 1\right )}^{\frac {2}{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.10, size = 12, normalized size = 0.18 \[ x \hypergeom \left (\left [-\frac {2}{3}, \frac {1}{3}\right ], \left [\frac {4}{3}\right ], x^{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.30, size = 105, normalized size = 1.57 \[ -\frac {2}{9} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (\frac {2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x} - 1\right )}\right ) - \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{3 \, x^{2} {\left (\frac {x^{3} - 1}{x^{3}} - 1\right )}} + \frac {2}{9} \, \log \left (\frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x} + 1\right ) - \frac {1}{9} \, \log \left (-\frac {{\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x} + \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 10, normalized size = 0.15 \[ x\,{{}}_2{\mathrm {F}}_1\left (-\frac {2}{3},\frac {1}{3};\ \frac {4}{3};\ x^3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.02, size = 31, normalized size = 0.46 \[ \frac {x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{3} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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