Optimal. Leaf size=132 \[ -\frac {\log \left (x^3+1\right )}{3 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{\sqrt [3]{2}}-\frac {1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )+\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {2^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}} \]
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Rubi [C] time = 0.01, antiderivative size = 21, normalized size of antiderivative = 0.16, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {429} \[ x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-x^3\right ) \]
Warning: Unable to verify antiderivative.
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Rule 429
Rubi steps
\begin {align*} \int \frac {\left (1-x^3\right )^{2/3}}{1+x^3} \, dx &=x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-x^3\right )\\ \end {align*}
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Mathematica [C] time = 0.10, size = 111, normalized size = 0.84 \[ -\frac {4 x \left (1-x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-x^3\right )}{\left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac {4}{3};-\frac {2}{3},2;\frac {7}{3};x^3,-x^3\right )+2 F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};x^3,-x^3\right )\right )-4 F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-x^3\right )\right )} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.63, size = 191, normalized size = 1.45 \[ -\frac {1}{3} \cdot 4^{\frac {1}{3}} \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 4^{\frac {1}{3}} \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{3} \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) + \frac {1}{3} \cdot 4^{\frac {1}{3}} \log \left (\frac {4^{\frac {2}{3}} x + 2 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) - \frac {1}{6} \cdot 4^{\frac {1}{3}} \log \left (\frac {2 \cdot 4^{\frac {1}{3}} x^{2} - 4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + 2 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) - \frac {1}{3} \, \log \left (\frac {x + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) + \frac {1}{6} \, \log \left (\frac {x^{2} - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) \]
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 \[ \int \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{3} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.60, size = 0, normalized size = 0.00 \[ \int \frac {\left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{3}+1}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{3} + 1}\,{d x} \]
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 \[ \int \frac {{\left (1-x^3\right )}^{2/3}}{x^3+1} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}}}{\left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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