Optimal. Leaf size=250 \[ \frac {2^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}-\frac {1}{2} x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )+\frac {\log \left ((1-x) (1+x)^2\right )}{6 \sqrt [3]{2}}+\frac {\log \left (1+\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )}{3 \sqrt [3]{2}}-\frac {1}{3} 2^{2/3} \log \left (1+\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}\right )-\frac {\log \left (-1+x+2^{2/3} \sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}} \]
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Rubi [A]
time = 0.08, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {495, 371,
502, 2174, 206, 31, 648, 631, 210, 642} \begin {gather*} -\frac {1}{2} x^2 \, _2F_1\left (\frac {1}{3},\frac {2}{3};\frac {5}{3};x^3\right )+\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}-\frac {1}{3} 2^{2/3} \log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )-\frac {\log \left (2^{2/3} \sqrt [3]{1-x^3}+x-1\right )}{2 \sqrt [3]{2}}+\frac {2^{2/3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{\sqrt [3]{2} \sqrt {3}}+\frac {\log \left ((1-x) (x+1)^2\right )}{6 \sqrt [3]{2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 371
Rule 495
Rule 502
Rule 631
Rule 642
Rule 648
Rule 2174
Rubi steps
\begin {align*} \int \frac {x \left (1-x^3\right )^{2/3}}{1+x^3} \, dx &=\frac {1}{2} x^2 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};x^3,-x^3\right )\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 5 in
optimal.
time = 10.01, size = 26, normalized size = 0.10 \begin {gather*} \frac {1}{2} x^2 F_1\left (\frac {2}{3};-\frac {2}{3},1;\frac {5}{3};x^3,-x^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-1)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {x \left (-x^{3}+1\right )^{\frac {2}{3}}}{x^{3}+1}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 1.61, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \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 = -0.00 \begin {gather*} \int \frac {x\,{\left (1-x^3\right )}^{2/3}}{x^3+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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