Optimal. Leaf size=106 \[ \frac{x}{2 \sqrt [3]{1-x^3}}+\frac{\log \left (x^3+1\right )}{12 \sqrt [3]{2}}-\frac{\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{4 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}} \]
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Rubi [C] time = 0.018378, antiderivative size = 38, normalized size of antiderivative = 0.36, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {510} \[ \frac{x^4 \, _2F_1\left (\frac{4}{3},\frac{4}{3};\frac{7}{3};\frac{2 x^3}{x^3+1}\right )}{4 \left (x^3+1\right )^{4/3}} \]
Warning: Unable to verify antiderivative.
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Rule 510
Rubi steps
\begin{align*} \int \frac{x^3}{\left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx &=\frac{x^4 \, _2F_1\left (\frac{4}{3},\frac{4}{3};\frac{7}{3};\frac{2 x^3}{1+x^3}\right )}{4 \left (1+x^3\right )^{4/3}}\\ \end{align*}
Mathematica [C] time = 0.0092013, size = 38, normalized size = 0.36 \[ \frac{x^4 \, _2F_1\left (\frac{4}{3},\frac{4}{3};\frac{7}{3};\frac{2 x^3}{x^3+1}\right )}{4 \left (x^3+1\right )^{4/3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.052, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{{x}^{3}+1} \left ( -{x}^{3}+1 \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 21.7687, size = 876, normalized size = 8.26 \begin{align*} -\frac{2 \, \sqrt{6} 2^{\frac{1}{6}} \left (-1\right )^{\frac{1}{3}}{\left (x^{3} - 1\right )} \arctan \left (\frac{2^{\frac{1}{6}}{\left (6 \, \sqrt{6} 2^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (5 \, x^{7} + 4 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 12 \, \sqrt{6} \left (-1\right )^{\frac{1}{3}}{\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \sqrt{6} 2^{\frac{1}{3}}{\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}\right )}}{6 \,{\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 2 \cdot 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{3} - 1\right )} \log \left (\frac{6 \cdot 2^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} - 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{3} + 1\right )} + 6 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x}{x^{3} + 1}\right ) + 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (x^{3} - 1\right )} \log \left (-\frac{3 \cdot 2^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (5 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 2^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}}{\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} + 12 \,{\left (2 \, x^{5} - x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) + 36 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x}{72 \,{\left (x^{3} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac{4}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{4}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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