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 = 0.02, 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*}
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Mathematica [C] time = 0.01, 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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fricas [B] time = 3.55, size = 318, normalized size = 3.00 \[ -\frac {2 \, \sqrt {6} 2^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} - 1\right )} \arctan \left (\frac {2^{\frac {1}{6}} {\left (6 \, \sqrt {6} 2^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 12 \, \sqrt {6} \left (-1\right )^{\frac {1}{3}} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {6} 2^{\frac {1}{3}} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}\right )}}{6 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 2 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} - 1\right )} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + 1\right )} + 6 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}\right ) + 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} - 1\right )} \log \left (-\frac {3 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (5 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} + 12 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) + 36 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{72 \, {\left (x^{3} - 1\right )}} \]
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 \[ \int \frac {x^{3}}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {4}{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 4.23, size = 627, normalized size = 5.92 \[ \text {result too large to display} \]
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 \[ \int \frac {x^{3}}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {4}{3}}}\,{d x} \]
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 \[ \int \frac {x^3}{{\left (1-x^3\right )}^{4/3}\,\left (x^3+1\right )} \,d x \]
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 \[ \int \frac {x^{3}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {4}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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