Optimal. Leaf size=291 \[ -\frac{1}{2} \sqrt [3]{1-x^3} x+\frac{\log \left (2^{2/3}-\frac{1-x}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}-\frac{\log \left (\frac{2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{6\ 2^{2/3}}+\frac{\log \left (\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}-\frac{\log \left (\frac{(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac{2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}\right )}{12\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{\frac{\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt{3}}\right )}{2\ 2^{2/3} \sqrt{3}} \]
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Rubi [C] time = 0.0189698, antiderivative size = 26, normalized size of antiderivative = 0.09, 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} \[ \frac{1}{7} x^7 F_1\left (\frac{7}{3};\frac{2}{3},1;\frac{10}{3};x^3,-x^3\right ) \]
Warning: Unable to verify antiderivative.
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Rule 510
Rubi steps
\begin{align*} \int \frac{x^6}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx &=\frac{1}{7} x^7 F_1\left (\frac{7}{3};\frac{2}{3},1;\frac{10}{3};x^3,-x^3\right )\\ \end{align*}
Mathematica [C] time = 0.118928, size = 115, normalized size = 0.4 \[ \frac{1}{2} x \sqrt [3]{1-x^3} \left (-\frac{4 F_1\left (\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};x^3,-x^3\right )}{\left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac{4}{3};-\frac{1}{3},2;\frac{7}{3};x^3,-x^3\right )+F_1\left (\frac{4}{3};\frac{2}{3},1;\frac{7}{3};x^3,-x^3\right )\right )-4 F_1\left (\frac{1}{3};-\frac{1}{3},1;\frac{4}{3};x^3,-x^3\right )\right )}-1\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.056, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{6}}{{x}^{3}+1} \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{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^{6}}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 17.7998, size = 949, normalized size = 3.26 \begin{align*} \frac{1}{36} \cdot 4^{\frac{1}{6}} \sqrt{3} \arctan \left (-\frac{4^{\frac{1}{6}}{\left (6 \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (x^{16} - 33 \, x^{13} + 110 \, x^{10} - 110 \, x^{7} + 33 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 48 \, \sqrt{3}{\left (x^{14} - 2 \, x^{11} - 6 \, x^{8} - 2 \, x^{5} + x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 4^{\frac{1}{3}} \sqrt{3}{\left (x^{18} + 42 \, x^{15} - 417 \, x^{12} + 812 \, x^{9} - 417 \, x^{6} + 42 \, x^{3} + 1\right )}\right )}}{6 \,{\left (x^{18} - 102 \, x^{15} + 447 \, x^{12} - 628 \, x^{9} + 447 \, x^{6} - 102 \, x^{3} + 1\right )}}\right ) + \frac{1}{72} \cdot 4^{\frac{2}{3}} \log \left (-\frac{12 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x^{2} - 3 \cdot 4^{\frac{2}{3}}{\left (x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 4^{\frac{1}{3}}{\left (x^{6} + 2 \, x^{3} + 1\right )}}{x^{6} + 2 \, x^{3} + 1}\right ) - \frac{1}{144} \cdot 4^{\frac{2}{3}} \log \left (\frac{24 \cdot 4^{\frac{1}{3}}{\left (x^{8} - 4 \, x^{5} + x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} + 4^{\frac{2}{3}}{\left (x^{12} - 32 \, x^{9} + 78 \, x^{6} - 32 \, x^{3} + 1\right )} + 12 \,{\left (x^{10} - 11 \, x^{7} + 11 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x^{12} + 4 \, x^{9} + 6 \, x^{6} + 4 \, x^{3} + 1}\right ) - \frac{1}{2} \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x \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^{6}}{\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{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^{6}}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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