Optimal. Leaf size=124 \[ -\frac{\left (1-x^3\right )^{2/3}}{x^2}+\frac{1}{2 x^2 \sqrt [3]{1-x^3}}+\frac{\log \left (x^3+1\right )}{12 \sqrt [3]{2}}-\frac{\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{4 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2 \sqrt [3]{2} \sqrt{3}} \]
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Rubi [C] time = 8.12141, antiderivative size = 204, normalized size of antiderivative = 1.65, 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} \[ -\frac{-18 \left (x^3+1\right )^2 x^6 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-54 x^{12} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-84 x^9 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-30 x^6 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-7 \left (1-x^3\right )^2 \left (9 x^6+12 x^3+2\right ) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )+63 x^{12}-42 x^9-91 x^6+56 x^3+14}{14 x^5 \left (1-x^3\right )^{7/3}} \]
Warning: Unable to verify antiderivative.
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Rule 510
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx &=-\frac{14+56 x^3-91 x^6-42 x^9+63 x^{12}-7 \left (1-x^3\right )^2 \left (2+12 x^3+9 x^6\right ) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )-30 x^6 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-84 x^9 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-54 x^{12} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-18 x^6 \left (1+x^3\right )^2 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )}{14 x^5 \left (1-x^3\right )^{7/3}}\\ \end{align*}
Mathematica [C] time = 2.11795, size = 192, normalized size = 1.55 \[ \frac{18 \left (x^3+1\right )^2 x^6 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{7}{3}\right \},\left \{1,\frac{10}{3}\right \},\frac{2 x^3}{x^3-1}\right )+54 x^{12} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{2 x^3}{x^3-1}\right )+84 x^9 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{2 x^3}{x^3-1}\right )+30 x^6 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{2 x^3}{x^3-1}\right )+7 \left (x^3-1\right )^2 \left (9 x^6+12 x^3+2\right ) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 x^3}{x^3-1}\right )-63 x^{12}+42 x^9+91 x^6-56 x^3-14}{14 x^5 \left (1-x^3\right )^{7/3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ({x}^{3}+1 \right ) } \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{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{4}{3}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 23.2538, size = 900, normalized size = 7.26 \begin{align*} -\frac{2 \, \sqrt{6} 2^{\frac{1}{6}} \left (-1\right )^{\frac{1}{3}}{\left (x^{5} - x^{2}\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^{5} - x^{2}\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^{5} - x^{2}\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 (2 \, x^{3} - 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{72 \,{\left (x^{5} - x^{2}\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{1}{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{1}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{4}{3}} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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