Optimal. Leaf size=88 \[ \frac{\log \left (x^3+1\right )}{6\ 2^{2/3}}-\frac{\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}} \]
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Rubi [A] time = 0.0611541, antiderivative size = 122, normalized size of antiderivative = 1.39, number of steps used = 7, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {494, 292, 31, 634, 617, 204, 628} \[ \frac{\log \left (\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{6\ 2^{2/3}}-\frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 494
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx &=\operatorname{Subst}\left (\int \frac{x}{1+2 x^3} \, dx,x,\frac{x}{\sqrt [3]{1-x^3}}\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{1+\sqrt [3]{2} x} \, dx,x,\frac{x}{\sqrt [3]{1-x^3}}\right )}{3 \sqrt [3]{2}}+\frac{\operatorname{Subst}\left (\int \frac{1+\sqrt [3]{2} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{1-x^3}}\right )}{3 \sqrt [3]{2}}\\ &=-\frac{\log \left (1+\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{-\sqrt [3]{2}+2\ 2^{2/3} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac{x}{\sqrt [3]{1-x^3}}\right )}{2 \sqrt [3]{2}}\\ &=\frac{\log \left (1+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}-\frac{\log \left (1+\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{2^{2/3}}\\ &=-\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}+\frac{\log \left (1+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}-\frac{\log \left (1+\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\\ \end{align*}
Mathematica [C] time = 0.0066669, size = 38, normalized size = 0.43 \[ \frac{x^2 \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{2 x^3}{x^3+1}\right )}{2 \left (x^3+1\right )^{2/3}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{x}{{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}{{\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 [B] time = 13.8144, size = 797, normalized size = 9.06 \begin{align*} -\frac{1}{18} \cdot 4^{\frac{1}{6}} \sqrt{3} \left (-1\right )^{\frac{1}{3}} \arctan \left (-\frac{4^{\frac{1}{6}}{\left (6 \cdot 4^{\frac{2}{3}} \sqrt{3} \left (-1\right )^{\frac{2}{3}}{\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 12 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}}{\left (5 \, x^{7} + 4 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 4^{\frac{1}{3}} \sqrt{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 ) + \frac{1}{36} \cdot 4^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \log \left (-\frac{3 \cdot 4^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} - 4^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}}{\left (x^{3} + 1\right )} - 6 \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x}{x^{3} + 1}\right ) - \frac{1}{72} \cdot 4^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}} \log \left (\frac{6 \cdot 4^{\frac{1}{3}} \left (-1\right )^{\frac{2}{3}}{\left (5 \, x^{4} - x\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 4^{\frac{2}{3}} \left (-1\right )^{\frac{1}{3}}{\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 24 \,{\left (2 \, x^{5} - x^{2}\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x^{6} + 2 \, 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}{\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}{{\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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