Optimal. Leaf size=82 \[ -\frac{\log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}} \]
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Rubi [A] time = 0.0578244, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {444, 55, 617, 204, 31} \[ -\frac{\log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 444
Rule 55
Rule 617
Rule 204
Rule 31
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{1-x} (1+x)} \, dx,x,x^3\right )\\ &=-\frac{\log \left (1+x^3\right )}{6 \sqrt [3]{2}}+\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{2^{2/3}+\sqrt [3]{2} x+x^2} \, dx,x,\sqrt [3]{1-x^3}\right )-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{2}-x} \, dx,x,\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\\ &=-\frac{\log \left (1+x^3\right )}{6 \sqrt [3]{2}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+2^{2/3} \sqrt [3]{1-x^3}\right )}{\sqrt [3]{2}}\\ &=\frac{\tan ^{-1}\left (\frac{1+2^{2/3} \sqrt [3]{1-x^3}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}-\frac{\log \left (1+x^3\right )}{6 \sqrt [3]{2}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}\\ \end{align*}
Mathematica [A] time = 0.0595163, size = 73, normalized size = 0.89 \[ \frac{-\log \left (x^3+1\right )+3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{6 \sqrt [3]{2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.039, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{{x}^{3}+1}{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40032, size = 116, normalized size = 1.41 \begin{align*} \frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}} \arctan \left (\frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}}{\left (2^{\frac{1}{3}} + 2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}\right ) - \frac{1}{12} \cdot 2^{\frac{2}{3}} \log \left (2^{\frac{2}{3}} + 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}\right ) + \frac{1}{6} \cdot 2^{\frac{2}{3}} \log \left (-2^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.12013, size = 282, normalized size = 3.44 \begin{align*} \frac{1}{6} \, \sqrt{6} 2^{\frac{1}{6}} \arctan \left (\frac{1}{6} \cdot 2^{\frac{1}{6}}{\left (\sqrt{6} 2^{\frac{1}{3}} + 2 \, \sqrt{6}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}\right ) - \frac{1}{12} \cdot 2^{\frac{2}{3}} \log \left (2^{\frac{2}{3}} + 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}\right ) + \frac{1}{6} \cdot 2^{\frac{2}{3}} \log \left (-2^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\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^{2}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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