Optimal. Leaf size=53 \[ \frac{1}{3} \sqrt{\frac{1-x^3}{x^3+1}} \left (x^3+1\right )-\frac{2}{3} \tan ^{-1}\left (\sqrt{\frac{1-x^3}{x^3+1}}\right ) \]
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Rubi [A] time = 0.0291266, antiderivative size = 53, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {1960, 288, 204} \[ \frac{1}{3} \sqrt{\frac{1-x^3}{x^3+1}} \left (x^3+1\right )-\frac{2}{3} \tan ^{-1}\left (\sqrt{\frac{1-x^3}{x^3+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 1960
Rule 288
Rule 204
Rubi steps
\begin{align*} \int x^2 \sqrt{\frac{1-x^3}{1+x^3}} \, dx &=-\left (\frac{4}{3} \operatorname{Subst}\left (\int \frac{x^2}{\left (-1-x^2\right )^2} \, dx,x,\sqrt{\frac{1-x^3}{1+x^3}}\right )\right )\\ &=\frac{1}{3} \sqrt{\frac{1-x^3}{1+x^3}} \left (1+x^3\right )+\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,\sqrt{\frac{1-x^3}{1+x^3}}\right )\\ &=\frac{1}{3} \sqrt{\frac{1-x^3}{1+x^3}} \left (1+x^3\right )-\frac{2}{3} \tan ^{-1}\left (\sqrt{\frac{1-x^3}{1+x^3}}\right )\\ \end{align*}
Mathematica [A] time = 0.0317817, size = 86, normalized size = 1.62 \[ \frac{\sqrt{\frac{1-x^3}{x^3+1}} \sqrt{x^3+1} \left (\sqrt{x^3+1} \left (x^3-1\right )+2 \sqrt{1-x^3} \sin ^{-1}\left (\frac{\sqrt{1-x^3}}{\sqrt{2}}\right )\right )}{3 \left (x^3-1\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.081, size = 68, normalized size = 1.3 \begin{align*}{\frac{{x}^{3}+1}{3}\sqrt{-{\frac{{x}^{3}-1}{{x}^{3}+1}}}}-{\frac{\arcsin \left ({x}^{3} \right ) }{3\,{x}^{3}-3}\sqrt{-{\frac{{x}^{3}-1}{{x}^{3}+1}}}\sqrt{- \left ({x}^{3}+1 \right ) \left ({x}^{3}-1 \right ) }} \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 x^{2} \sqrt{-\frac{x^{3} - 1}{x^{3} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.46181, size = 139, normalized size = 2.62 \begin{align*} \frac{1}{3} \,{\left (x^{3} + 1\right )} \sqrt{-\frac{x^{3} - 1}{x^{3} + 1}} - \frac{2}{3} \, \arctan \left (\frac{{\left (x^{3} + 1\right )} \sqrt{-\frac{x^{3} - 1}{x^{3} + 1}} - 1}{x^{3}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17868, size = 30, normalized size = 0.57 \begin{align*} \frac{1}{3} \,{\left (\sqrt{-x^{6} + 1} + \arcsin \left (x^{3}\right )\right )} \mathrm{sgn}\left (x^{3} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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