Optimal. Leaf size=83 \[ -\frac{\log \left (x^3+1\right )}{6\ 2^{2/3}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}} \]
[Out]
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Rubi [A] time = 0.14188, antiderivative size = 83, 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 \[ -\frac{\log \left (x^3+1\right )}{6\ 2^{2/3}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[x^2/((1 - x^3)^(2/3)*(1 + x^3)),x]
[Out]
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Rubi in Sympy [A] time = 8.60666, size = 71, normalized size = 0.86 \[ - \frac{\sqrt [3]{2} \log{\left (x^{3} + 1 \right )}}{12} + \frac{\sqrt [3]{2} \log{\left (- \sqrt [3]{- x^{3} + 1} + \sqrt [3]{2} \right )}}{4} - \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2^{\frac{2}{3}} \sqrt [3]{- x^{3} + 1}}{3} + \frac{1}{3}\right ) \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(-x**3+1)**(2/3)/(x**3+1),x)
[Out]
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Mathematica [A] time = 0.0605552, size = 103, normalized size = 1.24 \[ -\frac{-2 \log \left (2-2^{2/3} \sqrt [3]{1-x^3}\right )+\log \left (\sqrt [3]{2} \left (1-x^3\right )^{2/3}+2^{2/3} \sqrt [3]{1-x^3}+2\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{6\ 2^{2/3}} \]
Antiderivative was successfully verified.
[In] Integrate[x^2/((1 - x^3)^(2/3)*(1 + x^3)),x]
[Out]
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Maple [F] time = 0.058, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2}}{{x}^{3}+1} \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{3}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(-x^3+1)^(2/3)/(x^3+1),x)
[Out]
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Maxima [A] time = 1.52364, size = 116, normalized size = 1.4 \[ -\frac{1}{6} \, \sqrt{3} 2^{\frac{1}{3}} \arctan \left (\frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}}{\left (2^{\frac{1}{3}} + 2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}\right ) - \frac{1}{12} \cdot 2^{\frac{1}{3}} \log \left (2^{\frac{2}{3}} + 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}\right ) + \frac{1}{6} \cdot 2^{\frac{1}{3}} \log \left (-2^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((x^3 + 1)*(-x^3 + 1)^(2/3)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213346, size = 122, normalized size = 1.47 \[ -\frac{1}{72} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (\sqrt{3} \log \left (4^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} + 2 \cdot 4^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 4\right ) - 2 \, \sqrt{3} \log \left (4^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 2\right ) + 6 \, \arctan \left (\frac{1}{3} \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((x^3 + 1)*(-x^3 + 1)^(2/3)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac{2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(-x**3+1)**(2/3)/(x**3+1),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/((x^3 + 1)*(-x^3 + 1)^(2/3)),x, algorithm="giac")
[Out]