3.571 \(\int \frac{x^2}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx\)

Optimal. Leaf size=82 \[ -\frac{\log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}} \]

[Out]

ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]]/(2^(1/3)*Sqrt[3]) - Log[1 + x^3]/(
6*2^(1/3)) + Log[2^(1/3) - (1 - x^3)^(1/3)]/(2*2^(1/3))

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Rubi [A]  time = 0.136137, 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 \[ -\frac{\log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/((1 - x^3)^(1/3)*(1 + x^3)),x]

[Out]

ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]]/(2^(1/3)*Sqrt[3]) - Log[1 + x^3]/(
6*2^(1/3)) + Log[2^(1/3) - (1 - x^3)^(1/3)]/(2*2^(1/3))

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Rubi in Sympy [A]  time = 8.6589, size = 71, normalized size = 0.87 \[ - \frac{2^{\frac{2}{3}} \log{\left (x^{3} + 1 \right )}}{12} + \frac{2^{\frac{2}{3}} \log{\left (- \sqrt [3]{- x^{3} + 1} + \sqrt [3]{2} \right )}}{4} + \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2^{\frac{2}{3}} \sqrt [3]{- x^{3} + 1}}{3} + \frac{1}{3}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(-x**3+1)**(1/3)/(x**3+1),x)

[Out]

-2**(2/3)*log(x**3 + 1)/12 + 2**(2/3)*log(-(-x**3 + 1)**(1/3) + 2**(1/3))/4 + 2*
*(2/3)*sqrt(3)*atan(sqrt(3)*(2**(2/3)*(-x**3 + 1)**(1/3)/3 + 1/3))/6

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Mathematica [A]  time = 0.0563128, size = 105, normalized size = 1.28 \[ \frac{2 \log \left (2-2^{2/3} \sqrt [3]{1-x^3}\right )-\log \left (\sqrt [3]{2} \left (1-x^3\right )^{2/3}+2^{2/3} \sqrt [3]{1-x^3}+2\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{6 \sqrt [3]{2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/((1 - x^3)^(1/3)*(1 + x^3)),x]

[Out]

(2*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]] + 2*Log[2 - 2^(2/3)*(1
- x^3)^(1/3)] - Log[2 + 2^(2/3)*(1 - x^3)^(1/3) + 2^(1/3)*(1 - x^3)^(2/3)])/(6*2
^(1/3))

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Maple [F]  time = 0., size = 0, normalized size = 0. \[ \int{\frac{{x}^{2}}{{x}^{3}+1}{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-x^3+1)^(1/3)/(x^3+1),x)

[Out]

int(x^2/(-x^3+1)^(1/3)/(x^3+1),x)

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Maxima [A]  time = 1.52662, size = 116, normalized size = 1.41 \[ \frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}} \arctan \left (\frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}}{\left (2^{\frac{1}{3}} + 2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}\right ) - \frac{1}{12} \cdot 2^{\frac{2}{3}} \log \left (2^{\frac{2}{3}} + 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}\right ) + \frac{1}{6} \cdot 2^{\frac{2}{3}} \log \left (-2^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((x^3 + 1)*(-x^3 + 1)^(1/3)),x, algorithm="maxima")

[Out]

1/6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-x^3 + 1)^(1/3))) -
1/12*2^(2/3)*log(2^(2/3) + 2^(1/3)*(-x^3 + 1)^(1/3) + (-x^3 + 1)^(2/3)) + 1/6*2
^(2/3)*log(-2^(1/3) + (-x^3 + 1)^(1/3))

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Fricas [A]  time = 0.214434, size = 120, normalized size = 1.46 \[ -\frac{1}{36} \, \sqrt{3} 2^{\frac{2}{3}}{\left (\sqrt{3} \log \left (2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} + 2\right ) - 2 \, \sqrt{3} \log \left (2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 2\right ) - 6 \, \arctan \left (\frac{1}{3} \, \sqrt{3} 2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((x^3 + 1)*(-x^3 + 1)^(1/3)),x, algorithm="fricas")

[Out]

-1/36*sqrt(3)*2^(2/3)*(sqrt(3)*log(2^(2/3)*(-x^3 + 1)^(1/3) + 2^(1/3)*(-x^3 + 1)
^(2/3) + 2) - 2*sqrt(3)*log(2^(2/3)*(-x^3 + 1)^(1/3) - 2) - 6*arctan(1/3*sqrt(3)
*2^(2/3)*(-x^3 + 1)^(1/3) + 1/3*sqrt(3)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(-x**3+1)**(1/3)/(x**3+1),x)

[Out]

Integral(x**2/((-(x - 1)*(x**2 + x + 1))**(1/3)*(x + 1)*(x**2 - x + 1)), x)

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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)^(1/3)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError