3.585 \(\int \frac{x^8}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx\)

Optimal. Leaf size=98 \[ \frac{1}{4} \left (1-x^3\right )^{4/3}-\frac{\log \left (x^3+1\right )}{6\ 2^{2/3}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}} \]

[Out]

(1 - x^3)^(4/3)/4 - ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[
3]) - Log[1 + x^3]/(6*2^(2/3)) + Log[2^(1/3) - (1 - x^3)^(1/3)]/(2*2^(2/3))

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Rubi [A]  time = 0.196147, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{1}{4} \left (1-x^3\right )^{4/3}-\frac{\log \left (x^3+1\right )}{6\ 2^{2/3}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x^8/((1 - x^3)^(2/3)*(1 + x^3)),x]

[Out]

(1 - x^3)^(4/3)/4 - ArcTan[(1 + 2^(2/3)*(1 - x^3)^(1/3))/Sqrt[3]]/(2^(2/3)*Sqrt[
3]) - Log[1 + x^3]/(6*2^(2/3)) + Log[2^(1/3) - (1 - x^3)^(1/3)]/(2*2^(2/3))

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Rubi in Sympy [A]  time = 10.8946, size = 82, normalized size = 0.84 \[ \frac{\left (- x^{3} + 1\right )^{\frac{4}{3}}}{4} - \frac{\sqrt [3]{2} \log{\left (x^{3} + 1 \right )}}{12} + \frac{\sqrt [3]{2} \log{\left (- \sqrt [3]{- x^{3} + 1} + \sqrt [3]{2} \right )}}{4} - \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2^{\frac{2}{3}} \sqrt [3]{- x^{3} + 1}}{3} + \frac{1}{3}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(-x**3+1)**(2/3)/(x**3+1),x)

[Out]

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

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Mathematica [C]  time = 0.0518276, size = 61, normalized size = 0.62 \[ \frac{\left (x^3-1\right )^2-2 \left (\frac{x^3-1}{x^3+1}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{2}{x^3+1}\right )}{4 \left (1-x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/((1 - x^3)^(2/3)*(1 + x^3)),x]

[Out]

((-1 + x^3)^2 - 2*((-1 + x^3)/(1 + x^3))^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3,
2/(1 + x^3)])/(4*(1 - x^3)^(2/3))

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Maple [F]  time = 0.075, size = 0, normalized size = 0. \[ \int{\frac{{x}^{8}}{{x}^{3}+1} \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(-x^3+1)^(2/3)/(x^3+1),x)

[Out]

int(x^8/(-x^3+1)^(2/3)/(x^3+1),x)

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Maxima [A]  time = 1.51999, size = 131, normalized size = 1.34 \[ -\frac{1}{6} \, \sqrt{3} 2^{\frac{1}{3}} \arctan \left (\frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}}{\left (2^{\frac{1}{3}} + 2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}\right ) + \frac{1}{4} \,{\left (-x^{3} + 1\right )}^{\frac{4}{3}} - \frac{1}{12} \cdot 2^{\frac{1}{3}} \log \left (2^{\frac{2}{3}} + 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}\right ) + \frac{1}{6} \cdot 2^{\frac{1}{3}} \log \left (-2^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/((x^3 + 1)*(-x^3 + 1)^(2/3)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.221375, size = 153, normalized size = 1.56 \[ -\frac{1}{144} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (3 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (x^{3} - 1\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 2 \, \sqrt{3} \log \left (4^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} + 2 \cdot 4^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 4\right ) - 4 \, \sqrt{3} \log \left (4^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 2\right ) + 12 \, \arctan \left (\frac{1}{3} \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/((x^3 + 1)*(-x^3 + 1)^(2/3)),x, algorithm="fricas")

[Out]

-1/144*4^(2/3)*sqrt(3)*(3*4^(1/3)*sqrt(3)*(x^3 - 1)*(-x^3 + 1)^(1/3) + 2*sqrt(3)
*log(4^(2/3)*(-x^3 + 1)^(2/3) + 2*4^(1/3)*(-x^3 + 1)^(1/3) + 4) - 4*sqrt(3)*log(
4^(1/3)*(-x^3 + 1)^(1/3) - 2) + 12*arctan(1/3*4^(1/3)*sqrt(3)*(-x^3 + 1)^(1/3) +
1/3*sqrt(3)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac{2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(-x**3+1)**(2/3)/(x**3+1),x)

[Out]

Integral(x**8/((-(x - 1)*(x**2 + x + 1))**(2/3)*(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^8/((x^3 + 1)*(-x^3 + 1)^(2/3)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError