3.584 \(\int \frac{x^{11}}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx\)

Optimal. Leaf size=125 \[ -\frac{1}{7} \left (1-x^3\right )^{7/3}+\frac{1}{4} \left (1-x^3\right )^{4/3}-\sqrt [3]{1-x^3}+\frac{\log \left (x^3+1\right )}{6\ 2^{2/3}}-\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}} \]

[Out]

-(1 - x^3)^(1/3) + (1 - x^3)^(4/3)/4 - (1 - x^3)^(7/3)/7 + 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 = 0.229906, antiderivative size = 125, 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 \[ -\frac{1}{7} \left (1-x^3\right )^{7/3}+\frac{1}{4} \left (1-x^3\right )^{4/3}-\sqrt [3]{1-x^3}+\frac{\log \left (x^3+1\right )}{6\ 2^{2/3}}-\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x^11/((1 - x^3)^(2/3)*(1 + x^3)),x]

[Out]

-(1 - x^3)^(1/3) + (1 - x^3)^(4/3)/4 - (1 - x^3)^(7/3)/7 + 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 = 11.6213, size = 100, normalized size = 0.8 \[ - \frac{\left (- x^{3} + 1\right )^{\frac{7}{3}}}{7} + \frac{\left (- x^{3} + 1\right )^{\frac{4}{3}}}{4} - \sqrt [3]{- x^{3} + 1} + \frac{\sqrt [3]{2} \log{\left (x^{3} + 1 \right )}}{12} - \frac{\sqrt [3]{2} \log{\left (- \sqrt [3]{- x^{3} + 1} + \sqrt [3]{2} \right )}}{4} + \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2^{\frac{2}{3}} \sqrt [3]{- x^{3} + 1}}{3} + \frac{1}{3}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**11/(-x**3+1)**(2/3)/(x**3+1),x)

[Out]

-(-x**3 + 1)**(7/3)/7 + (-x**3 + 1)**(4/3)/4 - (-x**3 + 1)**(1/3) + 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 = 0.0573451, size = 70, normalized size = 0.56 \[ \frac{14 \left (\frac{x^3-1}{x^3+1}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{2}{x^3+1}\right )+4 x^9-5 x^6+26 x^3-25}{28 \left (1-x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^11/((1 - x^3)^(2/3)*(1 + x^3)),x]

[Out]

(-25 + 26*x^3 - 5*x^6 + 4*x^9 + 14*((-1 + x^3)/(1 + x^3))^(2/3)*Hypergeometric2F
1[2/3, 2/3, 5/3, 2/(1 + x^3)])/(28*(1 - x^3)^(2/3))

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Maple [F]  time = 0.1, size = 0, normalized size = 0. \[ \int{\frac{{x}^{11}}{{x}^{3}+1} \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^11/(-x^3+1)^(2/3)/(x^3+1),x)

[Out]

int(x^11/(-x^3+1)^(2/3)/(x^3+1),x)

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Maxima [A]  time = 1.52887, size = 161, normalized size = 1.29 \[ -\frac{1}{7} \,{\left (-x^{3} + 1\right )}^{\frac{7}{3}} + \frac{1}{6} \, \sqrt{3} 2^{\frac{1}{3}} \arctan \left (\frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}}{\left (2^{\frac{1}{3}} + 2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}\right ) + \frac{1}{4} \,{\left (-x^{3} + 1\right )}^{\frac{4}{3}} + \frac{1}{12} \cdot 2^{\frac{1}{3}} \log \left (2^{\frac{2}{3}} + 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}\right ) - \frac{1}{6} \cdot 2^{\frac{1}{3}} \log \left (-2^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right ) -{\left (-x^{3} + 1\right )}^{\frac{1}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/((x^3 + 1)*(-x^3 + 1)^(2/3)),x, algorithm="maxima")

[Out]

-1/7*(-x^3 + 1)^(7/3) + 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)) - (-x^3 + 1)^(1/3)

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Fricas [A]  time = 0.218497, size = 198, normalized size = 1.58 \[ -\frac{1}{1008} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (3 \cdot 4^{\frac{1}{3}} \sqrt{3}{\left (4 \, x^{6} - x^{3} + 25\right )}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 14 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (4^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 2 \cdot 4^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 4 \, \left (-1\right )^{\frac{2}{3}}\right ) - 28 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (4^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 2 \, \left (-1\right )^{\frac{1}{3}}\right ) - 84 \, \left (-1\right )^{\frac{1}{3}} \arctan \left (-\frac{1}{3} \, \left (-1\right )^{\frac{2}{3}}{\left (4^{\frac{1}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \sqrt{3} \left (-1\right )^{\frac{1}{3}}\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^11/((x^3 + 1)*(-x^3 + 1)^(2/3)),x, algorithm="fricas")

[Out]

-1/1008*4^(2/3)*sqrt(3)*(3*4^(1/3)*sqrt(3)*(4*x^6 - x^3 + 25)*(-x^3 + 1)^(1/3) +
14*sqrt(3)*(-1)^(1/3)*log(4^(2/3)*(-x^3 + 1)^(2/3) - 2*4^(1/3)*(-1)^(1/3)*(-x^3
+ 1)^(1/3) + 4*(-1)^(2/3)) - 28*sqrt(3)*(-1)^(1/3)*log(4^(1/3)*(-x^3 + 1)^(1/3)
+ 2*(-1)^(1/3)) - 84*(-1)^(1/3)*arctan(-1/3*(-1)^(2/3)*(4^(1/3)*sqrt(3)*(-x^3 +
1)^(1/3) - sqrt(3)*(-1)^(1/3))))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{11}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac{2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**11/(-x**3+1)**(2/3)/(x**3+1),x)

[Out]

Integral(x**11/((-(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^11/((x^3 + 1)*(-x^3 + 1)^(2/3)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError