3.568 \(\int \frac{x^{11}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx\)

Optimal. Leaf size=128 \[ -\frac{1}{8} \left (1-x^3\right )^{8/3}+\frac{1}{5} \left (1-x^3\right )^{5/3}-\frac{1}{2} \left (1-x^3\right )^{2/3}+\frac{\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}} \]

[Out]

-(1 - x^3)^(2/3)/2 + (1 - x^3)^(5/3)/5 - (1 - x^3)^(8/3)/8 - 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 = 0.210563, antiderivative size = 128, 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 \[ -\frac{1}{8} \left (1-x^3\right )^{8/3}+\frac{1}{5} \left (1-x^3\right )^{5/3}-\frac{1}{2} \left (1-x^3\right )^{2/3}+\frac{\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

-(1 - x^3)^(2/3)/2 + (1 - x^3)^(5/3)/5 - (1 - x^3)^(8/3)/8 - 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 = 11.5524, size = 102, normalized size = 0.8 \[ - \frac{\left (- x^{3} + 1\right )^{\frac{8}{3}}}{8} + \frac{\left (- x^{3} + 1\right )^{\frac{5}{3}}}{5} - \frac{\left (- x^{3} + 1\right )^{\frac{2}{3}}}{2} + \frac{2^{\frac{2}{3}} \log{\left (x^{3} + 1 \right )}}{12} - \frac{2^{\frac{2}{3}} \log{\left (- \sqrt [3]{- x^{3} + 1} + \sqrt [3]{2} \right )}}{4} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2^{\frac{2}{3}} \sqrt [3]{- x^{3} + 1}}{3} + \frac{1}{3}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-x**3 + 1)**(8/3)/8 + (-x**3 + 1)**(5/3)/5 - (-x**3 + 1)**(2/3)/2 + 2**(2/3)*l
og(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 [C]  time = 0.0498002, size = 70, normalized size = 0.55 \[ \frac{40 \sqrt [3]{\frac{x^3-1}{x^3+1}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{2}{x^3+1}\right )+5 x^9-7 x^6+19 x^3-17}{40 \sqrt [3]{1-x^3}} \]

Antiderivative was successfully verified.

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

[Out]

(-17 + 19*x^3 - 7*x^6 + 5*x^9 + 40*((-1 + x^3)/(1 + x^3))^(1/3)*Hypergeometric2F
1[1/3, 1/3, 4/3, 2/(1 + x^3)])/(40*(1 - x^3)^(1/3))

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Maple [F]  time = 0.068, size = 0, normalized size = 0. \[ \int{\frac{{x}^{11}}{{x}^{3}+1}{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.51185, size = 161, normalized size = 1.26 \[ -\frac{1}{8} \,{\left (-x^{3} + 1\right )}^{\frac{8}{3}} - \frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}} \arctan \left (\frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}}{\left (2^{\frac{1}{3}} + 2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}\right ) + \frac{1}{5} \,{\left (-x^{3} + 1\right )}^{\frac{5}{3}} + \frac{1}{12} \cdot 2^{\frac{2}{3}} \log \left (2^{\frac{2}{3}} + 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}\right ) - \frac{1}{6} \cdot 2^{\frac{2}{3}} \log \left (-2^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right ) - \frac{1}{2} \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(-x^3 + 1)^(8/3) - 1/6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3)
+ 2*(-x^3 + 1)^(1/3))) + 1/5*(-x^3 + 1)^(5/3) + 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)) - 1/2*(-x^3 + 1)^(2/3)

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Fricas [A]  time = 0.214583, size = 196, normalized size = 1.53 \[ -\frac{1}{720} \, \sqrt{3} 2^{\frac{2}{3}}{\left (3 \, \sqrt{3} 2^{\frac{1}{3}}{\left (5 \, x^{6} - 2 \, x^{3} + 17\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} + 20 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (2^{\frac{2}{3}} \left (-1\right )^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 2 \, \left (-1\right )^{\frac{1}{3}}\right ) - 40 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 2 \, \left (-1\right )^{\frac{2}{3}}\right ) - 120 \, \left (-1\right )^{\frac{1}{3}} \arctan \left (-\frac{1}{3} \, \left (-1\right )^{\frac{1}{3}}{\left (\sqrt{3} 2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \sqrt{3} \left (-1\right )^{\frac{2}{3}}\right )}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/720*sqrt(3)*2^(2/3)*(3*sqrt(3)*2^(1/3)*(5*x^6 - 2*x^3 + 17)*(-x^3 + 1)^(2/3)
+ 20*sqrt(3)*(-1)^(1/3)*log(2^(2/3)*(-1)^(2/3)*(-x^3 + 1)^(1/3) + 2^(1/3)*(-x^3
+ 1)^(2/3) - 2*(-1)^(1/3)) - 40*sqrt(3)*(-1)^(1/3)*log(2^(2/3)*(-x^3 + 1)^(1/3)
- 2*(-1)^(2/3)) - 120*(-1)^(1/3)*arctan(-1/3*(-1)^(1/3)*(sqrt(3)*2^(2/3)*(-x^3 +
1)^(1/3) + sqrt(3)*(-1)^(2/3))))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: NotImplementedError