3.567 \(\int \frac{x^{14}}{\sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx\)

Optimal. Leaf size=127 \[ \frac{1}{11} \left (1-x^3\right )^{11/3}-\frac{1}{4} \left (1-x^3\right )^{8/3}+\frac{2}{5} \left (1-x^3\right )^{5/3}-\frac{\log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}} \]

[Out]

(2*(1 - x^3)^(5/3))/5 - (1 - x^3)^(8/3)/4 + (1 - x^3)^(11/3)/11 + 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)) + L
og[2^(1/3) - (1 - x^3)^(1/3)]/(2*2^(1/3))

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Rubi [A]  time = 0.233371, antiderivative size = 127, 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 \[ \frac{1}{11} \left (1-x^3\right )^{11/3}-\frac{1}{4} \left (1-x^3\right )^{8/3}+\frac{2}{5} \left (1-x^3\right )^{5/3}-\frac{\log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x^14/((1 - x^3)^(1/3)*(1 + x^3)),x]

[Out]

(2*(1 - x^3)^(5/3))/5 - (1 - x^3)^(8/3)/4 + (1 - x^3)^(11/3)/11 + 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)) + L
og[2^(1/3) - (1 - x^3)^(1/3)]/(2*2^(1/3))

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Rubi in Sympy [A]  time = 12.7424, size = 104, normalized size = 0.82 \[ \frac{\left (- x^{3} + 1\right )^{\frac{11}{3}}}{11} - \frac{\left (- x^{3} + 1\right )^{\frac{8}{3}}}{4} + \frac{2 \left (- x^{3} + 1\right )^{\frac{5}{3}}}{5} - \frac{2^{\frac{2}{3}} \log{\left (x^{3} + 1 \right )}}{12} + \frac{2^{\frac{2}{3}} \log{\left (- \sqrt [3]{- x^{3} + 1} + \sqrt [3]{2} \right )}}{4} + \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2^{\frac{2}{3}} \sqrt [3]{- x^{3} + 1}}{3} + \frac{1}{3}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**14/(-x**3+1)**(1/3)/(x**3+1),x)

[Out]

(-x**3 + 1)**(11/3)/11 - (-x**3 + 1)**(8/3)/4 + 2*(-x**3 + 1)**(5/3)/5 - 2**(2/3
)*log(x**3 + 1)/12 + 2**(2/3)*log(-(-x**3 + 1)**(1/3) + 2**(1/3))/4 + 2**(2/3)*s
qrt(3)*atan(sqrt(3)*(2**(2/3)*(-x**3 + 1)**(1/3)/3 + 1/3))/6

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Mathematica [C]  time = 0.0733497, size = 74, normalized size = 0.58 \[ \frac{\left (x^3-1\right )^2 \left (20 x^6+15 x^3+53\right )-220 \sqrt [3]{\frac{x^3-1}{x^3+1}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{2}{x^3+1}\right )}{220 \sqrt [3]{1-x^3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^14/((1 - x^3)^(1/3)*(1 + x^3)),x]

[Out]

((-1 + x^3)^2*(53 + 15*x^3 + 20*x^6) - 220*((-1 + x^3)/(1 + x^3))^(1/3)*Hypergeo
metric2F1[1/3, 1/3, 4/3, 2/(1 + x^3)])/(220*(1 - x^3)^(1/3))

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Maple [F]  time = 0.085, size = 0, normalized size = 0. \[ \int{\frac{{x}^{14}}{{x}^{3}+1}{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^14/(-x^3+1)^(1/3)/(x^3+1),x)

[Out]

int(x^14/(-x^3+1)^(1/3)/(x^3+1),x)

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Maxima [A]  time = 1.67377, size = 161, normalized size = 1.27 \[ \frac{1}{11} \,{\left (-x^{3} + 1\right )}^{\frac{11}{3}} - \frac{1}{4} \,{\left (-x^{3} + 1\right )}^{\frac{8}{3}} + \frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}} \arctan \left (\frac{1}{6} \, \sqrt{3} 2^{\frac{2}{3}}{\left (2^{\frac{1}{3}} + 2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right )}\right ) + \frac{2}{5} \,{\left (-x^{3} + 1\right )}^{\frac{5}{3}} - \frac{1}{12} \cdot 2^{\frac{2}{3}} \log \left (2^{\frac{2}{3}} + 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}\right ) + \frac{1}{6} \cdot 2^{\frac{2}{3}} \log \left (-2^{\frac{1}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/((x^3 + 1)*(-x^3 + 1)^(1/3)),x, algorithm="maxima")

[Out]

1/11*(-x^3 + 1)^(11/3) - 1/4*(-x^3 + 1)^(8/3) + 1/6*sqrt(3)*2^(2/3)*arctan(1/6*s
qrt(3)*2^(2/3)*(2^(1/3) + 2*(-x^3 + 1)^(1/3))) + 2/5*(-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)*lo
g(-2^(1/3) + (-x^3 + 1)^(1/3))

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Fricas [A]  time = 0.214159, size = 167, normalized size = 1.31 \[ -\frac{1}{3960} \, \sqrt{3} 2^{\frac{2}{3}}{\left (3 \, \sqrt{3} 2^{\frac{1}{3}}{\left (20 \, x^{9} - 5 \, x^{6} + 38 \, x^{3} - 53\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} + 110 \, \sqrt{3} \log \left (2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} + 2\right ) - 220 \, \sqrt{3} \log \left (2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 2\right ) - 660 \, \arctan \left (\frac{1}{3} \, \sqrt{3} 2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^14/((x^3 + 1)*(-x^3 + 1)^(1/3)),x, algorithm="fricas")

[Out]

-1/3960*sqrt(3)*2^(2/3)*(3*sqrt(3)*2^(1/3)*(20*x^9 - 5*x^6 + 38*x^3 - 53)*(-x^3
+ 1)^(2/3) + 110*sqrt(3)*log(2^(2/3)*(-x^3 + 1)^(1/3) + 2^(1/3)*(-x^3 + 1)^(2/3)
+ 2) - 220*sqrt(3)*log(2^(2/3)*(-x^3 + 1)^(1/3) - 2) - 660*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. \[ \int \frac{x^{14}}{\sqrt [3]{- \left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**14/(-x**3+1)**(1/3)/(x**3+1),x)

[Out]

Integral(x**14/((-(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^14/((x^3 + 1)*(-x^3 + 1)^(1/3)),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError