3.590 \(\int \frac{x^4}{\left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx\)

Optimal. Leaf size=207 \[ -\frac{1}{3} \log \left (\frac{x}{\sqrt [3]{1-x^3}}+1\right )+\frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}+\frac{1}{6} \log \left (-\frac{x}{\sqrt [3]{1-x^3}}+\frac{x^2}{\left (1-x^3\right )^{2/3}}+1\right )-\frac{\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )}{6\ 2^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.308982, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409 \[ -\frac{1}{3} \log \left (\frac{x}{\sqrt [3]{1-x^3}}+1\right )+\frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}+\frac{1}{6} \log \left (-\frac{x}{\sqrt [3]{1-x^3}}+\frac{x^2}{\left (1-x^3\right )^{2/3}}+1\right )-\frac{\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )}{6\ 2^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/((1 - x^3)^(2/3)*(1 + x^3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 27.0951, size = 180, normalized size = 0.87 \[ - \frac{\log{\left (\frac{x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{3} + \frac{\sqrt [3]{2} \log{\left (\frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{6} + \frac{\log{\left (\frac{x^{2}}{\left (- x^{3} + 1\right )^{\frac{2}{3}}} - \frac{x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{6} - \frac{\sqrt [3]{2} \log{\left (\frac{2^{\frac{2}{3}} x^{2}}{\left (- x^{3} + 1\right )^{\frac{2}{3}}} - \frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{12} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3 \sqrt [3]{- x^{3} + 1}} - \frac{1}{3}\right ) \right )}}{3} + \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{- x^{3} + 1}} + \frac{1}{3}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(-x**3+1)**(2/3)/(x**3+1),x)

[Out]

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

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Mathematica [C]  time = 0.19284, size = 115, normalized size = 0.56 \[ -\frac{8 x^5 F_1\left (\frac{5}{3};\frac{2}{3},1;\frac{8}{3};x^3,-x^3\right )}{5 \left (1-x^3\right )^{2/3} \left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac{8}{3};\frac{2}{3},2;\frac{11}{3};x^3,-x^3\right )-2 F_1\left (\frac{8}{3};\frac{5}{3},1;\frac{11}{3};x^3,-x^3\right )\right )-8 F_1\left (\frac{5}{3};\frac{2}{3},1;\frac{8}{3};x^3,-x^3\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^4/((1 - x^3)^(2/3)*(1 + x^3)),x]

[Out]

(-8*x^5*AppellF1[5/3, 2/3, 1, 8/3, x^3, -x^3])/(5*(1 - x^3)^(2/3)*(1 + x^3)*(-8*
AppellF1[5/3, 2/3, 1, 8/3, x^3, -x^3] + x^3*(3*AppellF1[8/3, 2/3, 2, 11/3, x^3,
-x^3] - 2*AppellF1[8/3, 5/3, 1, 11/3, x^3, -x^3])))

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Maple [F]  time = 0.059, size = 0, normalized size = 0. \[ \int{\frac{{x}^{4}}{{x}^{3}+1} \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(-x^3+1)^(2/3)/(x^3+1),x)

[Out]

int(x^4/(-x^3+1)^(2/3)/(x^3+1),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((x^3 + 1)*(-x^3 + 1)^(2/3)),x, algorithm="maxima")

[Out]

integrate(x^4/((x^3 + 1)*(-x^3 + 1)^(2/3)), x)

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Fricas [A]  time = 0.2203, size = 274, normalized size = 1.32 \[ -\frac{1}{72} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (2 \cdot 4^{\frac{1}{3}} \sqrt{3} \log \left (\frac{x +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x}\right ) - 4^{\frac{1}{3}} \sqrt{3} \log \left (\frac{x^{2} -{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2}}\right ) - 2 \, \sqrt{3} \log \left (\frac{2 \, x + 4^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x}\right ) + \sqrt{3} \log \left (\frac{4 \, x^{2} - 2 \cdot 4^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x + 4^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2}}\right ) + 6 \cdot 4^{\frac{1}{3}} \arctan \left (-\frac{\sqrt{3} x - 2 \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{3 \, x}\right ) - 6 \, \arctan \left (-\frac{\sqrt{3} x - 4^{\frac{1}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{3 \, x}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((x^3 + 1)*(-x^3 + 1)^(2/3)),x, algorithm="fricas")

[Out]

-1/72*4^(2/3)*sqrt(3)*(2*4^(1/3)*sqrt(3)*log((x + (-x^3 + 1)^(1/3))/x) - 4^(1/3)
*sqrt(3)*log((x^2 - (-x^3 + 1)^(1/3)*x + (-x^3 + 1)^(2/3))/x^2) - 2*sqrt(3)*log(
(2*x + 4^(1/3)*(-x^3 + 1)^(1/3))/x) + sqrt(3)*log((4*x^2 - 2*4^(1/3)*(-x^3 + 1)^
(1/3)*x + 4^(2/3)*(-x^3 + 1)^(2/3))/x^2) + 6*4^(1/3)*arctan(-1/3*(sqrt(3)*x - 2*
sqrt(3)*(-x^3 + 1)^(1/3))/x) - 6*arctan(-1/3*(sqrt(3)*x - 4^(1/3)*sqrt(3)*(-x^3
+ 1)^(1/3))/x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac{2}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(-x**3+1)**(2/3)/(x**3+1),x)

[Out]

Integral(x**4/((-(x - 1)*(x**2 + x + 1))**(2/3)*(x + 1)*(x**2 - x + 1)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (x^{3} + 1\right )}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/((x^3 + 1)*(-x^3 + 1)^(2/3)),x, algorithm="giac")

[Out]

integrate(x^4/((x^3 + 1)*(-x^3 + 1)^(2/3)), x)