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

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

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

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Rubi [A]  time = 0.277401, 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 \[ \frac{1}{3} \log \left (\frac{x}{\sqrt [3]{1-x^3}}+1\right )-\frac{\log \left (\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+1\right )}{3 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}-\frac{1}{6} \log \left (-\frac{x}{\sqrt [3]{1-x^3}}+\frac{x^2}{\left (1-x^3\right )^{2/3}}+1\right )+\frac{\log \left (-\frac{\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}+\frac{2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}+1\right )}{6 \sqrt [3]{2}} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 30.5225, size = 180, normalized size = 0.87 \[ \frac{\log{\left (\frac{x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{3} - \frac{2^{\frac{2}{3}} \log{\left (\frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{6} - \frac{\log{\left (\frac{x^{2}}{\left (- x^{3} + 1\right )^{\frac{2}{3}}} - \frac{x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{6} + \frac{2^{\frac{2}{3}} \log{\left (\frac{2^{\frac{2}{3}} x^{2}}{\left (- x^{3} + 1\right )^{\frac{2}{3}}} - \frac{\sqrt [3]{2} x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{12} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3 \sqrt [3]{- x^{3} + 1}} - \frac{1}{3}\right ) \right )}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 \sqrt [3]{2} x}{3 \sqrt [3]{- x^{3} + 1}} + \frac{1}{3}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x/(-x**3 + 1)**(1/3) + 1)/3 - 2**(2/3)*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**(2/3)*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**(2/3)*sqrt(3)*atan(sqr
t(3)*(-2*2**(1/3)*x/(3*(-x**3 + 1)**(1/3)) + 1/3))/6

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Mathematica [C]  time = 0.112097, size = 115, normalized size = 0.56 \[ -\frac{7 x^4 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};x^3,-x^3\right )}{4 \sqrt [3]{1-x^3} \left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac{7}{3};\frac{1}{3},2;\frac{10}{3};x^3,-x^3\right )-F_1\left (\frac{7}{3};\frac{4}{3},1;\frac{10}{3};x^3,-x^3\right )\right )-7 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};x^3,-x^3\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-7*x^4*AppellF1[4/3, 1/3, 1, 7/3, x^3, -x^3])/(4*(1 - x^3)^(1/3)*(1 + x^3)*(-7*
AppellF1[4/3, 1/3, 1, 7/3, x^3, -x^3] + x^3*(3*AppellF1[7/3, 1/3, 2, 10/3, x^3,
-x^3] - AppellF1[7/3, 4/3, 1, 10/3, x^3, -x^3])))

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \[ \int{\frac{{x}^{3}}{{x}^{3}+1}{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [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^3/((x^3 + 1)*(-x^3 + 1)^(1/3)),x, algorithm="fricas")

[Out]

Exception raised: NotImplementedError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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