3.573 \(\int \frac{1}{x^4 \sqrt [3]{1-x^3} \left (1+x^3\right )} \, dx\)

Optimal. Leaf size=157 \[ -\frac{\left (1-x^3\right )^{2/3}}{3 x^3}-\frac{\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac{1}{3} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}+\frac{\log (x)}{3} \]

[Out]

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

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Rubi [A]  time = 0.285132, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{\left (1-x^3\right )^{2/3}}{3 x^3}-\frac{\log \left (x^3+1\right )}{6 \sqrt [3]{2}}-\frac{1}{3} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}-\frac{2 \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}+\frac{\log (x)}{3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.7974, size = 136, normalized size = 0.87 \[ \frac{\log{\left (x^{3} \right )}}{9} - \frac{2^{\frac{2}{3}} \log{\left (x^{3} + 1 \right )}}{12} - \frac{\log{\left (- \sqrt [3]{- x^{3} + 1} + 1 \right )}}{3} + \frac{2^{\frac{2}{3}} \log{\left (- \sqrt [3]{- x^{3} + 1} + \sqrt [3]{2} \right )}}{4} + \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2^{\frac{2}{3}} \sqrt [3]{- x^{3} + 1}}{3} + \frac{1}{3}\right ) \right )}}{6} - \frac{2 \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{- x^{3} + 1}}{3} + \frac{1}{3}\right ) \right )}}{9} - \frac{\left (- x^{3} + 1\right )^{\frac{2}{3}}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**3)/9 - 2**(2/3)*log(x**3 + 1)/12 - log(-(-x**3 + 1)**(1/3) + 1)/3 + 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 - 2*sqrt(3)*atan(sqrt(3)*(2*(-x**3 + 1)**(1/3
)/3 + 1/3))/9 - (-x**3 + 1)**(2/3)/(3*x**3)

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Mathematica [C]  time = 0.332608, size = 209, normalized size = 1.33 \[ \frac{-\frac{4 x^6 F_1\left (1;\frac{1}{3},1;2;x^3,-x^3\right )}{\left (x^3+1\right ) \left (x^3 \left (3 F_1\left (2;\frac{1}{3},2;3;x^3,-x^3\right )-F_1\left (2;\frac{4}{3},1;3;x^3,-x^3\right )\right )-6 F_1\left (1;\frac{1}{3},1;2;x^3,-x^3\right )\right )}+\frac{7 x^6 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};\frac{1}{x^3},-\frac{1}{x^3}\right )}{\left (x^3+1\right ) \left (7 x^3 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};\frac{1}{x^3},-\frac{1}{x^3}\right )-3 F_1\left (\frac{7}{3};\frac{1}{3},2;\frac{10}{3};\frac{1}{x^3},-\frac{1}{x^3}\right )+F_1\left (\frac{7}{3};\frac{4}{3},1;\frac{10}{3};\frac{1}{x^3},-\frac{1}{x^3}\right )\right )}+2 x^3-2}{6 x^3 \sqrt [3]{1-x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-2 + 2*x^3 - (4*x^6*AppellF1[1, 1/3, 1, 2, x^3, -x^3])/((1 + x^3)*(-6*AppellF1[
1, 1/3, 1, 2, x^3, -x^3] + x^3*(3*AppellF1[2, 1/3, 2, 3, x^3, -x^3] - AppellF1[2
, 4/3, 1, 3, x^3, -x^3]))) + (7*x^6*AppellF1[4/3, 1/3, 1, 7/3, x^(-3), -x^(-3)])
/((1 + x^3)*(7*x^3*AppellF1[4/3, 1/3, 1, 7/3, 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)]))
)/(6*x^3*(1 - x^3)^(1/3))

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Maple [F]  time = 0.09, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4} \left ({x}^{3}+1 \right ) }{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.220024, size = 274, normalized size = 1.75 \[ \frac{\sqrt{3} 2^{\frac{2}{3}}{\left (2 \, \sqrt{3} 2^{\frac{1}{3}} x^{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) - 4 \, \sqrt{3} 2^{\frac{1}{3}} x^{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1\right ) - 3 \, \sqrt{3} x^{3} \log \left (2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} + 2\right ) + 6 \, \sqrt{3} x^{3} \log \left (2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 2\right ) - 12 \cdot 2^{\frac{1}{3}} x^{3} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) + 18 \, x^{3} \arctan \left (\frac{1}{3} \, \sqrt{3} 2^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right ) - 6 \, \sqrt{3} 2^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}}\right )}}{108 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/108*sqrt(3)*2^(2/3)*(2*sqrt(3)*2^(1/3)*x^3*log((-x^3 + 1)^(2/3) + (-x^3 + 1)^(
1/3) + 1) - 4*sqrt(3)*2^(1/3)*x^3*log((-x^3 + 1)^(1/3) - 1) - 3*sqrt(3)*x^3*log(
2^(2/3)*(-x^3 + 1)^(1/3) + 2^(1/3)*(-x^3 + 1)^(2/3) + 2) + 6*sqrt(3)*x^3*log(2^(
2/3)*(-x^3 + 1)^(1/3) - 2) - 12*2^(1/3)*x^3*arctan(2/3*sqrt(3)*(-x^3 + 1)^(1/3)
+ 1/3*sqrt(3)) + 18*x^3*arctan(1/3*sqrt(3)*2^(2/3)*(-x^3 + 1)^(1/3) + 1/3*sqrt(3
)) - 6*sqrt(3)*2^(1/3)*(-x^3 + 1)^(2/3))/x^3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**4*(-(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(1/((x^3 + 1)*(-x^3 + 1)^(1/3)*x^4),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError