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

Optimal. Leaf size=137 \[ \frac{\log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}-\frac{\log (x)}{2} \]

[Out]

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

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Rubi [A]  time = 0.223675, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318 \[ \frac{\log \left (x^3+1\right )}{6 \sqrt [3]{2}}+\frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2 \sqrt [3]{2}}+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt [3]{2} \sqrt{3}}-\frac{\log (x)}{2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.2751, size = 121, normalized size = 0.88 \[ - \frac{\log{\left (x^{3} \right )}}{6} + \frac{2^{\frac{2}{3}} \log{\left (x^{3} + 1 \right )}}{12} + \frac{\log{\left (- \sqrt [3]{- x^{3} + 1} + 1 \right )}}{2} - \frac{2^{\frac{2}{3}} \log{\left (- \sqrt [3]{- x^{3} + 1} + \sqrt [3]{2} \right )}}{4} - \frac{2^{\frac{2}{3}} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2^{\frac{2}{3}} \sqrt [3]{- x^{3} + 1}}{3} + \frac{1}{3}\right ) \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{- x^{3} + 1}}{3} + \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.191695, size = 111, normalized size = 0.81 \[ -\frac{7 x^3 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};\frac{1}{x^3},-\frac{1}{x^3}\right )}{4 \sqrt [3]{1-x^3} \left (x^3+1\right ) \left (7 x^3 F_1\left (\frac{4}{3};\frac{1}{3},1;\frac{7}{3};\frac{1}{x^3},-\frac{1}{x^3}\right )-3 F_1\left (\frac{7}{3};\frac{1}{3},2;\frac{10}{3};\frac{1}{x^3},-\frac{1}{x^3}\right )+F_1\left (\frac{7}{3};\frac{4}{3},1;\frac{10}{3};\frac{1}{x^3},-\frac{1}{x^3}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-7*x^3*AppellF1[4/3, 1/3, 1, 7/3, x^(-3), -x^(-3)])/(4*(1 - x^3)^(1/3)*(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, 1
0/3, x^(-3), -x^(-3)] + AppellF1[7/3, 4/3, 1, 10/3, x^(-3), -x^(-3)]))

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{\frac{1}{x \left ({x}^{3}+1 \right ) }{\frac{1}{\sqrt [3]{-{x}^{3}+1}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: NotImplementedError

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*(-(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),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError