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

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

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

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Rubi [A]  time = 0.228312, 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 \[ \frac{\log \left (x^3+1\right )}{6\ 2^{2/3}}+\frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac{\log \left (\sqrt [3]{2}-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{\tan ^{-1}\left (\frac{2^{2/3} \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3}}-\frac{\log (x)}{2} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 11.2759, size = 121, normalized size = 0.88 \[ - \frac{\log{\left (x^{3} \right )}}{6} + \frac{\sqrt [3]{2} \log{\left (x^{3} + 1 \right )}}{12} + \frac{\log{\left (- \sqrt [3]{- x^{3} + 1} + 1 \right )}}{2} - \frac{\sqrt [3]{2} \log{\left (- \sqrt [3]{- x^{3} + 1} + \sqrt [3]{2} \right )}}{4} + \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2^{\frac{2}{3}} \sqrt [3]{- x^{3} + 1}}{3} + \frac{1}{3}\right ) \right )}}{6} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{- x^{3} + 1}}{3} + \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x**3)/6 + 2**(1/3)*log(x**3 + 1)/12 + log(-(-x**3 + 1)**(1/3) + 1)/2 - 2**(
1/3)*log(-(-x**3 + 1)**(1/3) + 2**(1/3))/4 + 2**(1/3)*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 = 0.23698, size = 113, normalized size = 0.82 \[ -\frac{8 x^3 F_1\left (\frac{5}{3};\frac{2}{3},1;\frac{8}{3};\frac{1}{x^3},-\frac{1}{x^3}\right )}{5 \left (1-x^3\right )^{2/3} \left (x^3+1\right ) \left (8 x^3 F_1\left (\frac{5}{3};\frac{2}{3},1;\frac{8}{3};\frac{1}{x^3},-\frac{1}{x^3}\right )-3 F_1\left (\frac{8}{3};\frac{2}{3},2;\frac{11}{3};\frac{1}{x^3},-\frac{1}{x^3}\right )+2 F_1\left (\frac{8}{3};\frac{5}{3},1;\frac{11}{3};\frac{1}{x^3},-\frac{1}{x^3}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [F]  time = 0.065, size = 0, normalized size = 0. \[ \int{\frac{1}{x \left ({x}^{3}+1 \right ) } \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.220461, size = 258, normalized size = 1.88 \[ -\frac{1}{72} \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (\sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (4^{\frac{2}{3}}{\left (-x^{3} + 1\right )}^{\frac{2}{3}} - 2 \cdot 4^{\frac{1}{3}} \left (-1\right )^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 4 \, \left (-1\right )^{\frac{2}{3}}\right ) - 2 \, \sqrt{3} \left (-1\right )^{\frac{1}{3}} \log \left (4^{\frac{1}{3}}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 2 \, \left (-1\right )^{\frac{1}{3}}\right ) + 4^{\frac{1}{3}} \sqrt{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) - 2 \cdot 4^{\frac{1}{3}} \sqrt{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1\right ) - 6 \, \left (-1\right )^{\frac{1}{3}} \arctan \left (-\frac{1}{3} \, \left (-1\right )^{\frac{2}{3}}{\left (4^{\frac{1}{3}} \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \sqrt{3} \left (-1\right )^{\frac{1}{3}}\right )}\right ) + 6 \cdot 4^{\frac{1}{3}} \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*(-(x - 1)*(x**2 + x + 1))**(2/3)*(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)^(2/3)*x),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError