3.36 \(\int \frac{1}{x \sqrt [3]{1-x^3}} \, dx\)

Optimal. Leaf size=55 \[ \frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\log (x)}{2} \]

[Out]

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

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Rubi [A]  time = 0.0699828, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )+\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\log (x)}{2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(1 - x^3)^(1/3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 2.29865, size = 48, normalized size = 0.87 \[ - \frac{\log{\left (x^{3} \right )}}{6} + \frac{\log{\left (- \sqrt [3]{- x^{3} + 1} + 1 \right )}}{2} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{- x^{3} + 1}}{3} + \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0189008, size = 39, normalized size = 0.71 \[ -\frac{\sqrt [3]{\frac{x^3-1}{x^3}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{1}{x^3}\right )}{\sqrt [3]{1-x^3}} \]

Antiderivative was successfully verified.

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

[Out]

-((((-1 + x^3)/x^3)^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, x^(-3)])/(1 - x^3)^(1
/3))

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Maple [C]  time = 0.057, size = 65, normalized size = 1.2 \[{\frac{\sqrt{3}\Gamma \left ({\frac{2}{3}} \right ) }{6\,\pi } \left ({\frac{2\,\pi \,\sqrt{3}}{3\,\Gamma \left ( 2/3 \right ) } \left ( -{\frac{\pi \,\sqrt{3}}{6}}-{\frac{3\,\ln \left ( 3 \right ) }{2}}+3\,\ln \left ( x \right ) +i\pi \right ) }+{\frac{2\,\pi \,\sqrt{3}{x}^{3}}{9\,\Gamma \left ( 2/3 \right ) }{\mbox{$_3$F$_2$}(1,1,{\frac{4}{3}};\,2,2;\,{x}^{3})}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-x^3+1)^(1/3),x)

[Out]

1/6/Pi*3^(1/2)*GAMMA(2/3)*(2/3*(-1/6*Pi*3^(1/2)-3/2*ln(3)+3*ln(x)+I*Pi)*Pi*3^(1/
2)/GAMMA(2/3)+2/9*Pi*3^(1/2)/GAMMA(2/3)*x^3*hypergeom([1,1,4/3],[2,2],x^3))

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Maxima [A]  time = 1.47398, size = 84, normalized size = 1.53 \[ \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) - \frac{1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{3} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.214686, size = 96, normalized size = 1.75 \[ -\frac{1}{18} \, \sqrt{3}{\left (\sqrt{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) - 2 \, \sqrt{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1\right ) - 6 \, \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.60516, size = 32, normalized size = 0.58 \[ - \frac{e^{- \frac{i \pi }{3}} \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{1}{x^{3}}} \right )}}{3 x \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-exp(-I*pi/3)*gamma(1/3)*hyper((1/3, 1/3), (4/3,), x**(-3))/(3*x*gamma(4/3))

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GIAC/XCAS [A]  time = 0.20865, size = 85, normalized size = 1.55 \[ \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) - \frac{1}{6} \,{\rm ln}\left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{3} \,{\rm ln}\left ({\left |{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-x^3 + 1)^(1/3) + 1)) - 1/6*ln((-x^3 + 1)^(2/
3) + (-x^3 + 1)^(1/3) + 1) + 1/3*ln(abs((-x^3 + 1)^(1/3) - 1))