3.33 \(\int \frac{1}{x \sqrt [3]{1-x^2}} \, dx\)

Optimal. Leaf size=58 \[ \frac{3}{4} \log \left (1-\sqrt [3]{1-x^2}\right )+\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )-\frac{\log (x)}{2} \]

[Out]

(Sqrt[3]*ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]])/2 - Log[x]/2 + (3*Log[1 - (1 -
x^2)^(1/3)])/4

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Rubi [A]  time = 0.0778211, antiderivative size = 58, 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 \[ \frac{3}{4} \log \left (1-\sqrt [3]{1-x^2}\right )+\frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^2}+1}{\sqrt{3}}\right )-\frac{\log (x)}{2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(1 - x^2)^(1/3)),x]

[Out]

(Sqrt[3]*ArcTan[(1 + 2*(1 - x^2)^(1/3))/Sqrt[3]])/2 - Log[x]/2 + (3*Log[1 - (1 -
x^2)^(1/3)])/4

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Rubi in Sympy [A]  time = 2.45576, size = 49, normalized size = 0.84 \[ - \frac{\log{\left (x^{2} \right )}}{4} + \frac{3 \log{\left (- \sqrt [3]{- x^{2} + 1} + 1 \right )}}{4} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{- x^{2} + 1}}{3} + \frac{1}{3}\right ) \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x**2)/4 + 3*log(-(-x**2 + 1)**(1/3) + 1)/4 + sqrt(3)*atan(sqrt(3)*(2*(-x**2
+ 1)**(1/3)/3 + 1/3))/2

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Mathematica [C]  time = 0.0209582, size = 41, normalized size = 0.71 \[ -\frac{3 \sqrt [3]{\frac{x^2-1}{x^2}} \, _2F_1\left (\frac{1}{3},\frac{1}{3};\frac{4}{3};\frac{1}{x^2}\right )}{2 \sqrt [3]{1-x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*((-1 + x^2)/x^2)^(1/3)*Hypergeometric2F1[1/3, 1/3, 4/3, x^(-2)])/(2*(1 - x^2
)^(1/3))

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Maple [C]  time = 0.056, size = 65, normalized size = 1.1 \[{\frac{\sqrt{3}\Gamma \left ({\frac{2}{3}} \right ) }{4\,\pi } \left ({\frac{2\,\pi \,\sqrt{3}}{3\,\Gamma \left ( 2/3 \right ) } \left ( -{\frac{\pi \,\sqrt{3}}{6}}-{\frac{3\,\ln \left ( 3 \right ) }{2}}+2\,\ln \left ( x \right ) +i\pi \right ) }+{\frac{2\,\pi \,\sqrt{3}{x}^{2}}{9\,\Gamma \left ( 2/3 \right ) }{\mbox{$_3$F$_2$}(1,1,{\frac{4}{3}};\,2,2;\,{x}^{2})}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(-x^2+1)^(1/3),x)

[Out]

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

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Maxima [A]  time = 1.49653, size = 84, normalized size = 1.45 \[ \frac{1}{2} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) - \frac{1}{4} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac{2}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{2} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-x^2 + 1)^(1/3) + 1)) - 1/4*log((-x^2 + 1)^(2
/3) + (-x^2 + 1)^(1/3) + 1) + 1/2*log((-x^2 + 1)^(1/3) - 1)

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Fricas [A]  time = 0.21369, size = 84, normalized size = 1.45 \[ \frac{1}{2} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) - \frac{1}{4} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac{2}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{2} \, \log \left ({\left (-x^{2} + 1\right )}^{\frac{1}{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-x^2 + 1)^(1/3) + 1)) - 1/4*log((-x^2 + 1)^(2
/3) + (-x^2 + 1)^(1/3) + 1) + 1/2*log((-x^2 + 1)^(1/3) - 1)

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Sympy [A]  time = 1.60745, size = 36, normalized size = 0.62 \[ - \frac{e^{- \frac{i \pi }{3}} \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{1}{x^{2}}} \right )}}{2 x^{\frac{2}{3}} \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-exp(-I*pi/3)*gamma(1/3)*hyper((1/3, 1/3), (4/3,), x**(-2))/(2*x**(2/3)*gamma(4/
3))

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GIAC/XCAS [A]  time = 0.20711, size = 86, normalized size = 1.48 \[ \frac{1}{2} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left (-x^{2} + 1\right )}^{\frac{2}{3}} +{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{2} \,{\rm ln}\left (-{\left (-x^{2} + 1\right )}^{\frac{1}{3}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-x^2 + 1)^(1/3) + 1)) - 1/4*ln((-x^2 + 1)^(2/
3) + (-x^2 + 1)^(1/3) + 1) + 1/2*ln(-(-x^2 + 1)^(1/3) + 1)