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

Optimal. Leaf size=67 \[ \sqrt [3]{1-x^3}+\frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\log (x)}{2} \]

[Out]

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

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Rubi [A]  time = 0.0840064, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \sqrt [3]{1-x^3}+\frac{1}{2} \log \left (1-\sqrt [3]{1-x^3}\right )-\frac{\tan ^{-1}\left (\frac{2 \sqrt [3]{1-x^3}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\log (x)}{2} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^3)^(1/3)/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 2.76762, size = 56, normalized size = 0.84 \[ \sqrt [3]{- x^{3} + 1} - \frac{\log{\left (x^{3} \right )}}{6} + \frac{\log{\left (- \sqrt [3]{- x^{3} + 1} + 1 \right )}}{2} - \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{- x^{3} + 1}}{3} + \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(-x**3 + 1)**(1/3) - 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 = 0.0246976, size = 48, normalized size = 0.72 \[ \frac{-\left (1-\frac{1}{x^3}\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};\frac{1}{x^3}\right )-2 x^3+2}{2 \left (1-x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x^3)^(1/3)/x,x]

[Out]

(2 - 2*x^3 - (1 - x^(-3))^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3, x^(-3)])/(2*(1
- x^3)^(2/3))

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Maple [C]  time = 0.073, size = 49, normalized size = 0.7 \[ -{\frac{1}{9\,\Gamma \left ( 2/3 \right ) } \left ( -3\, \left ( 3+1/6\,\pi \,\sqrt{3}-3/2\,\ln \left ( 3 \right ) +3\,\ln \left ( x \right ) +i\pi \right ) \Gamma \left ( 2/3 \right ) +\Gamma \left ({\frac{2}{3}} \right ){x}^{3}{\mbox{$_3$F$_2$}({\frac{2}{3}},1,1;\,2,2;\,{x}^{3})} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^3+1)^(1/3)/x,x)

[Out]

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

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Maxima [A]  time = 1.53367, size = 96, normalized size = 1.43 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \frac{1}{6} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{3} \, \log \left ({\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-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)) + (-x^3 + 1)^(1/3) - 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 = 0.208472, size = 115, normalized size = 1.72 \[ -\frac{1}{18} \, \sqrt{3}{\left (\sqrt{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) - 2 \, \sqrt{3} \log \left ({\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1\right ) - 6 \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 6 \, \arctan \left (\frac{2}{3} \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + \frac{1}{3} \, \sqrt{3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-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*sqrt(3)*(-x^3 + 1)^(1/3) + 6*arctan(2/3*sqrt(3)*(-
x^3 + 1)^(1/3) + 1/3*sqrt(3)))

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Sympy [A]  time = 1.67805, size = 37, normalized size = 0.55 \[ - \frac{x e^{\frac{i \pi }{3}} \Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, - \frac{1}{3} \\ \frac{2}{3} \end{matrix}\middle |{\frac{1}{x^{3}}} \right )}}{3 \Gamma \left (\frac{2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**3+1)**(1/3)/x,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.24, size = 97, normalized size = 1.45 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right )}\right ) +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - \frac{1}{6} \,{\rm ln}\left ({\left (-x^{3} + 1\right )}^{\frac{2}{3}} +{\left (-x^{3} + 1\right )}^{\frac{1}{3}} + 1\right ) + \frac{1}{3} \,{\rm ln}\left ({\left |{\left (-x^{3} + 1\right )}^{\frac{1}{3}} - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-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)) + (-x^3 + 1)^(1/3) - 1
/6*ln((-x^3 + 1)^(2/3) + (-x^3 + 1)^(1/3) + 1) + 1/3*ln(abs((-x^3 + 1)^(1/3) - 1
))