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

Optimal. Leaf size=49 \[ \frac{1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

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

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Rubi [A]  time = 0.0162811, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^3)^(-1/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 3.58718, size = 71, normalized size = 1.45 \[ \frac{\log{\left (\frac{x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{3} - \frac{\log{\left (\frac{x^{2}}{\left (- x^{3} + 1\right )^{\frac{2}{3}}} - \frac{x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3 \sqrt [3]{- x^{3} + 1}} - \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0709057, size = 86, normalized size = 1.76 \[ \frac{1}{3} \log \left (\frac{x}{\sqrt [3]{1-x^3}}+1\right )+\frac{\tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{1-x^3}}-1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{1}{6} \log \left (-\frac{x}{\sqrt [3]{1-x^3}}+\frac{x^2}{\left (1-x^3\right )^{2/3}}+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x^3)^(-1/3),x]

[Out]

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

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Maple [C]  time = 0.034, size = 12, normalized size = 0.2 \[ x{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{1}{3}};\,{\frac{4}{3}};\,{x}^{3})} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-x^3+1)^(1/3),x)

[Out]

x*hypergeom([1/3,1/3],[4/3],x^3)

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Maxima [A]  time = 1.50195, size = 105, normalized size = 2.14 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (\frac{2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x} - 1\right )}\right ) + \frac{1}{3} \, \log \left (\frac{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x} + 1\right ) - \frac{1}{6} \, \log \left (-\frac{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x} + \frac{{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.216066, size = 122, normalized size = 2.49 \[ \frac{1}{18} \, \sqrt{3}{\left (2 \, \sqrt{3} \log \left (\frac{x +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x}\right ) - \sqrt{3} \log \left (\frac{x^{2} -{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2}}\right ) - 6 \, \arctan \left (-\frac{\sqrt{3} x - 2 \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{3 \, x}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.60956, size = 29, normalized size = 0.59 \[ \frac{x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{x^{3} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-x**3+1)**(1/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-x^3 + 1)^(-1/3), x)