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

Optimal. Leaf size=46 \[ \frac{\tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{1}{2} \log \left (\sqrt [3]{x^3+2}-x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0168941, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{\tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{1}{2} \log \left (\sqrt [3]{x^3+2}-x\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2 + x^3)^(-1/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.34997, size = 71, normalized size = 1.54 \[ - \frac{\log{\left (- \frac{x}{\sqrt [3]{x^{3} + 2}} + 1 \right )}}{3} + \frac{\log{\left (\frac{x^{2}}{\left (x^{3} + 2\right )^{\frac{2}{3}}} + \frac{x}{\sqrt [3]{x^{3} + 2}} + 1 \right )}}{6} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3 \sqrt [3]{x^{3} + 2}} + \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0597671, size = 78, normalized size = 1.7 \[ -\frac{1}{3} \log \left (1-\frac{x}{\sqrt [3]{x^3+2}}\right )+\frac{\tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )}{\sqrt{3}}+\frac{1}{6} \log \left (\frac{x}{\sqrt [3]{x^3+2}}+\frac{x^2}{\left (x^3+2\right )^{2/3}}+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + x^3)^(-1/3),x]

[Out]

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

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Maple [C]  time = 0.041, size = 18, normalized size = 0.4 \[{\frac{x{2}^{{\frac{2}{3}}}}{2}{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{1}{3}};\,{\frac{4}{3}};\,-{\frac{{x}^{3}}{2}})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^3+2)^(1/3),x)

[Out]

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

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Maxima [A]  time = 1.58662, size = 93, normalized size = 2.02 \[ -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (\frac{2 \,{\left (x^{3} + 2\right )}^{\frac{1}{3}}}{x} + 1\right )}\right ) + \frac{1}{6} \, \log \left (\frac{{\left (x^{3} + 2\right )}^{\frac{1}{3}}}{x} + \frac{{\left (x^{3} + 2\right )}^{\frac{2}{3}}}{x^{2}} + 1\right ) - \frac{1}{3} \, \log \left (\frac{{\left (x^{3} + 2\right )}^{\frac{1}{3}}}{x} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.244586, size = 113, normalized size = 2.46 \[ -\frac{1}{18} \, \sqrt{3}{\left (2 \, \sqrt{3} \log \left (-\frac{x -{\left (x^{3} + 2\right )}^{\frac{1}{3}}}{x}\right ) - \sqrt{3} \log \left (\frac{x^{2} +{\left (x^{3} + 2\right )}^{\frac{1}{3}} x +{\left (x^{3} + 2\right )}^{\frac{2}{3}}}{x^{2}}\right ) + 6 \, \arctan \left (\frac{\sqrt{3} x + 2 \, \sqrt{3}{\left (x^{3} + 2\right )}^{\frac{1}{3}}}{3 \, x}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.22593, size = 34, normalized size = 0.74 \[ \frac{2^{\frac{2}{3}} x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{x^{3} e^{i \pi }}{2}} \right )}}{6 \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**3+2)**(1/3),x)

[Out]

2**(2/3)*x*gamma(1/3)*hyper((1/3, 1/3), (4/3,), x**3*exp_polar(I*pi)/2)/(6*gamma
(4/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((x^3 + 2)^(-1/3), x)