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

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 = 0.0050076, 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, Rules used = {239} \[ \frac{\tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{1}{2} \log \left (\sqrt [3]{x^3+2}-x\right ) \]

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

Rule 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [3]{2+x^3}} \, dx &=\frac{\tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{2+x^3}}}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{1}{2} \log \left (-x+\sqrt [3]{2+x^3}\right )\\ \end{align*}

Mathematica [A]  time = 0.0331848, size = 78, normalized size = 1.7 \[ -\frac{1}{3} \log \left (1-\frac{x}{\sqrt [3]{x^3+2}}\right )+\frac{1}{6} \log \left (\frac{x^2}{\left (x^3+2\right )^{2/3}}+\frac{x}{\sqrt [3]{x^3+2}}+1\right )+\frac{\tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{x^3+2}}+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

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 = 0.013, size = 18, normalized size = 0.4 \begin{align*}{\frac{x{2}^{{\frac{2}{3}}}}{2}{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{1}{3}};\,{\frac{4}{3}};\,-{\frac{{x}^{3}}{2}})}} \end{align*}

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 = 1.52163, size = 93, normalized size = 2.02 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(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 [B]  time = 1.53686, size = 220, normalized size = 4.78 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (\frac{\sqrt{3} x + 2 \, \sqrt{3}{\left (x^{3} + 2\right )}^{\frac{1}{3}}}{3 \, x}\right ) - \frac{1}{3} \, \log \left (-\frac{x -{\left (x^{3} + 2\right )}^{\frac{1}{3}}}{x}\right ) + \frac{1}{6} \, \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^3+2)^(1/3),x, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 1.00409, size = 34, normalized size = 0.74 \begin{align*} \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 )} \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (x^{3} + 2\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^3+2)^(1/3),x, algorithm="giac")

[Out]

integrate((x^3 + 2)^(-1/3), x)