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 = 0.0050845, 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, Rules used = {239} \[ \frac{1}{2} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{\sqrt{3}} \]

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

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

Mathematica [A]  time = 0.0398614, size = 86, normalized size = 1.76 \[ -\frac{1}{6} \log \left (\frac{x^2}{\left (1-x^3\right )^{2/3}}-\frac{x}{\sqrt [3]{1-x^3}}+1\right )+\frac{1}{3} \log \left (\frac{x}{\sqrt [3]{1-x^3}}+1\right )+\frac{\tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{1-x^3}}-1}{\sqrt{3}}\right )}{\sqrt{3}} \]

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 + Lo
g[1 + x/(1 - x^3)^(1/3)]/3

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Maple [C]  time = 0.023, size = 12, normalized size = 0.2 \begin{align*} x{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{1}{3}};\,{\frac{4}{3}};\,{x}^{3})} \end{align*}

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 = 1.43961, size = 105, normalized size = 2.14 \begin{align*} -\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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-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 [B]  time = 2.03169, size = 225, normalized size = 4.59 \begin{align*} -\frac{1}{3} \, \sqrt{3} \arctan \left (-\frac{\sqrt{3} x - 2 \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{3 \, x}\right ) + \frac{1}{3} \, \log \left (\frac{x +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x}\right ) - \frac{1}{6} \, \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 ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 0.893634, size = 29, normalized size = 0.59 \begin{align*} \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 )} \end{align*}

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-x^3 + 1)^(-1/3), x)