3.106 \(\int (1-x^3)^{2/3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0098674, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {195, 239} \[ \frac{1}{3} \left (1-x^3\right )^{2/3} x+\frac{1}{3} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac{2 \tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

Int[(1 - x^3)^(2/3),x]

[Out]

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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

Mathematica [C]  time = 0.102595, size = 101, normalized size = 1.51 \[ \frac{3 (x-1) \left (1-x^3\right )^{2/3} F_1\left (\frac{5}{3};-\frac{2}{3},-\frac{2}{3};\frac{8}{3};-\frac{x-1}{1-(-1)^{2/3}},-\frac{x-1}{1+\sqrt [3]{-1}}\right )}{5 \left (\frac{x-1}{1+\sqrt [3]{-1}}+1\right )^{2/3} \left (\frac{x-1}{1-(-1)^{2/3}}+1\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(1 - x^3)^(2/3),x]

[Out]

(3*(-1 + x)*(1 - x^3)^(2/3)*AppellF1[5/3, -2/3, -2/3, 8/3, -((-1 + x)/(1 - (-1)^(2/3))), -((-1 + x)/(1 + (-1)^
(1/3)))])/(5*(1 + (-1 + x)/(1 + (-1)^(1/3)))^(2/3)*(1 + (-1 + x)/(1 - (-1)^(2/3)))^(2/3))

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Maple [C]  time = 0.025, size = 12, normalized size = 0.2 \begin{align*} x{\mbox{$_2$F$_1$}(-{\frac{2}{3}},{\frac{1}{3}};\,{\frac{4}{3}};\,{x}^{3})} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-x^3+1)^(2/3),x)

[Out]

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

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Maxima [B]  time = 1.39687, size = 142, normalized size = 2.12 \begin{align*} -\frac{2}{9} \, \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{{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{3 \, x^{2}{\left (\frac{x^{3} - 1}{x^{3}} - 1\right )}} + \frac{2}{9} \, \log \left (\frac{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x} + 1\right ) - \frac{1}{9} \, \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((-x^3+1)^(2/3),x, algorithm="maxima")

[Out]

-2/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-x^3 + 1)^(1/3)/x - 1)) - 1/3*(-x^3 + 1)^(2/3)/(x^2*((x^3 - 1)/x^3 - 1)) +
2/9*log((-x^3 + 1)^(1/3)/x + 1) - 1/9*log(-(-x^3 + 1)^(1/3)/x + (-x^3 + 1)^(2/3)/x^2 + 1)

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Fricas [A]  time = 1.59905, size = 258, normalized size = 3.85 \begin{align*} \frac{1}{3} \,{\left (-x^{3} + 1\right )}^{\frac{2}{3}} x - \frac{2}{9} \, \sqrt{3} \arctan \left (-\frac{\sqrt{3} x - 2 \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{3 \, x}\right ) + \frac{2}{9} \, \log \left (\frac{x +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x}\right ) - \frac{1}{9} \, \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((-x^3+1)^(2/3),x, algorithm="fricas")

[Out]

1/3*(-x^3 + 1)^(2/3)*x - 2/9*sqrt(3)*arctan(-1/3*(sqrt(3)*x - 2*sqrt(3)*(-x^3 + 1)^(1/3))/x) + 2/9*log((x + (-
x^3 + 1)^(1/3))/x) - 1/9*log((x^2 - (-x^3 + 1)^(1/3)*x + (-x^3 + 1)^(2/3))/x^2)

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Sympy [C]  time = 1.15963, size = 31, normalized size = 0.46 \begin{align*} \frac{x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{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((-x**3+1)**(2/3),x)

[Out]

x*gamma(1/3)*hyper((-2/3, 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{\left (-x^{3} + 1\right )}^{\frac{2}{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-x^3 + 1)^(2/3), x)