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

Optimal. Leaf size=107 \[ -\frac{1}{9} \log \left (\frac{x}{\sqrt [3]{1-x^3}}+1\right )-\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{1}{3} \sqrt [3]{1-x^3} x^2+\frac{1}{18} \log \left (-\frac{x}{\sqrt [3]{1-x^3}}+\frac{x^2}{\left (1-x^3\right )^{2/3}}+1\right ) \]

[Out]

(x^2*(1 - x^3)^(1/3))/3 - ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]]/(3*Sqrt[3]
) + Log[1 + x^2/(1 - x^3)^(2/3) - x/(1 - x^3)^(1/3)]/18 - Log[1 + x/(1 - x^3)^(1
/3)]/9

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Rubi [A]  time = 0.10215, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{1}{9} \log \left (\frac{x}{\sqrt [3]{1-x^3}}+1\right )-\frac{\tan ^{-1}\left (\frac{1-\frac{2 x}{\sqrt [3]{1-x^3}}}{\sqrt{3}}\right )}{3 \sqrt{3}}+\frac{1}{3} \sqrt [3]{1-x^3} x^2+\frac{1}{18} \log \left (-\frac{x}{\sqrt [3]{1-x^3}}+\frac{x^2}{\left (1-x^3\right )^{2/3}}+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[x*(1 - x^3)^(1/3),x]

[Out]

(x^2*(1 - x^3)^(1/3))/3 - ArcTan[(1 - (2*x)/(1 - x^3)^(1/3))/Sqrt[3]]/(3*Sqrt[3]
) + Log[1 + x^2/(1 - x^3)^(2/3) - x/(1 - x^3)^(1/3)]/18 - Log[1 + x/(1 - x^3)^(1
/3)]/9

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Rubi in Sympy [A]  time = 4.77537, size = 85, normalized size = 0.79 \[ \frac{x^{2} \sqrt [3]{- x^{3} + 1}}{3} - \frac{\log{\left (\frac{x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{9} + \frac{\log{\left (\frac{x^{2}}{\left (- x^{3} + 1\right )^{\frac{2}{3}}} - \frac{x}{\sqrt [3]{- x^{3} + 1}} + 1 \right )}}{18} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x}{3 \sqrt [3]{- x^{3} + 1}} - \frac{1}{3}\right ) \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*(-x**3 + 1)**(1/3)/3 - log(x/(-x**3 + 1)**(1/3) + 1)/9 + log(x**2/(-x**3 +
1)**(2/3) - x/(-x**3 + 1)**(1/3) + 1)/18 + sqrt(3)*atan(sqrt(3)*(2*x/(3*(-x**3 +
1)**(1/3)) - 1/3))/9

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Mathematica [C]  time = 0.0183334, size = 34, normalized size = 0.32 \[ \frac{1}{6} x^2 \left (\, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};x^3\right )+2 \sqrt [3]{1-x^3}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x*(1 - x^3)^(1/3),x]

[Out]

(x^2*(2*(1 - x^3)^(1/3) + Hypergeometric2F1[2/3, 2/3, 5/3, x^3]))/6

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Maple [C]  time = 0.069, size = 69, normalized size = 0.6 \[ -{\frac{{x}^{2} \left ({x}^{3}-1 \right ) }{3} \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{3}}}}+{\frac{{x}^{2}}{6} \left ({x}^{3}-1 \right ) ^{{\frac{2}{3}}} \left ( -{\it signum} \left ({x}^{3}-1 \right ) \right ) ^{{\frac{2}{3}}}{\mbox{$_2$F$_1$}({\frac{2}{3}},{\frac{2}{3}};\,{\frac{5}{3}};\,{x}^{3})} \left ({\it signum} \left ({x}^{3}-1 \right ) \right ) ^{-{\frac{2}{3}}} \left ( -{x}^{3}+1 \right ) ^{-{\frac{2}{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(-x^3+1)^(1/3),x)

[Out]

-1/3*x^2*(x^3-1)/(-x^3+1)^(2/3)+1/6*(x^3-1)^(2/3)/signum(x^3-1)^(2/3)*(-signum(x
^3-1))^(2/3)*x^2*hypergeom([2/3,2/3],[5/3],x^3)/(-x^3+1)^(2/3)

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Maxima [A]  time = 1.66624, size = 142, normalized size = 1.33 \[ -\frac{1}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (\frac{2 \,{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x} - 1\right )}\right ) - \frac{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{3 \, x{\left (\frac{x^{3} - 1}{x^{3}} - 1\right )}} - \frac{1}{9} \, \log \left (\frac{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x} + 1\right ) + \frac{1}{18} \, \log \left (-\frac{{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x} + \frac{{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2}} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-x^3 + 1)^(1/3)/x - 1)) - 1/3*(-x^3 + 1)^(1/
3)/(x*((x^3 - 1)/x^3 - 1)) - 1/9*log((-x^3 + 1)^(1/3)/x + 1) + 1/18*log(-(-x^3 +
1)^(1/3)/x + (-x^3 + 1)^(2/3)/x^2 + 1)

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Fricas [A]  time = 0.209585, size = 143, normalized size = 1.34 \[ \frac{1}{54} \, \sqrt{3}{\left (6 \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x^{2} - 2 \, \sqrt{3} \log \left (\frac{x +{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{x}\right ) + \sqrt{3} \log \left (\frac{x^{2} -{\left (-x^{3} + 1\right )}^{\frac{1}{3}} x +{\left (-x^{3} + 1\right )}^{\frac{2}{3}}}{x^{2}}\right ) - 6 \, \arctan \left (-\frac{\sqrt{3} x - 2 \, \sqrt{3}{\left (-x^{3} + 1\right )}^{\frac{1}{3}}}{3 \, x}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/54*sqrt(3)*(6*sqrt(3)*(-x^3 + 1)^(1/3)*x^2 - 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*arct
an(-1/3*(sqrt(3)*x - 2*sqrt(3)*(-x^3 + 1)^(1/3))/x))

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Sympy [A]  time = 1.78304, size = 32, normalized size = 0.3 \[ \frac{x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{x^{3} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*(-x**3+1)**(1/3),x)

[Out]

x**2*gamma(2/3)*hyper((-1/3, 2/3), (5/3,), x**3*exp_polar(2*I*pi))/(3*gamma(5/3)
)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-x^3 + 1)^(1/3)*x, x)