3.306 \(\int \sqrt [3]{x \left (1-x^2\right )} \, dx\)

Optimal. Leaf size=93 \[ \frac{1}{2} \sqrt [3]{x \left (1-x^2\right )} x-\frac{1}{4} \log \left (\sqrt [3]{x \left (1-x^2\right )}+x\right )+\frac{\tan ^{-1}\left (\frac{2 x-\sqrt [3]{x \left (1-x^2\right )}}{\sqrt{3} \sqrt [3]{x \left (1-x^2\right )}}\right )}{2 \sqrt{3}}+\frac{\log (x)}{12} \]

[Out]

(x*(x*(1 - x^2))^(1/3))/2 + ArcTan[(2*x - (x*(1 - x^2))^(1/3))/(Sqrt[3]*(x*(1 -
x^2))^(1/3))]/(2*Sqrt[3]) + Log[x]/12 - Log[x + (x*(1 - x^2))^(1/3)]/4

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Rubi [B]  time = 0.263093, antiderivative size = 200, normalized size of antiderivative = 2.15, number of steps used = 12, number of rules used = 12, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.923 \[ \frac{1}{2} \sqrt [3]{x-x^3} x+\frac{\left (1-x^2\right )^{2/3} x^{2/3} \log \left (\frac{x^{4/3}}{\left (1-x^2\right )^{2/3}}-\frac{x^{2/3}}{\sqrt [3]{1-x^2}}+1\right )}{12 \left (x-x^3\right )^{2/3}}-\frac{\left (1-x^2\right )^{2/3} x^{2/3} \log \left (\frac{x^{2/3}}{\sqrt [3]{1-x^2}}+1\right )}{6 \left (x-x^3\right )^{2/3}}-\frac{\left (1-x^2\right )^{2/3} x^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 x^{2/3}}{\sqrt [3]{1-x^2}}}{\sqrt{3}}\right )}{2 \sqrt{3} \left (x-x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[(x*(1 - x^2))^(1/3),x]

[Out]

(x*(x - x^3)^(1/3))/2 - (x^(2/3)*(1 - x^2)^(2/3)*ArcTan[(1 - (2*x^(2/3))/(1 - x^
2)^(1/3))/Sqrt[3]])/(2*Sqrt[3]*(x - x^3)^(2/3)) + (x^(2/3)*(1 - x^2)^(2/3)*Log[1
+ x^(4/3)/(1 - x^2)^(2/3) - x^(2/3)/(1 - x^2)^(1/3)])/(12*(x - x^3)^(2/3)) - (x
^(2/3)*(1 - x^2)^(2/3)*Log[1 + x^(2/3)/(1 - x^2)^(1/3)])/(6*(x - x^3)^(2/3))

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Rubi in Sympy [A]  time = 8.95156, size = 162, normalized size = 1.74 \[ \frac{x \sqrt [3]{- x^{3} + x}}{2} - \frac{\sqrt [3]{- x^{3} + x} \log{\left (\frac{x^{\frac{2}{3}}}{\sqrt [3]{- x^{2} + 1}} + 1 \right )}}{6 \sqrt [3]{x} \sqrt [3]{- x^{2} + 1}} + \frac{\sqrt [3]{- x^{3} + x} \log{\left (\frac{x^{\frac{4}{3}}}{\left (- x^{2} + 1\right )^{\frac{2}{3}}} - \frac{x^{\frac{2}{3}}}{\sqrt [3]{- x^{2} + 1}} + 1 \right )}}{12 \sqrt [3]{x} \sqrt [3]{- x^{2} + 1}} + \frac{\sqrt{3} \sqrt [3]{- x^{3} + x} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{\frac{2}{3}}}{3 \sqrt [3]{- x^{2} + 1}} - \frac{1}{3}\right ) \right )}}{6 \sqrt [3]{x} \sqrt [3]{- x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*(-x**3 + x)**(1/3)/2 - (-x**3 + x)**(1/3)*log(x**(2/3)/(-x**2 + 1)**(1/3) + 1)
/(6*x**(1/3)*(-x**2 + 1)**(1/3)) + (-x**3 + x)**(1/3)*log(x**(4/3)/(-x**2 + 1)**
(2/3) - x**(2/3)/(-x**2 + 1)**(1/3) + 1)/(12*x**(1/3)*(-x**2 + 1)**(1/3)) + sqrt
(3)*(-x**3 + x)**(1/3)*atan(sqrt(3)*(2*x**(2/3)/(3*(-x**2 + 1)**(1/3)) - 1/3))/(
6*x**(1/3)*(-x**2 + 1)**(1/3))

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Mathematica [C]  time = 0.028443, size = 56, normalized size = 0.6 \[ \frac{x \sqrt [3]{x-x^3} \left (-\left (1-x^2\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};x^2\right )+2 x^2-2\right )}{4 \left (x^2-1\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[(x*(1 - x^2))^(1/3),x]

[Out]

(x*(x - x^3)^(1/3)*(-2 + 2*x^2 - (1 - x^2)^(2/3)*Hypergeometric2F1[2/3, 2/3, 5/3
, x^2]))/(4*(-1 + x^2))

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Maple [C]  time = 0.043, size = 15, normalized size = 0.2 \[{\frac{3}{4}{x}^{{\frac{4}{3}}}{\mbox{$_2$F$_1$}(-{\frac{1}{3}},{\frac{2}{3}};\,{\frac{5}{3}};\,{x}^{2})}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x*(-x^2+1))^(1/3),x)

[Out]

3/4*x^(4/3)*hypergeom([-1/3,2/3],[5/3],x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-(x^2 - 1)*x)^(1/3), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt [3]{x \left (- x^{2} + 1\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((x*(-x**2 + 1))**(1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((-(x^2 - 1)*x)^(1/3), x)