3.43 \(\int \frac{x}{\left (2^{2/3}-x\right ) \sqrt{1-x^3}} \, dx\)

Optimal. Leaf size=160 \[ \frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}-\frac{2\ 2^{2/3} \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{1-x^3}}\right )}{3 \sqrt{3}} \]

[Out]

(-2*2^(2/3)*ArcTan[(Sqrt[3]*(1 - 2^(1/3)*x))/Sqrt[1 - x^3]])/(3*Sqrt[3]) + (2*Sq
rt[2 + Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*EllipticF[ArcSin
[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/(3*3^(1/4)*Sqrt[(1 - x)/
(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3])

_______________________________________________________________________________________

Rubi [A]  time = 0.362124, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}-\frac{2\ 2^{2/3} \tan ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{1-x^3}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*2^(2/3)*ArcTan[(Sqrt[3]*(1 - 2^(1/3)*x))/Sqrt[1 - x^3]])/(3*Sqrt[3]) + (2*Sq
rt[2 + Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 + Sqrt[3] - x)^2]*EllipticF[ArcSin
[(1 - Sqrt[3] - x)/(1 + Sqrt[3] - x)], -7 - 4*Sqrt[3]])/(3*3^(1/4)*Sqrt[(1 - x)/
(1 + Sqrt[3] - x)^2]*Sqrt[1 - x^3])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 139.543, size = 473, normalized size = 2.96 \[ - \frac{2 \cdot 2^{\frac{2}{3}} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) \operatorname{atan}{\left (\frac{3^{\frac{3}{4}} \sqrt{1 + \sqrt [3]{2}} \sqrt{1 - \frac{\left (x - 1 + \sqrt{3}\right )^{2}}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- 4 \sqrt{3} + 8}}{6 \sqrt{-1 + \sqrt [3]{2}} \sqrt{- 4 \sqrt{3} + 7 + \frac{\left (x - 1 + \sqrt{3}\right )^{2}}{\left (- x + 1 + \sqrt{3}\right )^{2}}}} \right )}}{\sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{-1 + \sqrt [3]{2}} \left (1 + \sqrt [3]{2}\right )^{\frac{3}{2}} \sqrt{- 4 \sqrt{3} + 8} \sqrt{- x^{3} + 1}} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (1 + \sqrt{3}\right ) \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- x^{3} + 1} \left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right )} - \frac{4 \cdot 2^{\frac{2}{3}} \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) \Pi \left (\frac{\left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right )^{2}}{\left (-1 + 2^{\frac{2}{3}} + \sqrt{3}\right )^{2}}; \operatorname{asin}{\left (\frac{x - 1 + \sqrt{3}}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- 4 \sqrt{3} + 7} \sqrt{- x^{3} + 1} \left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right ) \left (- \sqrt{3} - 2^{\frac{2}{3}} + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*2**(2/3)*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(-sqrt(3) + 2)*(-x +
1)*atan(3**(3/4)*sqrt(1 + 2**(1/3))*sqrt(1 - (x - 1 + sqrt(3))**2/(-x + 1 + sqrt
(3))**2)*sqrt(-4*sqrt(3) + 8)/(6*sqrt(-1 + 2**(1/3))*sqrt(-4*sqrt(3) + 7 + (x -
1 + sqrt(3))**2/(-x + 1 + sqrt(3))**2)))/(sqrt((-x + 1)/(-x + 1 + sqrt(3))**2)*s
qrt(-1 + 2**(1/3))*(1 + 2**(1/3))**(3/2)*sqrt(-4*sqrt(3) + 8)*sqrt(-x**3 + 1)) +
2*3**(3/4)*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(3))**2)*(1 + sqrt(3))*sqrt(sqrt(3
) + 2)*(-x + 1)*elliptic_f(asin((-x - sqrt(3) + 1)/(-x + 1 + sqrt(3))), -7 - 4*s
qrt(3))/(3*sqrt((-x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(-x**3 + 1)*(-2**(2/3) + 1 +
sqrt(3))) - 4*2**(2/3)*3**(1/4)*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(3))**2)*sqrt
(-sqrt(3) + 2)*(-x + 1)*elliptic_pi((-2**(2/3) + 1 + sqrt(3))**2/(-1 + 2**(2/3)
+ sqrt(3))**2, asin((x - 1 + sqrt(3))/(-x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(sqrt
((-x + 1)/(-x + 1 + sqrt(3))**2)*sqrt(-4*sqrt(3) + 7)*sqrt(-x**3 + 1)*(-2**(2/3)
+ 1 + sqrt(3))*(-sqrt(3) - 2**(2/3) + 1))

_______________________________________________________________________________________

Mathematica [C]  time = 0.532433, size = 209, normalized size = 1.31 \[ \frac{2 \sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \left (-\frac{\left (x+\sqrt [3]{-1}\right ) \sqrt{\frac{(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac{i 2^{2/3} \sqrt{x^2+x+1} \Pi \left (\frac{i \sqrt{3}}{\sqrt [3]{-1}+2^{2/3}};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [3]{-1}+2^{2/3}}\right )}{\sqrt{1-x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*(-((((-1)^(1/3) + x)*Sqrt[((-1)^(1/3) + (-1)^(
2/3)*x)/(1 + (-1)^(1/3))]*EllipticF[ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/
3))]], (-1)^(1/3)])/Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]) + (I*2^(2/3)*Sqrt
[1 + x + x^2]*EllipticPi[(I*Sqrt[3])/((-1)^(1/3) + 2^(2/3)), ArcSin[Sqrt[(1 - (-
1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/((-1)^(1/3) + 2^(2/3))))/Sqrt[1 - x
^3]

_______________________________________________________________________________________

Maple [A]  time = 0.035, size = 253, normalized size = 1.6 \[{{\frac{2\,i}{3}}\sqrt{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}}+{\frac{{\frac{2\,i}{3}}{2}^{{\frac{2}{3}}}\sqrt{3}}{-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}-{2}^{{\frac{2}{3}}}}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3}-{2}^{{\frac{2}{3}}}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*I*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((-1+x)/(-3/2+1/2*I*3^(1/2
)))^(1/2)*(-I*(x+1/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(-x^3+1)^(1/2)*EllipticF(1/3*
3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),(I*3^(1/2)/(-3/2+1/2*I*3^(1/2)))
^(1/2))+2/3*I*2^(2/3)*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((-1+x)/(-
3/2+1/2*I*3^(1/2)))^(1/2)*(-I*(x+1/2+1/2*I*3^(1/2))*3^(1/2))^(1/2)/(-x^3+1)^(1/2
)/(-1/2+1/2*I*3^(1/2)-2^(2/3))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3
^(1/2))^(1/2),I*3^(1/2)/(-1/2+1/2*I*3^(1/2)-2^(2/3)),(I*3^(1/2)/(-3/2+1/2*I*3^(1
/2)))^(1/2))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-integrate(x/(sqrt(-x^3 + 1)*(x - 2^(2/3))), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-\frac{x}{\sqrt{-x^{3} + 1}{\left (x - 2^{\frac{2}{3}}\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-x/(sqrt(-x^3 + 1)*(x - 2^(2/3))), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-Integral(x/(x*sqrt(-x**3 + 1) - 2**(2/3)*sqrt(-x**3 + 1)), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(-x/(sqrt(-x^3 + 1)*(x - 2^(2/3))), x)