3.113 \(\int \frac{x}{\left (1+\sqrt{3}-x\right ) \sqrt{1-x^3}} \, dx\)

Optimal. Leaf size=152 \[ \frac{\sqrt{2} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{1-x^3}}\right )}{3^{3/4}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.471326, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16 \[ \frac{\sqrt{2} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}-\frac{\sqrt{2} \tan ^{-1}\left (\frac{\sqrt{3+2 \sqrt{3}} (1-x)}{\sqrt{1-x^3}}\right )}{3^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 13.0498, size = 78, normalized size = 0.51 \[ \frac{2 \tilde{\infty } \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*zoo*sqrt((x**2 + x + 1)/(-x + 1 + sqrt(3))**2)*(-x + 1)*elliptic_f(asin((-x -
sqrt(3) + 1)/(-x + 1 + sqrt(3))), -7 - 4*sqrt(3))/(sqrt((-x + 1)/(-x + 1 + sqrt(
3))**2)*sqrt(-x**3 + 1))

_______________________________________________________________________________________

Mathematica [C]  time = 1.03893, size = 232, normalized size = 1.53 \[ \frac{2 i \sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \left (2 \left (1+\sqrt{3}\right ) \sqrt{x^2+x+1} \Pi \left (\frac{2 i \sqrt{3}}{3+(2+i) \sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )+\frac{i \sqrt{\frac{(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} \left (\left (3+(2+i) \sqrt{3}\right ) x+(1+2 i) \sqrt{3}+3 i\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}\right )}{\left (3+(2+i) \sqrt{3}\right ) \sqrt{1-x^3}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.033, size = 257, normalized size = 1.7 \[{\frac{-{\frac{2\,i}{3}} \left ( -1-\sqrt{3} \right ) \sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}-\sqrt{3}}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticPi} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}-\sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}}+{{\frac{2\,i}{3}}\sqrt{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*I*(-1-3^(1/2))*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)/
(-3/2+1/2*I*3^(1/2)-3^(1/2))*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)/(-3/2+1/2*I*3^(1/2)-3^(1/2)),(I*3^(1/2)/(-3/2+1/2*I*3^(1/2
)))^(1/2))+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)*Ell
ipticF(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))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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