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

Optimal. Leaf size=163 \[ \frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}-\frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{x^3-1}}\right )}{3 \sqrt{3}} \]

[Out]

(-2*2^(2/3)*ArcTanh[(Sqrt[3]*(1 - 2^(1/3)*x))/Sqrt[-1 + x^3]])/(3*Sqrt[3]) + (2*
Sqrt[2 - Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*EllipticF[ArcS
in[(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])

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Rubi [A]  time = 0.348498, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}-\frac{2\ 2^{2/3} \tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{2} x\right )}{\sqrt{x^3-1}}\right )}{3 \sqrt{3}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*2^(2/3)*ArcTanh[(Sqrt[3]*(1 - 2^(1/3)*x))/Sqrt[-1 + x^3]])/(3*Sqrt[3]) + (2*
Sqrt[2 - Sqrt[3]]*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*EllipticF[ArcS
in[(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])

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Rubi in Sympy [A]  time = 143.185, size = 444, normalized size = 2.72 \[ - \frac{2^{\frac{2}{3}} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- x + 1\right ) \operatorname{atanh}{\left (\frac{3^{\frac{3}{4}} \sqrt{1 + \sqrt [3]{2}} \sqrt{\sqrt{3} + 2} \sqrt{- \frac{\left (- x + 1 + \sqrt{3}\right )^{2}}{\left (x - 1 + \sqrt{3}\right )^{2}} + 1}}{3 \sqrt{-1 + \sqrt [3]{2}} \sqrt{\frac{\left (- x + 1 + \sqrt{3}\right )^{2}}{\left (x - 1 + \sqrt{3}\right )^{2}} + 4 \sqrt{3} + 7}} \right )}}{\sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{-1 + \sqrt [3]{2}} \left (1 + \sqrt [3]{2}\right )^{\frac{3}{2}} \sqrt{x^{3} - 1}} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- \sqrt{3} + 1\right ) \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1} \left (- \sqrt{3} - 2^{\frac{2}{3}} + 1\right )} + \frac{4 \cdot 2^{\frac{2}{3}} \sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) \Pi \left (\frac{\left (-1 + 2^{\frac{2}{3}} + \sqrt{3}\right )^{2}}{\left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right )^{2}}; \operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{x - 1 + \sqrt{3}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{\sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{4 \sqrt{3} + 7} \sqrt{x^{3} - 1} \left (- 2^{\frac{2}{3}} + 1 + \sqrt{3}\right ) \left (- \sqrt{3} - 2^{\frac{2}{3}} + 1\right )} \]

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**(2/3)*sqrt((x**2 + x + 1)/(-x - sqrt(3) + 1)**2)*(-x + 1)*atanh(3**(3/4)*sqr
t(1 + 2**(1/3))*sqrt(sqrt(3) + 2)*sqrt(-(-x + 1 + sqrt(3))**2/(x - 1 + sqrt(3))*
*2 + 1)/(3*sqrt(-1 + 2**(1/3))*sqrt((-x + 1 + sqrt(3))**2/(x - 1 + sqrt(3))**2 +
4*sqrt(3) + 7)))/(sqrt((x - 1)/(-x - sqrt(3) + 1)**2)*sqrt(-1 + 2**(1/3))*(1 +
2**(1/3))**(3/2)*sqrt(x**3 - 1)) + 2*3**(3/4)*sqrt((x**2 + x + 1)/(-x - sqrt(3)
+ 1)**2)*(-sqrt(3) + 1)*sqrt(-sqrt(3) + 2)*(-x + 1)*elliptic_f(asin((-x + 1 + sq
rt(3))/(-x - sqrt(3) + 1)), -7 + 4*sqrt(3))/(3*sqrt((x - 1)/(-x - sqrt(3) + 1)**
2)*sqrt(x**3 - 1)*(-sqrt(3) - 2**(2/3) + 1)) + 4*2**(2/3)*3**(1/4)*sqrt((x**2 +
x + 1)/(-x - sqrt(3) + 1)**2)*sqrt(sqrt(3) + 2)*(-x + 1)*elliptic_pi((-1 + 2**(2
/3) + sqrt(3))**2/(-2**(2/3) + 1 + sqrt(3))**2, asin((-x + 1 + sqrt(3))/(x - 1 +
sqrt(3))), -7 + 4*sqrt(3))/(sqrt((x - 1)/(-x - sqrt(3) + 1)**2)*sqrt(4*sqrt(3)
+ 7)*sqrt(x**3 - 1)*(-2**(2/3) + 1 + sqrt(3))*(-sqrt(3) - 2**(2/3) + 1))

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Mathematica [C]  time = 0.488098, size = 207, normalized size = 1.27 \[ \frac{2 \sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \left (-\frac{\left (x+\sqrt [3]{-1}\right ) \sqrt{\frac{(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}}+\frac{i 2^{2/3} \sqrt{x^2+x+1} \Pi \left (\frac{i \sqrt{3}}{\sqrt [3]{-1}+2^{2/3}};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt [3]{-1}+2^{2/3}}\right )}{\sqrt{x^3-1}} \]

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]

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Maple [B]  time = 0.029, size = 262, normalized size = 1.6 \[ -2\,{\frac{-3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) }-2\,{\frac{{2}^{2/3} \left ( -3/2-i/2\sqrt{3} \right ) }{\sqrt{{x}^{3}-1} \left ( -{2}^{2/3}+1 \right ) }\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticPi} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},{\frac{3/2+i/2\sqrt{3}}{-{2}^{2/3}+1}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) } \]

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

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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)

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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)

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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)

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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)