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

Optimal. Leaf size=120 \[ -\frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

[Out]

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

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Rubi [A]  time = 0.0354077, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{2 \sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[-1 + x^3],x]

[Out]

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

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Rubi in Sympy [A]  time = 1.89606, size = 95, normalized size = 0.79 \[ - \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0421034, size = 71, normalized size = 0.59 \[ \frac{2 i \sqrt{(-1)^{5/6} (x-1)} \sqrt{x^2+x+1} F\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{\sqrt [4]{3} \sqrt{x^3-1}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/Sqrt[-1 + x^3],x]

[Out]

((2*I)*Sqrt[(-1)^(5/6)*(-1 + x)]*Sqrt[1 + x + x^2]*EllipticF[ArcSin[Sqrt[-(-1)^(
5/6) - I*x]/3^(1/4)], (-1)^(1/3)])/(3^(1/4)*Sqrt[-1 + x^3])

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Maple [A]  time = 0.022, size = 116, normalized size = 1. \[ 2\,{\frac{-3/2-i/2\sqrt{3}}{\sqrt{{x}^{3}-1}}\sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2-i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}}\sqrt{{\frac{x+1/2+i/2\sqrt{3}}{3/2+i/2\sqrt{3}}}}{\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-3/2-i/2\sqrt{3}}}},\sqrt{{\frac{3/2+i/2\sqrt{3}}{3/2-i/2\sqrt{3}}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(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))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(x^3 - 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/sqrt(x^3 - 1), x)

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Sympy [A]  time = 1.76128, size = 26, normalized size = 0.22 \[ - \frac{i x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{x^{3}} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**3-1)**(1/2),x)

[Out]

-I*x*gamma(1/3)*hyper((1/3, 1/2), (4/3,), x**3)/(3*gamma(4/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/sqrt(x^3 - 1), x)