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

Optimal. Leaf size=139 \[ \frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}-\frac{2}{3} \left (1+\sqrt{3}\right ) \tanh ^{-1}\left (\sqrt{1-x^3}\right ) \]

[Out]

(-2*(1 + Sqrt[3])*ArcTanh[Sqrt[1 - x^3]])/3 + (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 = 0.112637, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{2 \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x+\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x-\sqrt{3}+1}{-x+\sqrt{3}+1}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{\frac{1-x}{\left (-x+\sqrt{3}+1\right )^2}} \sqrt{1-x^3}}-\frac{2}{3} \left (1+\sqrt{3}\right ) \tanh ^{-1}\left (\sqrt{1-x^3}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(-2*(1 + Sqrt[3])*ArcTanh[Sqrt[1 - x^3]])/3 + (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 = 12.8204, size = 117, normalized size = 0.84 \[ - \left (\frac{2}{3} + \frac{2 \sqrt{3}}{3}\right ) \operatorname{atanh}{\left (\sqrt{- x^{3} + 1} \right )} + \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x - \sqrt{3} + 1}{- x + 1 + \sqrt{3}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x + 1}{\left (- x + 1 + \sqrt{3}\right )^{2}}} \sqrt{- x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.817931, size = 157, normalized size = 1.13 \[ -\frac{2 \tanh ^{-1}\left (\sqrt{1-x^3}\right )}{\sqrt{3}}-\frac{2}{3} \tanh ^{-1}\left (\sqrt{1-x^3}\right )-\frac{2 \sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \left (x+\sqrt [3]{-1}\right ) \sqrt{\frac{(-1)^{2/3} x+\sqrt [3]{-1}}{1+\sqrt [3]{-1}}} F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \sqrt{1-x^3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.049, size = 125, normalized size = 0.9 \[ -{\frac{2+2\,\sqrt{3}}{3}{\it Artanh} \left ( \sqrt{-{x}^{3}+1} \right ) }+{{\frac{2\,i}{3}}\sqrt{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{-{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{-{x}^{3}+1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*arctanh((-x^3+1)^(1/2))*(1+3^(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)*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))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.07263, size = 99, normalized size = 0.71 \[ - \frac{x \Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{x^{3} e^{2 i \pi }} \right )}}{3 \Gamma \left (\frac{4}{3}\right )} + \begin{cases} - \frac{2 \operatorname{acosh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} & \text{for}\: \left |{\frac{1}{x^{3}}}\right | > 1 \\\frac{2 i \operatorname{asin}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} & \text{otherwise} \end{cases} + \sqrt{3} \left (\begin{cases} - \frac{2 \operatorname{acosh}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} & \text{for}\: \left |{\frac{1}{x^{3}}}\right | > 1 \\\frac{2 i \operatorname{asin}{\left (\frac{1}{x^{\frac{3}{2}}} \right )}}{3} & \text{otherwise} \end{cases}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x*gamma(1/3)*hyper((1/3, 1/2), (4/3,), x**3*exp_polar(2*I*pi))/(3*gamma(4/3)) +
Piecewise((-2*acosh(x**(-3/2))/3, Abs(x**(-3)) > 1), (2*I*asin(x**(-3/2))/3, Tr
ue)) + sqrt(3)*Piecewise((-2*acosh(x**(-3/2))/3, Abs(x**(-3)) > 1), (2*I*asin(x*
*(-3/2))/3, True))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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