3.490 \(\int \frac{1}{x^2 \sqrt{-1+x^3}} \, dx\)

Optimal. Leaf size=269 \[ \frac{\sqrt{x^3-1}}{-x-\sqrt{3}+1}+\frac{\sqrt{x^3-1}}{x}+\frac{\sqrt{2} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2 \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

[Out]

Sqrt[-1 + x^3]/(1 - Sqrt[3] - x) + Sqrt[-1 + x^3]/x - (3^(1/4)*Sqrt[2 + Sqrt[3]]
*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*EllipticE[ArcSin[(1 + Sqrt[3] -
x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt[3]])/(2*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]
*Sqrt[-1 + x^3]) + (Sqrt[2]*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*Elli
pticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqr
t[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3])

_______________________________________________________________________________________

Rubi [A]  time = 0.160687, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ \frac{\sqrt{x^3-1}}{-x-\sqrt{3}+1}+\frac{\sqrt{x^3-1}}{x}+\frac{\sqrt{2} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}}-\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2 \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[-1 + x^3]),x]

[Out]

Sqrt[-1 + x^3]/(1 - Sqrt[3] - x) + Sqrt[-1 + x^3]/x - (3^(1/4)*Sqrt[2 + Sqrt[3]]
*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*EllipticE[ArcSin[(1 + Sqrt[3] -
x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt[3]])/(2*Sqrt[-((1 - x)/(1 - Sqrt[3] - x)^2)]
*Sqrt[-1 + x^3]) + (Sqrt[2]*(1 - x)*Sqrt[(1 + x + x^2)/(1 - Sqrt[3] - x)^2]*Elli
pticF[ArcSin[(1 + Sqrt[3] - x)/(1 - Sqrt[3] - x)], -7 + 4*Sqrt[3]])/(3^(1/4)*Sqr
t[-((1 - x)/(1 - Sqrt[3] - x)^2)]*Sqrt[-1 + x^3])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 11.9169, size = 207, normalized size = 0.77 \[ \frac{\sqrt{x^{3} - 1}}{- x - \sqrt{3} + 1} - \frac{\sqrt [4]{3} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (- x + 1\right ) E\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{2 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1}} + \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1}} + \frac{\sqrt{x^{3} - 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x**3 - 1)/(-x - sqrt(3) + 1) - 3**(1/4)*sqrt((x**2 + x + 1)/(-x - sqrt(3) +
1)**2)*sqrt(sqrt(3) + 2)*(-x + 1)*elliptic_e(asin((-x + 1 + sqrt(3))/(-x - sqrt
(3) + 1)), -7 + 4*sqrt(3))/(2*sqrt((x - 1)/(-x - sqrt(3) + 1)**2)*sqrt(x**3 - 1)
) + sqrt(2)*3**(3/4)*sqrt((x**2 + x + 1)/(-x - sqrt(3) + 1)**2)*(-x + 1)*ellipti
c_f(asin((-x + 1 + sqrt(3))/(-x - sqrt(3) + 1)), -7 + 4*sqrt(3))/(3*sqrt((x - 1)
/(-x - sqrt(3) + 1)**2)*sqrt(x**3 - 1)) + sqrt(x**3 - 1)/x

_______________________________________________________________________________________

Mathematica [C]  time = 0.242693, size = 130, normalized size = 0.48 \[ \frac{\sqrt{x^3-1}}{x}+\frac{(-1)^{2/3} \sqrt{(-1)^{5/6} (x-1)} \sqrt{x^2+x+1} \left ((-1)^{5/6} F\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )+\sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{\sqrt [4]{3} \sqrt{x^3-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.028, size = 185, normalized size = 0.7 \[{\frac{1}{x}\sqrt{{x}^{3}-1}}-{(-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3})\sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }} \left ( \left ({\frac{3}{2}}-{\frac{i}{2}}\sqrt{3} \right ){\it EllipticE} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ) + \left ( -{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ){\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ) \right ){\frac{1}{\sqrt{{x}^{3}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(x^3-1)^(1/2),x)

[Out]

(x^3-1)^(1/2)/x-(-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)*((3/2-1/2*I*3^(1/2))*EllipticE(((-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))+(-1/2+1/2*I*3^(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)))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^3 - 1)*x^2), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(x^3 - 1)*x^2), x)

_______________________________________________________________________________________

Sympy [A]  time = 2.18406, size = 29, normalized size = 0.11 \[ - \frac{i \Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{1}{2} \\ \frac{2}{3} \end{matrix}\middle |{x^{3}} \right )}}{3 x \Gamma \left (\frac{2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^3 - 1)*x^2), x)