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

Optimal. Leaf size=257 \[ \frac{\sqrt{-x^3-1}}{x}-\frac{\sqrt{-x^3-1}}{x-\sqrt{3}+1}-\frac{\sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}+\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2 \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

[Out]

Sqrt[-1 - x^3]/x - Sqrt[-1 - x^3]/(1 - Sqrt[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])

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Rubi [A]  time = 0.162286, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{\sqrt{-x^3-1}}{x}-\frac{\sqrt{-x^3-1}}{x-\sqrt{3}+1}-\frac{\sqrt{2} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}}+\frac{\sqrt [4]{3} \sqrt{2+\sqrt{3}} (x+1) \sqrt{\frac{x^2-x+1}{\left (x-\sqrt{3}+1\right )^2}} E\left (\sin ^{-1}\left (\frac{x+\sqrt{3}+1}{x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2 \sqrt{-\frac{x+1}{\left (x-\sqrt{3}+1\right )^2}} \sqrt{-x^3-1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[-1 - x^3]),x]

[Out]

Sqrt[-1 - x^3]/x - Sqrt[-1 - x^3]/(1 - Sqrt[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])

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Rubi in Sympy [A]  time = 11.8695, size = 218, normalized size = 0.85 \[ - \frac{\sqrt{- x^{3} - 1}}{x - \sqrt{3} + 1} + \frac{\sqrt [4]{3} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (x + 1\right ) E\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{2 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1}} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt{\frac{x^{2} - x + 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \left (x + 1\right ) F\left (\operatorname{asin}{\left (\frac{x + 1 + \sqrt{3}}{x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{3 \sqrt{\frac{- x - 1}{\left (x - \sqrt{3} + 1\right )^{2}}} \sqrt{- x^{3} - 1}} + \frac{\sqrt{- x^{3} - 1}}{x} \]

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)*elliptic_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

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Mathematica [C]  time = 0.382241, size = 156, normalized size = 0.61 \[ \frac{-\frac{3 \left (x^3+1\right )}{x}+(-1)^{5/6} 3^{3/4} \sqrt{-(-1)^{5/6}+i x} \sqrt{-\sqrt [3]{-1} x^2-(-1)^{2/3} x+1} \left (\sqrt [3]{-1} F\left (\sin ^{-1}\left (\frac{\sqrt{-\sqrt [6]{-1} \left (x+(-1)^{2/3}\right )}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )-i \sqrt{3} E\left (\sin ^{-1}\left (\frac{\sqrt{-\sqrt [6]{-1} \left (x+(-1)^{2/3}\right )}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )\right )}{3 \sqrt{-x^3-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

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

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Maple [A]  time = 0.033, size = 174, normalized size = 0.7 \[{\frac{1}{x}\sqrt{-{x}^{3}-1}}-{{\frac{i}{3}}\sqrt{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}\sqrt{{\frac{1+x}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}}\sqrt{-i \left ( x-{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}} \left ( \left ({\frac{3}{2}}+{\frac{i}{2}}\sqrt{3} \right ){\it EllipticE} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ) -{\it EllipticF} \left ({\frac{\sqrt{3}}{3}\sqrt{i \left ( x-{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) \sqrt{3}}},\sqrt{{\frac{i\sqrt{3}}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}}} \right ) \right ){\frac{1}{\sqrt{-{x}^{3}-1}}}} \]

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-1/3*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)
*((3/2+1/2*I*3^(1/2))*EllipticE(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))-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. \[ \int \frac{1}{\sqrt{-x^{3} - 1} x^{2}}\,{d x} \]

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)

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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)

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Sympy [A]  time = 1.99073, size = 34, normalized size = 0.13 \[ - \frac{i \Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{1}{2} \\ \frac{2}{3} \end{matrix}\middle |{x^{3} e^{i \pi }} \right )}}{3 x \Gamma \left (\frac{2}{3}\right )} \]

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*exp_polar(I*pi))/(3*x*gamma(2/3))

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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)