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

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

[Out]

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

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Rubi [A]  time = 0.0672633, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\sqrt{x^3-1}}{2 x^2}-\frac{\sqrt{2-\sqrt{3}} (1-x) \sqrt{\frac{x^2+x+1}{\left (-x-\sqrt{3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac{-x+\sqrt{3}+1}{-x-\sqrt{3}+1}\right )|-7+4 \sqrt{3}\right )}{2 \sqrt [4]{3} \sqrt{-\frac{1-x}{\left (-x-\sqrt{3}+1\right )^2}} \sqrt{x^3-1}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*Sqrt[-1 + x^3]),x]

[Out]

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

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Rubi in Sympy [A]  time = 4.08005, size = 105, normalized size = 0.76 \[ - \frac{3^{\frac{3}{4}} \sqrt{\frac{x^{2} + x + 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (- x + 1\right ) F\left (\operatorname{asin}{\left (\frac{- x + 1 + \sqrt{3}}{- x - \sqrt{3} + 1} \right )}\middle | -7 + 4 \sqrt{3}\right )}{6 \sqrt{\frac{x - 1}{\left (- x - \sqrt{3} + 1\right )^{2}}} \sqrt{x^{3} - 1}} + \frac{\sqrt{x^{3} - 1}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-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))/(6*sqrt
((x - 1)/(-x - sqrt(3) + 1)**2)*sqrt(x**3 - 1)) + sqrt(x**3 - 1)/(2*x**2)

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Mathematica [C]  time = 0.0582846, size = 90, normalized size = 0.65 \[ \frac{\sqrt{x^3-1}}{2 x^2}+\frac{i \sqrt{(-1)^{5/6} (x-1)} \sqrt{x^2+x+1} F\left (\sin ^{-1}\left (\frac{\sqrt{-i x-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{2 \sqrt [4]{3} \sqrt{x^3-1}} \]

Warning: Unable to verify antiderivative.

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

[Out]

Sqrt[-1 + x^3]/(2*x^2) + ((I/2)*Sqrt[(-1)^(5/6)*(-1 + x)]*Sqrt[1 + x + x^2]*Elli
pticF[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 = 0.027, size = 129, normalized size = 0.9 \[{\frac{1}{2\,{x}^{2}}\sqrt{{x}^{3}-1}}+{\frac{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}{2}\sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}}\sqrt{{\frac{1}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}-{\frac{i}{2}}\sqrt{3} \right ) }}\sqrt{{\frac{1}{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}} \left ( x+{\frac{1}{2}}+{\frac{i}{2}}\sqrt{3} \right ) }}{\it EllipticF} \left ( \sqrt{{\frac{-1+x}{-{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}},\sqrt{{\frac{{\frac{3}{2}}+{\frac{i}{2}}\sqrt{3}}{{\frac{3}{2}}-{\frac{i}{2}}\sqrt{3}}}} \right ){\frac{1}{\sqrt{{x}^{3}-1}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(x^3-1)^(1/2),x)

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^3 - 1)*x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(1/(sqrt(x^3 - 1)*x^3), x)

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Sympy [A]  time = 2.18907, size = 31, normalized size = 0.22 \[ - \frac{i \Gamma \left (- \frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{1}{2} \\ \frac{1}{3} \end{matrix}\middle |{x^{3}} \right )}}{3 x^{2} \Gamma \left (\frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(sqrt(x^3 - 1)*x^3), x)