3.518 \(\int \frac{\sqrt{-1+\frac{1}{x^2}}}{x \left (-1+x^2\right )^2} \, dx\)

Optimal. Leaf size=21 \[ \frac{1}{\sqrt{\frac{1}{x^2}-1}}-\sqrt{\frac{1}{x^2}-1} \]

[Out]

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

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Rubi [A]  time = 0.0303385, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{1}{\sqrt{\frac{1}{x^2}-1}}-\sqrt{\frac{1}{x^2}-1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 2.24651, size = 20, normalized size = 0.95 \[ - \sqrt{-1 + \frac{1}{x^{2}}} + \frac{1}{\sqrt{-1 + \frac{1}{x^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0142098, size = 24, normalized size = 1.14 \[ \frac{\sqrt{\frac{1}{x^2}-1} \left (1-2 x^2\right )}{x^2-1} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 29, normalized size = 1.4 \[ -{\frac{2\,{x}^{2}-1}{{x}^{2}-1}\sqrt{-{\frac{{x}^{2}-1}{{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*x^2-1)*(-(x^2-1)/x^2)^(1/2)/(x^2-1)

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Maxima [A]  time = 0.73913, size = 41, normalized size = 1.95 \[ -\frac{{\left (2 \, x^{2} - 1\right )} \sqrt{x + 1} \sqrt{-x + 1}}{x^{3} - x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.263873, size = 38, normalized size = 1.81 \[ -\frac{{\left (2 \, x^{2} - 1\right )} \sqrt{-\frac{x^{2} - 1}{x^{2}}}}{x^{2} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(2*x^2 - 1)*sqrt(-(x^2 - 1)/x^2)/(x^2 - 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (-1 + \frac{1}{x}\right ) \left (1 + \frac{1}{x}\right )}}{x \left (x - 1\right )^{2} \left (x + 1\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt((-1 + 1/x)*(1 + 1/x))/(x*(x - 1)**2*(x + 1)**2), x)

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GIAC/XCAS [A]  time = 0.267366, size = 78, normalized size = 3.71 \[ -\frac{\sqrt{-x^{2} + 1} x{\rm sign}\left (x\right )}{x^{2} - 1} + \frac{x{\rm sign}\left (x\right )}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}} - \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}{\rm sign}\left (x\right )}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(-x^2 + 1)*x*sign(x)/(x^2 - 1) + 1/2*x*sign(x)/(sqrt(-x^2 + 1) - 1) - 1/2*(
sqrt(-x^2 + 1) - 1)*sign(x)/x