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

Optimal. Leaf size=9 \[ \sqrt{\frac{1}{x^2}-1} \]

[Out]

Sqrt[-1 + x^(-2)]

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Rubi [A]  time = 0.0154961, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \sqrt{\frac{1}{x^2}-1} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[-1 + x^(-2)]

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Rubi in Sympy [A]  time = 1.47169, size = 8, normalized size = 0.89 \[ \sqrt{-1 + \frac{1}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-1 + x**(-2))

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Mathematica [A]  time = 0.00928303, size = 9, normalized size = 1. \[ \sqrt{\frac{1}{x^2}-1} \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[-1 + x^(-2)]

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Maple [A]  time = 0.005, size = 13, normalized size = 1.4 \[ \sqrt{-{\frac{{x}^{2}-1}{{x}^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.736675, size = 22, normalized size = 2.44 \[ \frac{\sqrt{x + 1} \sqrt{-x + 1}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x + 1)*sqrt(-x + 1)/x

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Fricas [A]  time = 0.263645, size = 16, normalized size = 1.78 \[ \sqrt{-\frac{x^{2} - 1}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-(x^2 - 1)/x^2)

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Sympy [A]  time = 4.06615, size = 8, normalized size = 0.89 \[ \sqrt{-1 + \frac{1}{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(-1 + x**(-2))

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GIAC/XCAS [A]  time = 0.266895, size = 50, normalized size = 5.56 \[ -\frac{x{\rm sign}\left (x\right )}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}} + \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}{\rm sign}\left (x\right )}{2 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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