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