Optimal. Leaf size=34 \[ \sqrt{\frac{1}{x^2}-1}-\frac{2}{\sqrt{\frac{1}{x^2}-1}}-\frac{1}{3 \left (\frac{1}{x^2}-1\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.038637, antiderivative size = 34, 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 \[ \sqrt{\frac{1}{x^2}-1}-\frac{2}{\sqrt{\frac{1}{x^2}-1}}-\frac{1}{3 \left (\frac{1}{x^2}-1\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[-1 + x^(-2)]/(x*(-1 + x^2)^3),x]
[Out]
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Rubi in Sympy [A] time = 2.71833, size = 34, normalized size = 1. \[ \sqrt{-1 + \frac{1}{x^{2}}} - \frac{2}{\sqrt{-1 + \frac{1}{x^{2}}}} - \frac{1}{3 \left (-1 + \frac{1}{x^{2}}\right )^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-1+1/x**2)**(1/2)/x/(x**2-1)**3,x)
[Out]
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Mathematica [A] time = 0.0173236, size = 32, normalized size = 0.94 \[ \frac{\sqrt{\frac{1}{x^2}-1} \left (8 x^4-12 x^2+3\right )}{3 \left (x^2-1\right )^2} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[-1 + x^(-2)]/(x*(-1 + x^2)^3),x]
[Out]
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Maple [A] time = 0.007, size = 34, normalized size = 1. \[{\frac{8\,{x}^{4}-12\,{x}^{2}+3}{3\, \left ({x}^{2}-1 \right ) ^{2}}\sqrt{-{\frac{{x}^{2}-1}{{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-1+1/x^2)^(1/2)/x/(x^2-1)^3,x)
[Out]
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Maxima [A] time = 0.741181, size = 51, normalized size = 1.5 \[ \frac{{\left (8 \, x^{4} - 12 \, x^{2} + 3\right )} \sqrt{x + 1} \sqrt{-x + 1}}{3 \,{\left (x^{5} - 2 \, x^{3} + x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(1/x^2 - 1)/((x^2 - 1)^3*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.267539, size = 51, normalized size = 1.5 \[ \frac{{\left (8 \, x^{4} - 12 \, x^{2} + 3\right )} \sqrt{-\frac{x^{2} - 1}{x^{2}}}}{3 \,{\left (x^{4} - 2 \, x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(1/x^2 - 1)/((x^2 - 1)^3*x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (-1 + \frac{1}{x}\right ) \left (1 + \frac{1}{x}\right )}}{x \left (x - 1\right )^{3} \left (x + 1\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-1+1/x**2)**(1/2)/x/(x**2-1)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.26911, size = 92, normalized size = 2.71 \[ -\frac{x{\rm sign}\left (x\right )}{2 \,{\left (\sqrt{-x^{2} + 1} - 1\right )}} + \frac{{\left (\sqrt{-x^{2} + 1} - 1\right )}{\rm sign}\left (x\right )}{2 \, x} - \frac{{\left (5 \, x^{2}{\rm sign}\left (x\right ) - 6 \,{\rm sign}\left (x\right )\right )} x}{3 \,{\left (x^{2} - 1\right )} \sqrt{-x^{2} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(1/x^2 - 1)/((x^2 - 1)^3*x),x, algorithm="giac")
[Out]