3.519 \(\int \frac{\sqrt{-1+\frac{1}{x^2}}}{x \left (-1+x^2\right )^3} \, dx\)

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]

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

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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]

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

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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]

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

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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]

(Sqrt[-1 + x^(-2)]*(3 - 12*x^2 + 8*x^4))/(3*(-1 + x^2)^2)

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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]

1/3*(8*x^4-12*x^2+3)*(-(x^2-1)/x^2)^(1/2)/(x^2-1)^2

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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]

1/3*(8*x^4 - 12*x^2 + 3)*sqrt(x + 1)*sqrt(-x + 1)/(x^5 - 2*x^3 + x)

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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]

1/3*(8*x^4 - 12*x^2 + 3)*sqrt(-(x^2 - 1)/x^2)/(x^4 - 2*x^2 + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (-1 + \frac{1}{x}\right ) \left (1 + \frac{1}{x}\right )}}{x \left (x - 1\right )^{3} \left (x + 1\right )^{3}}\, dx \]

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]

Integral(sqrt((-1 + 1/x)*(1 + 1/x))/(x*(x - 1)**3*(x + 1)**3), x)

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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]

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