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

Optimal. Leaf size=21 \[ \frac{x}{2 \left (1-x^2\right )}+\frac{1}{2} \tanh ^{-1}(x) \]

[Out]

x/(2*(1 - x^2)) + ArcTanh[x]/2

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Rubi [A]  time = 0.0106945, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{x}{2 \left (1-x^2\right )}+\frac{1}{2} \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(-1 + x^2)^(-2),x]

[Out]

x/(2*(1 - x^2)) + ArcTanh[x]/2

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Rubi in Sympy [A]  time = 1.18394, size = 12, normalized size = 0.57 \[ \frac{x}{2 \left (- x^{2} + 1\right )} + \frac{\operatorname{atanh}{\left (x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**2-1)**2,x)

[Out]

x/(2*(-x**2 + 1)) + atanh(x)/2

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Mathematica [A]  time = 0.0122346, size = 27, normalized size = 1.29 \[ \frac{1}{4} \left (-\frac{2 x}{x^2-1}-\log (1-x)+\log (x+1)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(-1 + x^2)^(-2),x]

[Out]

((-2*x)/(-1 + x^2) - Log[1 - x] + Log[1 + x])/4

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Maple [A]  time = 0.013, size = 28, normalized size = 1.3 \[ -{\frac{1}{-4+4\,x}}-{\frac{\ln \left ( -1+x \right ) }{4}}-{\frac{1}{4+4\,x}}+{\frac{\ln \left ( 1+x \right ) }{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2-1)^2,x)

[Out]

-1/4/(-1+x)-1/4*ln(-1+x)-1/4/(1+x)+1/4*ln(1+x)

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Maxima [A]  time = 0.804754, size = 31, normalized size = 1.48 \[ -\frac{x}{2 \,{\left (x^{2} - 1\right )}} + \frac{1}{4} \, \log \left (x + 1\right ) - \frac{1}{4} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*x/(x^2 - 1) + 1/4*log(x + 1) - 1/4*log(x - 1)

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Fricas [A]  time = 0.265184, size = 46, normalized size = 2.19 \[ \frac{{\left (x^{2} - 1\right )} \log \left (x + 1\right ) -{\left (x^{2} - 1\right )} \log \left (x - 1\right ) - 2 \, x}{4 \,{\left (x^{2} - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*((x^2 - 1)*log(x + 1) - (x^2 - 1)*log(x - 1) - 2*x)/(x^2 - 1)

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Sympy [A]  time = 0.218957, size = 20, normalized size = 0.95 \[ - \frac{x}{2 x^{2} - 2} - \frac{\log{\left (x - 1 \right )}}{4} + \frac{\log{\left (x + 1 \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**2-1)**2,x)

[Out]

-x/(2*x**2 - 2) - log(x - 1)/4 + log(x + 1)/4

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GIAC/XCAS [A]  time = 0.259816, size = 34, normalized size = 1.62 \[ -\frac{x}{2 \,{\left (x^{2} - 1\right )}} + \frac{1}{4} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{4} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*x/(x^2 - 1) + 1/4*ln(abs(x + 1)) - 1/4*ln(abs(x - 1))