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