3.96 \(\int \frac{1-x^2}{1-2 x^2+x^4} \, dx\)

Optimal. Leaf size=2 \[ \tanh ^{-1}(x) \]

[Out]

ArcTanh[x]

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Rubi [A]  time = 0.00582081, antiderivative size = 2, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \tanh ^{-1}(x) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^2)/(1 - 2*x^2 + x^4),x]

[Out]

ArcTanh[x]

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Rubi in Sympy [A]  time = 3.88464, size = 2, normalized size = 1. \[ \operatorname{atanh}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-x**2+1)/(x**4-2*x**2+1),x)

[Out]

atanh(x)

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Mathematica [B]  time = 0.00318543, size = 19, normalized size = 9.5 \[ \frac{1}{2} \log (x+1)-\frac{1}{2} \log (1-x) \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - x^2)/(1 - 2*x^2 + x^4),x]

[Out]

-Log[1 - x]/2 + Log[1 + x]/2

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Maple [A]  time = 0.001, size = 3, normalized size = 1.5 \[{\it Artanh} \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+1)/(x^4-2*x^2+1),x)

[Out]

arctanh(x)

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Maxima [A]  time = 0.735299, size = 18, normalized size = 9. \[ \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 1)/(x^4 - 2*x^2 + 1),x, algorithm="maxima")

[Out]

1/2*log(x + 1) - 1/2*log(x - 1)

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Fricas [A]  time = 0.280422, size = 18, normalized size = 9. \[ \frac{1}{2} \, \log \left (x + 1\right ) - \frac{1}{2} \, \log \left (x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 1)/(x^4 - 2*x^2 + 1),x, algorithm="fricas")

[Out]

1/2*log(x + 1) - 1/2*log(x - 1)

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Sympy [A]  time = 0.166231, size = 12, normalized size = 6. \[ - \frac{\log{\left (x - 1 \right )}}{2} + \frac{\log{\left (x + 1 \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x**2+1)/(x**4-2*x**2+1),x)

[Out]

-log(x - 1)/2 + log(x + 1)/2

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GIAC/XCAS [A]  time = 0.269551, size = 20, normalized size = 10. \[ \frac{1}{2} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(x^2 - 1)/(x^4 - 2*x^2 + 1),x, algorithm="giac")

[Out]

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