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

Optimal. Leaf size=14 \[ \frac{\tanh ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

[Out]

ArcTanh[Sqrt[2]*x]/Sqrt[2]

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Rubi [A]  time = 0.0143112, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\tanh ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTanh[Sqrt[2]*x]/Sqrt[2]

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Rubi in Sympy [A]  time = 4.57746, size = 14, normalized size = 1. \[ \frac{\sqrt{2} \operatorname{atanh}{\left (\sqrt{2} x \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*atanh(sqrt(2)*x)/2

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Mathematica [B]  time = 0.0112848, size = 32, normalized size = 2.29 \[ \frac{\log \left (2 x+\sqrt{2}\right )-\log \left (\sqrt{2}-2 x\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

(-Log[Sqrt[2] - 2*x] + Log[Sqrt[2] + 2*x])/(2*Sqrt[2])

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Maple [A]  time = 0.003, size = 12, normalized size = 0.9 \[{\frac{{\it Artanh} \left ( \sqrt{2}x \right ) \sqrt{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*arctanh(2^(1/2)*x)*2^(1/2)

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Maxima [A]  time = 0.826188, size = 35, normalized size = 2.5 \[ -\frac{1}{4} \, \sqrt{2} \log \left (\frac{2 \,{\left (2 \, x - \sqrt{2}\right )}}{4 \, x + 2 \, \sqrt{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.280236, size = 42, normalized size = 3. \[ \frac{1}{4} \, \sqrt{2} \log \left (\frac{\sqrt{2}{\left (2 \, x^{2} + 1\right )} + 4 \, x}{2 \, x^{2} - 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.16878, size = 32, normalized size = 2.29 \[ - \frac{\sqrt{2} \log{\left (x - \frac{\sqrt{2}}{2} \right )}}{4} + \frac{\sqrt{2} \log{\left (x + \frac{\sqrt{2}}{2} \right )}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.27209, size = 39, normalized size = 2.79 \[ \frac{1}{4} \, \sqrt{2}{\rm ln}\left ({\left | x + \frac{1}{2} \, \sqrt{2} \right |}\right ) - \frac{1}{4} \, \sqrt{2}{\rm ln}\left ({\left | x - \frac{1}{2} \, \sqrt{2} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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