3.75 \(\int \frac{1-2 x^2}{1-6 x^2+4 x^4} \, dx\)

Optimal. Leaf size=48 \[ \frac{\tanh ^{-1}\left (\frac{2 \sqrt{2} x+1}{\sqrt{5}}\right )}{\sqrt{10}}-\frac{\tanh ^{-1}\left (\frac{1-2 \sqrt{2} x}{\sqrt{5}}\right )}{\sqrt{10}} \]

[Out]

-(ArcTanh[(1 - 2*Sqrt[2]*x)/Sqrt[5]]/Sqrt[10]) + ArcTanh[(1 + 2*Sqrt[2]*x)/Sqrt[
5]]/Sqrt[10]

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Rubi [A]  time = 0.086317, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\tanh ^{-1}\left (\frac{2 \sqrt{2} x+1}{\sqrt{5}}\right )}{\sqrt{10}}-\frac{\tanh ^{-1}\left (\frac{1-2 \sqrt{2} x}{\sqrt{5}}\right )}{\sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 2*x^2)/(1 - 6*x^2 + 4*x^4),x]

[Out]

-(ArcTanh[(1 - 2*Sqrt[2]*x)/Sqrt[5]]/Sqrt[10]) + ArcTanh[(1 + 2*Sqrt[2]*x)/Sqrt[
5]]/Sqrt[10]

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Rubi in Sympy [A]  time = 8.77798, size = 53, normalized size = 1.1 \[ - \frac{\sqrt{10} \operatorname{atanh}{\left (\sqrt{10} \left (- \frac{2 x}{5} - \frac{\sqrt{2}}{10}\right ) \right )}}{10} - \frac{\sqrt{10} \operatorname{atanh}{\left (\sqrt{10} \left (- \frac{2 x}{5} + \frac{\sqrt{2}}{10}\right ) \right )}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-2*x**2+1)/(4*x**4-6*x**2+1),x)

[Out]

-sqrt(10)*atanh(sqrt(10)*(-2*x/5 - sqrt(2)/10))/10 - sqrt(10)*atanh(sqrt(10)*(-2
*x/5 + sqrt(2)/10))/10

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Mathematica [A]  time = 0.0306739, size = 42, normalized size = 0.88 \[ \frac{\log \left (2 x^2+\sqrt{10} x+1\right )-\log \left (-2 x^2+\sqrt{10} x-1\right )}{2 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]  Integrate[(1 - 2*x^2)/(1 - 6*x^2 + 4*x^4),x]

[Out]

(-Log[-1 + Sqrt[10]*x - 2*x^2] + Log[1 + Sqrt[10]*x + 2*x^2])/(2*Sqrt[10])

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Maple [B]  time = 0.018, size = 82, normalized size = 1.7 \[{\frac{2\,\sqrt{5} \left ( \sqrt{5}+1 \right ) }{10\,\sqrt{10}+10\,\sqrt{2}}{\it Artanh} \left ( 8\,{\frac{x}{2\,\sqrt{10}+2\,\sqrt{2}}} \right ) }+{\frac{ \left ( -2+2\,\sqrt{5} \right ) \sqrt{5}}{10\,\sqrt{10}-10\,\sqrt{2}}{\it Artanh} \left ( 8\,{\frac{x}{2\,\sqrt{10}-2\,\sqrt{2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-2*x^2+1)/(4*x^4-6*x^2+1),x)

[Out]

2/5*5^(1/2)*(5^(1/2)+1)/(2*10^(1/2)+2*2^(1/2))*arctanh(8*x/(2*10^(1/2)+2*2^(1/2)
))+2/5*(5^(1/2)-1)*5^(1/2)/(2*10^(1/2)-2*2^(1/2))*arctanh(8*x/(2*10^(1/2)-2*2^(1
/2)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{2 \, x^{2} - 1}{4 \, x^{4} - 6 \, x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x^2 - 1)/(4*x^4 - 6*x^2 + 1),x, algorithm="maxima")

[Out]

-integrate((2*x^2 - 1)/(4*x^4 - 6*x^2 + 1), x)

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Fricas [A]  time = 0.275535, size = 62, normalized size = 1.29 \[ \frac{1}{20} \, \sqrt{10} \log \left (\frac{40 \, x^{3} + \sqrt{10}{\left (4 \, x^{4} + 14 \, x^{2} + 1\right )} + 20 \, x}{4 \, x^{4} - 6 \, x^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x^2 - 1)/(4*x^4 - 6*x^2 + 1),x, algorithm="fricas")

[Out]

1/20*sqrt(10)*log((40*x^3 + sqrt(10)*(4*x^4 + 14*x^2 + 1) + 20*x)/(4*x^4 - 6*x^2
+ 1))

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Sympy [A]  time = 0.207049, size = 46, normalized size = 0.96 \[ - \frac{\sqrt{10} \log{\left (x^{2} - \frac{\sqrt{10} x}{2} + \frac{1}{2} \right )}}{20} + \frac{\sqrt{10} \log{\left (x^{2} + \frac{\sqrt{10} x}{2} + \frac{1}{2} \right )}}{20} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-2*x**2+1)/(4*x**4-6*x**2+1),x)

[Out]

-sqrt(10)*log(x**2 - sqrt(10)*x/2 + 1/2)/20 + sqrt(10)*log(x**2 + sqrt(10)*x/2 +
1/2)/20

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GIAC/XCAS [A]  time = 0.308836, size = 104, normalized size = 2.17 \[ \frac{1}{20} \, \sqrt{10}{\rm ln}\left ({\left | x + \frac{1}{4} \, \sqrt{10} + \frac{1}{4} \, \sqrt{2} \right |}\right ) + \frac{1}{20} \, \sqrt{10}{\rm ln}\left ({\left | x + \frac{1}{4} \, \sqrt{10} - \frac{1}{4} \, \sqrt{2} \right |}\right ) - \frac{1}{20} \, \sqrt{10}{\rm ln}\left ({\left | x - \frac{1}{4} \, \sqrt{10} + \frac{1}{4} \, \sqrt{2} \right |}\right ) - \frac{1}{20} \, \sqrt{10}{\rm ln}\left ({\left | x - \frac{1}{4} \, \sqrt{10} - \frac{1}{4} \, \sqrt{2} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(2*x^2 - 1)/(4*x^4 - 6*x^2 + 1),x, algorithm="giac")

[Out]

1/20*sqrt(10)*ln(abs(x + 1/4*sqrt(10) + 1/4*sqrt(2))) + 1/20*sqrt(10)*ln(abs(x +
1/4*sqrt(10) - 1/4*sqrt(2))) - 1/20*sqrt(10)*ln(abs(x - 1/4*sqrt(10) + 1/4*sqrt
(2))) - 1/20*sqrt(10)*ln(abs(x - 1/4*sqrt(10) - 1/4*sqrt(2)))