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

Optimal. Leaf size=44 \[ \frac{\tanh ^{-1}\left (\sqrt{5}-2 \sqrt{2} x\right )}{\sqrt{2}}-\frac{\tanh ^{-1}\left (2 \sqrt{2} x+\sqrt{5}\right )}{\sqrt{2}} \]

[Out]

ArcTanh[Sqrt[5] - 2*Sqrt[2]*x]/Sqrt[2] - ArcTanh[Sqrt[5] + 2*Sqrt[2]*x]/Sqrt[2]

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Rubi [A]  time = 0.0719869, antiderivative size = 44, 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 \[ \frac{\tanh ^{-1}\left (\sqrt{5}-2 \sqrt{2} x\right )}{\sqrt{2}}-\frac{\tanh ^{-1}\left (2 \sqrt{2} x+\sqrt{5}\right )}{\sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTanh[Sqrt[5] - 2*Sqrt[2]*x]/Sqrt[2] - ArcTanh[Sqrt[5] + 2*Sqrt[2]*x]/Sqrt[2]

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Rubi in Sympy [A]  time = 8.8038, size = 48, normalized size = 1.09 \[ - \frac{\sqrt{2} \operatorname{atanh}{\left (\sqrt{2} \left (2 x - \frac{\sqrt{10}}{2}\right ) \right )}}{2} - \frac{\sqrt{2} \operatorname{atanh}{\left (\sqrt{2} \left (2 x + \frac{\sqrt{10}}{2}\right ) \right )}}{2} \]

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

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Mathematica [A]  time = 0.020653, size = 42, normalized size = 0.95 \[ \frac{\log \left (-2 x^2+\sqrt{2} x+1\right )-\log \left (2 x^2+\sqrt{2} x-1\right )}{2 \sqrt{2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.048, size = 82, normalized size = 1.9 \[ -{\frac{ \left ( 10+2\,\sqrt{5} \right ) \sqrt{5}}{10\,\sqrt{10}+10\,\sqrt{2}}{\it Artanh} \left ( 8\,{\frac{x}{2\,\sqrt{10}+2\,\sqrt{2}}} \right ) }-{\frac{2\,\sqrt{5} \left ( -5+\sqrt{5} \right ) }{10\,\sqrt{10}-10\,\sqrt{2}}{\it Artanh} \left ( 8\,{\frac{x}{2\,\sqrt{10}-2\,\sqrt{2}}} \right ) } \]

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+5^(1/2))*5^(1/2)/(2*10^(1/2)+2*2^(1/2))*arctanh(8*x/(2*10^(1/2)+2*2^(1/2
)))-2/5*5^(1/2)*(-5+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 = 0.27736, size = 65, normalized size = 1.48 \[ \frac{1}{4} \, \sqrt{2} \log \left (-\frac{8 \, x^{3} - \sqrt{2}{\left (4 \, x^{4} - 2 \, x^{2} + 1\right )} - 4 \, x}{4 \, x^{4} - 6 \, x^{2} + 1}\right ) \]

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]

1/4*sqrt(2)*log(-(8*x^3 - sqrt(2)*(4*x^4 - 2*x^2 + 1) - 4*x)/(4*x^4 - 6*x^2 + 1)
)

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Sympy [A]  time = 0.207952, size = 46, normalized size = 1.05 \[ \frac{\sqrt{2} \log{\left (x^{2} - \frac{\sqrt{2} x}{2} - \frac{1}{2} \right )}}{4} - \frac{\sqrt{2} \log{\left (x^{2} + \frac{\sqrt{2} x}{2} - \frac{1}{2} \right )}}{4} \]

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

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GIAC/XCAS [A]  time = 0.308264, size = 104, normalized size = 2.36 \[ -\frac{1}{4} \, \sqrt{2}{\rm ln}\left ({\left | x + \frac{1}{4} \, \sqrt{10} + \frac{1}{4} \, \sqrt{2} \right |}\right ) + \frac{1}{4} \, \sqrt{2}{\rm ln}\left ({\left | x + \frac{1}{4} \, \sqrt{10} - \frac{1}{4} \, \sqrt{2} \right |}\right ) - \frac{1}{4} \, \sqrt{2}{\rm ln}\left ({\left | x - \frac{1}{4} \, \sqrt{10} + \frac{1}{4} \, \sqrt{2} \right |}\right ) + \frac{1}{4} \, \sqrt{2}{\rm ln}\left ({\left | x - \frac{1}{4} \, \sqrt{10} - \frac{1}{4} \, \sqrt{2} \right |}\right ) \]

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]

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