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

Optimal. Leaf size=47 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} x+1}{\sqrt{3}}\right )}{\sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{1-\sqrt{2} x}{\sqrt{3}}\right )}{\sqrt{6}} \]

[Out]

-(ArcTanh[(1 - Sqrt[2]*x)/Sqrt[3]]/Sqrt[6]) + ArcTanh[(1 + Sqrt[2]*x)/Sqrt[3]]/S
qrt[6]

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Rubi [A]  time = 0.0785488, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{2} x+1}{\sqrt{3}}\right )}{\sqrt{6}}-\frac{\tanh ^{-1}\left (\frac{1-\sqrt{2} x}{\sqrt{3}}\right )}{\sqrt{6}} \]

Antiderivative was successfully verified.

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

[Out]

-(ArcTanh[(1 - Sqrt[2]*x)/Sqrt[3]]/Sqrt[6]) + ArcTanh[(1 + Sqrt[2]*x)/Sqrt[3]]/S
qrt[6]

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Rubi in Sympy [A]  time = 8.3762, size = 49, normalized size = 1.04 \[ - \frac{\sqrt{6} \operatorname{atanh}{\left (\sqrt{6} \left (- \frac{x}{3} - \frac{\sqrt{2}}{6}\right ) \right )}}{6} - \frac{\sqrt{6} \operatorname{atanh}{\left (\sqrt{6} \left (- \frac{x}{3} + \frac{\sqrt{2}}{6}\right ) \right )}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(6)*atanh(sqrt(6)*(-x/3 - sqrt(2)/6))/6 - sqrt(6)*atanh(sqrt(6)*(-x/3 + sqr
t(2)/6))/6

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Mathematica [A]  time = 0.0305654, size = 40, normalized size = 0.85 \[ \frac{\log \left (x^2+\sqrt{6} x+1\right )-\log \left (-x^2+\sqrt{6} x-1\right )}{2 \sqrt{6}} \]

Antiderivative was successfully verified.

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

[Out]

(-Log[-1 + Sqrt[6]*x - x^2] + Log[1 + Sqrt[6]*x + x^2])/(2*Sqrt[6])

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Maple [A]  time = 0.016, size = 70, normalized size = 1.5 \[{\frac{ \left ( \sqrt{3}-1 \right ) \sqrt{3}}{3\,\sqrt{6}-3\,\sqrt{2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{6}-\sqrt{2}}} \right ) }+{\frac{ \left ( 1+\sqrt{3} \right ) \sqrt{3}}{3\,\sqrt{6}+3\,\sqrt{2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{6}+\sqrt{2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-x^2+1)/(x^4-4*x^2+1),x)

[Out]

1/3*(3^(1/2)-1)*3^(1/2)/(6^(1/2)-2^(1/2))*arctanh(2*x/(6^(1/2)-2^(1/2)))+1/3*(1+
3^(1/2))*3^(1/2)/(6^(1/2)+2^(1/2))*arctanh(2*x/(6^(1/2)+2^(1/2)))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.269136, size = 57, normalized size = 1.21 \[ \frac{1}{12} \, \sqrt{6} \log \left (\frac{12 \, x^{3} + \sqrt{6}{\left (x^{4} + 8 \, x^{2} + 1\right )} + 12 \, x}{x^{4} - 4 \, x^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*sqrt(6)*log((12*x^3 + sqrt(6)*(x^4 + 8*x^2 + 1) + 12*x)/(x^4 - 4*x^2 + 1))

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Sympy [A]  time = 0.217941, size = 39, normalized size = 0.83 \[ - \frac{\sqrt{6} \log{\left (x^{2} - \sqrt{6} x + 1 \right )}}{12} + \frac{\sqrt{6} \log{\left (x^{2} + \sqrt{6} x + 1 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(6)*log(x**2 - sqrt(6)*x + 1)/12 + sqrt(6)*log(x**2 + sqrt(6)*x + 1)/12

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GIAC/XCAS [A]  time = 0.277794, size = 53, normalized size = 1.13 \[ -\frac{1}{12} \, \sqrt{6}{\rm ln}\left (\frac{{\left | 2 \, x - 2 \, \sqrt{6} + \frac{2}{x} \right |}}{{\left | 2 \, x + 2 \, \sqrt{6} + \frac{2}{x} \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*sqrt(6)*ln(abs(2*x - 2*sqrt(6) + 2/x)/abs(2*x + 2*sqrt(6) + 2/x))