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

Optimal. Leaf size=67 \[ \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{3 \left (2+\sqrt{3}\right )}} \]

[Out]

ArcTanh[x/Sqrt[2 - Sqrt[3]]]/(2*Sqrt[3*(2 - Sqrt[3])]) - ArcTanh[x/Sqrt[2 + Sqrt
[3]]]/(2*Sqrt[3*(2 + Sqrt[3])])

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Rubi [A]  time = 0.102278, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{3 \left (2+\sqrt{3}\right )}} \]

Antiderivative was successfully verified.

[In]  Int[(1 - 4*x^2 + x^4)^(-1),x]

[Out]

ArcTanh[x/Sqrt[2 - Sqrt[3]]]/(2*Sqrt[3*(2 - Sqrt[3])]) - ArcTanh[x/Sqrt[2 + Sqrt
[3]]]/(2*Sqrt[3*(2 + Sqrt[3])])

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Rubi in Sympy [A]  time = 1.47789, size = 60, normalized size = 0.9 \[ \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{x}{\sqrt{- \sqrt{3} + 2}} \right )}}{6 \sqrt{- \sqrt{3} + 2}} - \frac{\sqrt{3} \operatorname{atanh}{\left (\frac{x}{\sqrt{\sqrt{3} + 2}} \right )}}{6 \sqrt{\sqrt{3} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.049046, size = 67, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2-\sqrt{3}}}\right )}{2 \sqrt{3 \left (2-\sqrt{3}\right )}}-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{2+\sqrt{3}}}\right )}{2 \sqrt{3 \left (2+\sqrt{3}\right )}} \]

Antiderivative was successfully verified.

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

[Out]

ArcTanh[x/Sqrt[2 - Sqrt[3]]]/(2*Sqrt[3*(2 - Sqrt[3])]) - ArcTanh[x/Sqrt[2 + Sqrt
[3]]]/(2*Sqrt[3*(2 + Sqrt[3])])

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Maple [A]  time = 0.022, size = 60, normalized size = 0.9 \[{\frac{\sqrt{3}}{3\,\sqrt{6}-3\,\sqrt{2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{6}-\sqrt{2}}} \right ) }-{\frac{\sqrt{3}}{3\,\sqrt{6}+3\,\sqrt{2}}{\it Artanh} \left ( 2\,{\frac{x}{\sqrt{6}+\sqrt{2}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^4-4*x^2+1),x)

[Out]

1/3*3^(1/2)/(6^(1/2)-2^(1/2))*arctanh(2*x/(6^(1/2)-2^(1/2)))-1/3*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. \[ \int \frac{1}{x^{4} - 4 \, x^{2} + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(x^4 - 4*x^2 + 1), x)

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Fricas [A]  time = 0.212098, size = 225, normalized size = 3.36 \[ -\frac{1}{12} \, \sqrt{\sqrt{3}{\left (2 \, \sqrt{3} - 3\right )}} \log \left (\sqrt{3} x + \sqrt{\sqrt{3}{\left (2 \, \sqrt{3} - 3\right )}}{\left (\sqrt{3} + 2\right )}\right ) + \frac{1}{12} \, \sqrt{\sqrt{3}{\left (2 \, \sqrt{3} - 3\right )}} \log \left (\sqrt{3} x - \sqrt{\sqrt{3}{\left (2 \, \sqrt{3} - 3\right )}}{\left (\sqrt{3} + 2\right )}\right ) - \frac{1}{12} \, \sqrt{\sqrt{3}{\left (2 \, \sqrt{3} + 3\right )}} \log \left (\sqrt{3} x + \sqrt{\sqrt{3}{\left (2 \, \sqrt{3} + 3\right )}}{\left (\sqrt{3} - 2\right )}\right ) + \frac{1}{12} \, \sqrt{\sqrt{3}{\left (2 \, \sqrt{3} + 3\right )}} \log \left (\sqrt{3} x - \sqrt{\sqrt{3}{\left (2 \, \sqrt{3} + 3\right )}}{\left (\sqrt{3} - 2\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/12*sqrt(sqrt(3)*(2*sqrt(3) - 3))*log(sqrt(3)*x + sqrt(sqrt(3)*(2*sqrt(3) - 3)
)*(sqrt(3) + 2)) + 1/12*sqrt(sqrt(3)*(2*sqrt(3) - 3))*log(sqrt(3)*x - sqrt(sqrt(
3)*(2*sqrt(3) - 3))*(sqrt(3) + 2)) - 1/12*sqrt(sqrt(3)*(2*sqrt(3) + 3))*log(sqrt
(3)*x + sqrt(sqrt(3)*(2*sqrt(3) + 3))*(sqrt(3) - 2)) + 1/12*sqrt(sqrt(3)*(2*sqrt
(3) + 3))*log(sqrt(3)*x - sqrt(sqrt(3)*(2*sqrt(3) + 3))*(sqrt(3) - 2))

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Sympy [A]  time = 0.788284, size = 24, normalized size = 0.36 \[ \operatorname{RootSum}{\left (2304 t^{4} - 192 t^{2} + 1, \left ( t \mapsto t \log{\left (384 t^{3} - 28 t + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**4-4*x**2+1),x)

[Out]

RootSum(2304*_t**4 - 192*_t**2 + 1, Lambda(_t, _t*log(384*_t**3 - 28*_t + x)))

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GIAC/XCAS [A]  time = 0.249576, size = 136, normalized size = 2.03 \[ \frac{1}{24} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left ({\left | x + \frac{1}{2} \, \sqrt{6} + \frac{1}{2} \, \sqrt{2} \right |}\right ) + \frac{1}{24} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )}{\rm ln}\left ({\left | x + \frac{1}{2} \, \sqrt{6} - \frac{1}{2} \, \sqrt{2} \right |}\right ) - \frac{1}{24} \,{\left (\sqrt{6} + 3 \, \sqrt{2}\right )}{\rm ln}\left ({\left | x - \frac{1}{2} \, \sqrt{6} + \frac{1}{2} \, \sqrt{2} \right |}\right ) - \frac{1}{24} \,{\left (\sqrt{6} - 3 \, \sqrt{2}\right )}{\rm ln}\left ({\left | x - \frac{1}{2} \, \sqrt{6} - \frac{1}{2} \, \sqrt{2} \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/24*(sqrt(6) - 3*sqrt(2))*ln(abs(x + 1/2*sqrt(6) + 1/2*sqrt(2))) + 1/24*(sqrt(6
) + 3*sqrt(2))*ln(abs(x + 1/2*sqrt(6) - 1/2*sqrt(2))) - 1/24*(sqrt(6) + 3*sqrt(2
))*ln(abs(x - 1/2*sqrt(6) + 1/2*sqrt(2))) - 1/24*(sqrt(6) - 3*sqrt(2))*ln(abs(x
- 1/2*sqrt(6) - 1/2*sqrt(2)))