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

Optimal. Leaf size=74 \[ -\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{1}{2} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{1}{2} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

[Out]

-ArcTan[Sqrt[3] - 2*x]/2 + ArcTan[Sqrt[3] + 2*x]/2 - Log[1 - Sqrt[3]*x + x^2]/(4
*Sqrt[3]) + Log[1 + Sqrt[3]*x + x^2]/(4*Sqrt[3])

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Rubi [A]  time = 0.0896653, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417 \[ -\frac{\log \left (x^2-\sqrt{3} x+1\right )}{4 \sqrt{3}}+\frac{\log \left (x^2+\sqrt{3} x+1\right )}{4 \sqrt{3}}-\frac{1}{2} \tan ^{-1}\left (\sqrt{3}-2 x\right )+\frac{1}{2} \tan ^{-1}\left (2 x+\sqrt{3}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(1 - x^2 + x^4)^(-1),x]

[Out]

-ArcTan[Sqrt[3] - 2*x]/2 + ArcTan[Sqrt[3] + 2*x]/2 - Log[1 - Sqrt[3]*x + x^2]/(4
*Sqrt[3]) + Log[1 + Sqrt[3]*x + x^2]/(4*Sqrt[3])

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Rubi in Sympy [A]  time = 16.7216, size = 63, normalized size = 0.85 \[ - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} + \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{2} + \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.121835, size = 77, normalized size = 1.04 \[ \frac{i \left (\sqrt{-1-i \sqrt{3}} \tan ^{-1}\left (\frac{1}{2} \left (1-i \sqrt{3}\right ) x\right )-\sqrt{-1+i \sqrt{3}} \tan ^{-1}\left (\frac{1}{2} \left (1+i \sqrt{3}\right ) x\right )\right )}{\sqrt{6}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(I*(Sqrt[-1 - I*Sqrt[3]]*ArcTan[((1 - I*Sqrt[3])*x)/2] - Sqrt[-1 + I*Sqrt[3]]*Ar
cTan[((1 + I*Sqrt[3])*x)/2]))/Sqrt[6]

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Maple [A]  time = 0.035, size = 57, normalized size = 0.8 \[{\frac{\arctan \left ( 2\,x-\sqrt{3} \right ) }{2}}+{\frac{\arctan \left ( 2\,x+\sqrt{3} \right ) }{2}}-{\frac{\ln \left ( 1+{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{12}}+{\frac{\ln \left ( 1+{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^4-x^2+1),x)

[Out]

1/2*arctan(2*x-3^(1/2))+1/2*arctan(2*x+3^(1/2))-1/12*ln(1+x^2-x*3^(1/2))*3^(1/2)
+1/12*ln(1+x^2+x*3^(1/2))*3^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(x^4 - x^2 + 1), x)

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Fricas [A]  time = 0.267376, size = 143, normalized size = 1.93 \[ -\frac{1}{12} \, \sqrt{3}{\left (4 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}}{2 \, \sqrt{3} x + 2 \, \sqrt{3} \sqrt{x^{2} + \sqrt{3} x + 1} + 3}\right ) + 4 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}}{2 \, \sqrt{3} x + 2 \, \sqrt{3} \sqrt{x^{2} - \sqrt{3} x + 1} - 3}\right ) - \log \left (x^{2} + \sqrt{3} x + 1\right ) + \log \left (x^{2} - \sqrt{3} x + 1\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.526391, size = 63, normalized size = 0.85 \[ - \frac{\sqrt{3} \log{\left (x^{2} - \sqrt{3} x + 1 \right )}}{12} + \frac{\sqrt{3} \log{\left (x^{2} + \sqrt{3} x + 1 \right )}}{12} + \frac{\operatorname{atan}{\left (2 x - \sqrt{3} \right )}}{2} + \frac{\operatorname{atan}{\left (2 x + \sqrt{3} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**4-x**2+1),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/(x^4 - x^2 + 1), x)