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

Optimal. Leaf size=20 \[ \frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

ArcTan[(1 + 2*x^2)/Sqrt[3]]/Sqrt[3]

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Rubi [A]  time = 0.0485539, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[x/(1 + x^2 + x^4),x]

[Out]

ArcTan[(1 + 2*x^2)/Sqrt[3]]/Sqrt[3]

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Rubi in Sympy [A]  time = 4.46774, size = 22, normalized size = 1.1 \[ \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} + \frac{1}{3}\right ) \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(3)*atan(sqrt(3)*(2*x**2/3 + 1/3))/3

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Mathematica [A]  time = 0.00907408, size = 20, normalized size = 1. \[ \frac{\tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(1 + x^2 + x^4),x]

[Out]

ArcTan[(1 + 2*x^2)/Sqrt[3]]/Sqrt[3]

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Maple [A]  time = 0.002, size = 19, normalized size = 1. \[{\frac{\sqrt{3}}{3}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(x^4+x^2+1),x)

[Out]

1/3*arctan(1/3*(2*x^2+1)*3^(1/2))*3^(1/2)

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Maxima [A]  time = 0.76745, size = 24, normalized size = 1.2 \[ \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 + 1))

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Fricas [A]  time = 0.274031, size = 24, normalized size = 1.2 \[ \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 + 1))

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Sympy [A]  time = 0.19504, size = 26, normalized size = 1.3 \[ \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{2}}{3} + \frac{\sqrt{3}}{3} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(x**4+x**2+1),x)

[Out]

sqrt(3)*atan(2*sqrt(3)*x**2/3 + sqrt(3)/3)/3

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GIAC/XCAS [A]  time = 0.285103, size = 24, normalized size = 1.2 \[ \frac{1}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x^2 + 1))