3.689 \(\int \frac{x}{2+3 x^4} \, dx\)

Optimal. Leaf size=21 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}} \]

[Out]

ArcTan[Sqrt[3/2]*x^2]/(2*Sqrt[6])

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Rubi [A]  time = 0.0201487, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]  Int[x/(2 + 3*x^4),x]

[Out]

ArcTan[Sqrt[3/2]*x^2]/(2*Sqrt[6])

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Rubi in Sympy [A]  time = 2.6205, size = 17, normalized size = 0.81 \[ \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(3*x**4+2),x)

[Out]

sqrt(6)*atan(sqrt(6)*x**2/2)/12

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Mathematica [A]  time = 0.0129919, size = 21, normalized size = 1. \[ \frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{2 \sqrt{6}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(2 + 3*x^4),x]

[Out]

ArcTan[Sqrt[3/2]*x^2]/(2*Sqrt[6])

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Maple [A]  time = 0.002, size = 15, normalized size = 0.7 \[{\frac{\sqrt{6}}{12}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(3*x^4+2),x)

[Out]

1/12*arctan(1/2*x^2*6^(1/2))*6^(1/2)

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Maxima [A]  time = 1.59247, size = 19, normalized size = 0.9 \[ \frac{1}{12} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(3*x^4 + 2),x, algorithm="maxima")

[Out]

1/12*sqrt(6)*arctan(1/2*sqrt(6)*x^2)

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Fricas [A]  time = 0.222981, size = 19, normalized size = 0.9 \[ \frac{1}{12} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(3*x^4 + 2),x, algorithm="fricas")

[Out]

1/12*sqrt(6)*arctan(1/2*sqrt(6)*x^2)

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Sympy [A]  time = 0.189759, size = 17, normalized size = 0.81 \[ \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(3*x**4+2),x)

[Out]

sqrt(6)*atan(sqrt(6)*x**2/2)/12

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GIAC/XCAS [A]  time = 0.224629, size = 19, normalized size = 0.9 \[ \frac{1}{12} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(3*x^4 + 2),x, algorithm="giac")

[Out]

1/12*sqrt(6)*arctan(1/2*sqrt(6)*x^2)