3.698 \(\int \frac{x}{\left (2+3 x^4\right )^2} \, dx\)

Optimal. Leaf size=38 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{8 \sqrt{6}}+\frac{x^2}{8 \left (3 x^4+2\right )} \]

[Out]

x^2/(8*(2 + 3*x^4)) + ArcTan[Sqrt[3/2]*x^2]/(8*Sqrt[6])

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Rubi [A]  time = 0.0352096, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{8 \sqrt{6}}+\frac{x^2}{8 \left (3 x^4+2\right )} \]

Antiderivative was successfully verified.

[In]  Int[x/(2 + 3*x^4)^2,x]

[Out]

x^2/(8*(2 + 3*x^4)) + ArcTan[Sqrt[3/2]*x^2]/(8*Sqrt[6])

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Rubi in Sympy [A]  time = 2.8662, size = 27, normalized size = 0.71 \[ \frac{x^{2}}{8 \left (3 x^{4} + 2\right )} + \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{48} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(3*x**4+2)**2,x)

[Out]

x**2/(8*(3*x**4 + 2)) + sqrt(6)*atan(sqrt(6)*x**2/2)/48

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Mathematica [A]  time = 0.030039, size = 38, normalized size = 1. \[ \frac{\tan ^{-1}\left (\sqrt{\frac{3}{2}} x^2\right )}{8 \sqrt{6}}+\frac{x^2}{8 \left (3 x^4+2\right )} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(2 + 3*x^4)^2,x]

[Out]

x^2/(8*(2 + 3*x^4)) + ArcTan[Sqrt[3/2]*x^2]/(8*Sqrt[6])

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Maple [A]  time = 0.01, size = 30, normalized size = 0.8 \[{\frac{{x}^{2}}{24\,{x}^{4}+16}}+{\frac{\sqrt{6}}{48}\arctan \left ({\frac{{x}^{2}\sqrt{6}}{2}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(3*x^4+2)^2,x)

[Out]

1/8*x^2/(3*x^4+2)+1/48*arctan(1/2*x^2*6^(1/2))*6^(1/2)

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Maxima [A]  time = 1.57935, size = 39, normalized size = 1.03 \[ \frac{1}{48} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) + \frac{x^{2}}{8 \,{\left (3 \, x^{4} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*sqrt(6)*arctan(1/2*sqrt(6)*x^2) + 1/8*x^2/(3*x^4 + 2)

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Fricas [A]  time = 0.229895, size = 53, normalized size = 1.39 \[ \frac{\sqrt{6}{\left (\sqrt{6} x^{2} +{\left (3 \, x^{4} + 2\right )} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right )\right )}}{48 \,{\left (3 \, x^{4} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*sqrt(6)*(sqrt(6)*x^2 + (3*x^4 + 2)*arctan(1/2*sqrt(6)*x^2))/(3*x^4 + 2)

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Sympy [A]  time = 0.323769, size = 27, normalized size = 0.71 \[ \frac{x^{2}}{24 x^{4} + 16} + \frac{\sqrt{6} \operatorname{atan}{\left (\frac{\sqrt{6} x^{2}}{2} \right )}}{48} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(3*x**4+2)**2,x)

[Out]

x**2/(24*x**4 + 16) + sqrt(6)*atan(sqrt(6)*x**2/2)/48

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GIAC/XCAS [A]  time = 0.224473, size = 39, normalized size = 1.03 \[ \frac{1}{48} \, \sqrt{6} \arctan \left (\frac{1}{2} \, \sqrt{6} x^{2}\right ) + \frac{x^{2}}{8 \,{\left (3 \, x^{4} + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/48*sqrt(6)*arctan(1/2*sqrt(6)*x^2) + 1/8*x^2/(3*x^4 + 2)