3.140 \(\int \frac{2 x+x^3}{1+x^2+x^4} \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )+\frac{1}{4} \log \left (x^4+x^2+1\right ) \]

[Out]

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

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Rubi [A]  time = 0.0701185, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )+\frac{1}{4} \log \left (x^4+x^2+1\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2*x + x^3)/(1 + x^2 + x^4),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.9051, size = 34, normalized size = 0.92 \[ \frac{\log{\left (x^{4} + x^{2} + 1 \right )}}{4} + \frac{\sqrt{3} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 x^{2}}{3} + \frac{1}{3}\right ) \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**3+2*x)/(x**4+x**2+1),x)

[Out]

log(x**4 + x**2 + 1)/4 + sqrt(3)*atan(sqrt(3)*(2*x**2/3 + 1/3))/2

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Mathematica [A]  time = 0.00857202, size = 37, normalized size = 1. \[ \frac{1}{2} \sqrt{3} \tan ^{-1}\left (\frac{2 x^2+1}{\sqrt{3}}\right )+\frac{1}{4} \log \left (x^4+x^2+1\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(2*x + x^3)/(1 + x^2 + x^4),x]

[Out]

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

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Maple [A]  time = 0.003, size = 31, normalized size = 0.8 \[{\frac{\ln \left ({x}^{4}+{x}^{2}+1 \right ) }{4}}+{\frac{\sqrt{3}}{2}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+1 \right ) \sqrt{3}}{3}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^3+2*x)/(x^4+x^2+1),x)

[Out]

1/4*ln(x^4+x^2+1)+1/2*arctan(1/3*(2*x^2+1)*3^(1/2))*3^(1/2)

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Maxima [A]  time = 0.795853, size = 72, normalized size = 1.95 \[ -\frac{1}{2} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x + 1\right )}\right ) + \frac{1}{2} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x - 1\right )}\right ) + \frac{1}{4} \, \log \left (x^{2} + x + 1\right ) + \frac{1}{4} \, \log \left (x^{2} - x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + 2*x)/(x^4 + x^2 + 1),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.249944, size = 41, normalized size = 1.11 \[ \frac{1}{2} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) + \frac{1}{4} \, \log \left (x^{4} + x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + 2*x)/(x^4 + x^2 + 1),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.22477, size = 37, normalized size = 1. \[ \frac{\log{\left (x^{4} + x^{2} + 1 \right )}}{4} + \frac{\sqrt{3} \operatorname{atan}{\left (\frac{2 \sqrt{3} x^{2}}{3} + \frac{\sqrt{3}}{3} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**3+2*x)/(x**4+x**2+1),x)

[Out]

log(x**4 + x**2 + 1)/4 + sqrt(3)*atan(2*sqrt(3)*x**2/3 + sqrt(3)/3)/2

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GIAC/XCAS [A]  time = 0.271782, size = 41, normalized size = 1.11 \[ \frac{1}{2} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (2 \, x^{2} + 1\right )}\right ) + \frac{1}{4} \,{\rm ln}\left (x^{4} + x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^3 + 2*x)/(x^4 + x^2 + 1),x, algorithm="giac")

[Out]

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