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

Optimal. Leaf size=13 \[ \frac{1}{2} \log \left (x^2+2\right )+\tan ^{-1}(x) \]

[Out]

ArcTan[x] + Log[2 + x^2]/2

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Rubi [A]  time = 0.0407511, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{1}{2} \log \left (x^2+2\right )+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[x] + Log[2 + x^2]/2

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Rubi in Sympy [A]  time = 17.3369, size = 10, normalized size = 0.77 \[ \frac{\log{\left (x^{2} + 2 \right )}}{2} + \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**2 + 2)/2 + atan(x)

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Mathematica [A]  time = 0.01097, size = 13, normalized size = 1. \[ \frac{1}{2} \log \left (x^2+2\right )+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

ArcTan[x] + Log[2 + x^2]/2

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Maple [A]  time = 0.007, size = 12, normalized size = 0.9 \[ \arctan \left ( x \right ) +{\frac{\ln \left ({x}^{2}+2 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.865083, size = 15, normalized size = 1.15 \[ \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(x) + 1/2*log(x^2 + 2)

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Fricas [A]  time = 0.248314, size = 15, normalized size = 1.15 \[ \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(x) + 1/2*log(x^2 + 2)

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Sympy [A]  time = 0.243299, size = 10, normalized size = 0.77 \[ \frac{\log{\left (x^{2} + 2 \right )}}{2} + \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**2 + 2)/2 + atan(x)

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GIAC/XCAS [A]  time = 0.261356, size = 15, normalized size = 1.15 \[ \arctan \left (x\right ) + \frac{1}{2} \,{\rm ln}\left (x^{2} + 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

arctan(x) + 1/2*ln(x^2 + 2)