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

Optimal. Leaf size=24 \[ -\frac{1}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right )+\tan ^{-1}(x) \]

[Out]

-1/(2*(1 + x^2)) + ArcTan[x] + Log[1 + x^2]/2

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Rubi [A]  time = 0.0354224, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ -\frac{1}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right )+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*(1 + x^2)) + ArcTan[x] + Log[1 + x^2]/2

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Rubi in Sympy [A]  time = 12.894, size = 20, normalized size = 0.83 \[ \frac{x^{2}}{2 \left (x^{2} + 1\right )} + \frac{\log{\left (x^{2} + 1 \right )}}{2} + \operatorname{atan}{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2/(2*(x**2 + 1)) + log(x**2 + 1)/2 + atan(x)

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Mathematica [A]  time = 0.0172592, size = 24, normalized size = 1. \[ -\frac{1}{2 \left (x^2+1\right )}+\frac{1}{2} \log \left (x^2+1\right )+\tan ^{-1}(x) \]

Antiderivative was successfully verified.

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

[Out]

-1/(2*(1 + x^2)) + ArcTan[x] + Log[1 + x^2]/2

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Maple [A]  time = 0.007, size = 21, normalized size = 0.9 \[ -{\frac{1}{2\,{x}^{2}+2}}+\arctan \left ( x \right ) +{\frac{\ln \left ({x}^{2}+1 \right ) }{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.88082, size = 27, normalized size = 1.12 \[ -\frac{1}{2 \,{\left (x^{2} + 1\right )}} + \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.246784, size = 43, normalized size = 1.79 \[ \frac{2 \,{\left (x^{2} + 1\right )} \arctan \left (x\right ) +{\left (x^{2} + 1\right )} \log \left (x^{2} + 1\right ) - 1}{2 \,{\left (x^{2} + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.241453, size = 19, normalized size = 0.79 \[ \frac{\log{\left (x^{2} + 1 \right )}}{2} + \operatorname{atan}{\left (x \right )} - \frac{1}{2 x^{2} + 2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x**2 + 1)/2 + atan(x) - 1/(2*x**2 + 2)

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GIAC/XCAS [A]  time = 0.261496, size = 27, normalized size = 1.12 \[ -\frac{1}{2 \,{\left (x^{2} + 1\right )}} + \arctan \left (x\right ) + \frac{1}{2} \,{\rm ln}\left (x^{2} + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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