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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]