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