Optimal. Leaf size=23 \[ \frac{1}{2} \log \left (x^2+4\right )-\frac{3}{2} \tan ^{-1}\left (\frac{x}{2}\right )+\tan ^{-1}(x) \]
[Out]
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Rubi [A] time = 0.0555356, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{1}{2} \log \left (x^2+4\right )-\frac{3}{2} \tan ^{-1}\left (\frac{x}{2}\right )+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[(1 + x - 2*x^2 + x^3)/(4 + 5*x^2 + x^4),x]
[Out]
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Rubi in Sympy [A] time = 22.5066, size = 19, normalized size = 0.83 \[ \frac{\log{\left (x^{2} + 4 \right )}}{2} - \frac{3 \operatorname{atan}{\left (\frac{x}{2} \right )}}{2} + \operatorname{atan}{\left (x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((x**3-2*x**2+x+1)/(x**4+5*x**2+4),x)
[Out]
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Mathematica [A] time = 0.0143496, size = 23, normalized size = 1. \[ \frac{1}{2} \log \left (x^2+4\right )-\frac{3}{2} \tan ^{-1}\left (\frac{x}{2}\right )+\tan ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Integrate[(1 + x - 2*x^2 + x^3)/(4 + 5*x^2 + x^4),x]
[Out]
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Maple [A] time = 0.006, size = 18, normalized size = 0.8 \[ -{\frac{3}{2}\arctan \left ({\frac{x}{2}} \right ) }+\arctan \left ( x \right ) +{\frac{\ln \left ({x}^{2}+4 \right ) }{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^3-2*x^2+x+1)/(x^4+5*x^2+4),x)
[Out]
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Maxima [A] time = 0.892481, size = 23, normalized size = 1. \[ -\frac{3}{2} \, \arctan \left (\frac{1}{2} \, x\right ) + \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^3 - 2*x^2 + x + 1)/(x^4 + 5*x^2 + 4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.255242, size = 23, normalized size = 1. \[ -\frac{3}{2} \, \arctan \left (\frac{1}{2} \, x\right ) + \arctan \left (x\right ) + \frac{1}{2} \, \log \left (x^{2} + 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^3 - 2*x^2 + x + 1)/(x^4 + 5*x^2 + 4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.493768, size = 19, normalized size = 0.83 \[ \frac{\log{\left (x^{2} + 4 \right )}}{2} - \frac{3 \operatorname{atan}{\left (\frac{x}{2} \right )}}{2} + \operatorname{atan}{\left (x \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x**3-2*x**2+x+1)/(x**4+5*x**2+4),x)
[Out]
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GIAC/XCAS [A] time = 0.261755, size = 23, normalized size = 1. \[ -\frac{3}{2} \, \arctan \left (\frac{1}{2} \, x\right ) + \arctan \left (x\right ) + \frac{1}{2} \,{\rm ln}\left (x^{2} + 4\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x^3 - 2*x^2 + x + 1)/(x^4 + 5*x^2 + 4),x, algorithm="giac")
[Out]