Optimal. Leaf size=57 \[ -\frac{1}{6} (d+4 f) \tanh ^{-1}\left (\frac{x}{2}\right )+\frac{1}{3} (d+f) \tanh ^{-1}(x)-\frac{1}{6} (e+g) \log \left (1-x^2\right )+\frac{1}{6} (e+4 g) \log \left (4-x^2\right ) \]
[Out]
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Rubi [A] time = 0.175102, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ -\frac{1}{6} (d+4 f) \tanh ^{-1}\left (\frac{x}{2}\right )+\frac{1}{3} (d+f) \tanh ^{-1}(x)-\frac{1}{6} (e+g) \log \left (1-x^2\right )+\frac{1}{6} (e+4 g) \log \left (4-x^2\right ) \]
Antiderivative was successfully verified.
[In] Int[(d + e*x + f*x^2 + g*x^3)/(4 - 5*x^2 + x^4),x]
[Out]
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Rubi in Sympy [A] time = 33.6666, size = 51, normalized size = 0.89 \[ - \left (\frac{d}{6} + \frac{2 f}{3}\right ) \operatorname{atanh}{\left (\frac{x}{2} \right )} + \left (\frac{d}{3} + \frac{f}{3}\right ) \operatorname{atanh}{\left (x \right )} - \left (\frac{e}{6} + \frac{g}{6}\right ) \log{\left (- x^{2} + 1 \right )} + \left (\frac{e}{6} + \frac{2 g}{3}\right ) \log{\left (- x^{2} + 4 \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x**3+f*x**2+e*x+d)/(x**4-5*x**2+4),x)
[Out]
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Mathematica [A] time = 0.0589006, size = 68, normalized size = 1.19 \[ \frac{1}{12} (-2 \log (1-x) (d+e+f+g)+\log (2-x) (d+2 e+4 f+8 g)+2 \log (x+1) (d-e+f-g)-\log (x+2) (d-2 e+4 f-8 g)) \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x + f*x^2 + g*x^3)/(4 - 5*x^2 + x^4),x]
[Out]
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Maple [B] time = 0.013, size = 114, normalized size = 2. \[ -{\frac{\ln \left ( 2+x \right ) d}{12}}+{\frac{\ln \left ( 2+x \right ) e}{6}}-{\frac{\ln \left ( 2+x \right ) f}{3}}+{\frac{2\,\ln \left ( 2+x \right ) g}{3}}-{\frac{\ln \left ( -1+x \right ) d}{6}}-{\frac{\ln \left ( -1+x \right ) e}{6}}-{\frac{\ln \left ( -1+x \right ) f}{6}}-{\frac{\ln \left ( -1+x \right ) g}{6}}+{\frac{\ln \left ( 1+x \right ) d}{6}}-{\frac{\ln \left ( 1+x \right ) e}{6}}+{\frac{\ln \left ( 1+x \right ) f}{6}}-{\frac{\ln \left ( 1+x \right ) g}{6}}+{\frac{\ln \left ( x-2 \right ) d}{12}}+{\frac{\ln \left ( x-2 \right ) e}{6}}+{\frac{\ln \left ( x-2 \right ) f}{3}}+{\frac{2\,\ln \left ( x-2 \right ) g}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4),x)
[Out]
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Maxima [A] time = 0.706383, size = 82, normalized size = 1.44 \[ -\frac{1}{12} \,{\left (d - 2 \, e + 4 \, f - 8 \, g\right )} \log \left (x + 2\right ) + \frac{1}{6} \,{\left (d - e + f - g\right )} \log \left (x + 1\right ) - \frac{1}{6} \,{\left (d + e + f + g\right )} \log \left (x - 1\right ) + \frac{1}{12} \,{\left (d + 2 \, e + 4 \, f + 8 \, g\right )} \log \left (x - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x^3 + f*x^2 + e*x + d)/(x^4 - 5*x^2 + 4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.48402, size = 82, normalized size = 1.44 \[ -\frac{1}{12} \,{\left (d - 2 \, e + 4 \, f - 8 \, g\right )} \log \left (x + 2\right ) + \frac{1}{6} \,{\left (d - e + f - g\right )} \log \left (x + 1\right ) - \frac{1}{6} \,{\left (d + e + f + g\right )} \log \left (x - 1\right ) + \frac{1}{12} \,{\left (d + 2 \, e + 4 \, f + 8 \, g\right )} \log \left (x - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x^3 + f*x^2 + e*x + d)/(x^4 - 5*x^2 + 4),x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x**3+f*x**2+e*x+d)/(x**4-5*x**2+4),x)
[Out]
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GIAC/XCAS [A] time = 0.291984, size = 93, normalized size = 1.63 \[ -\frac{1}{12} \,{\left (d + 4 \, f - 8 \, g - 2 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{1}{6} \,{\left (d + f - g - e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{6} \,{\left (d + f + g + e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{12} \,{\left (d + 4 \, f + 8 \, g + 2 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x^3 + f*x^2 + e*x + d)/(x^4 - 5*x^2 + 4),x, algorithm="giac")
[Out]