Optimal. Leaf size=57 \[ -\frac{1}{2} \log (1-x) (d+e+f+g)+\frac{1}{3} \log (2-x) (d+2 e+4 f+8 g)+\frac{1}{6} \log (x+1) (d-e+f-g)+g x \]
[Out]
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Rubi [A] time = 0.147466, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065 \[ -\frac{1}{2} \log (1-x) (d+e+f+g)+\frac{1}{3} \log (2-x) (d+2 e+4 f+8 g)+\frac{1}{6} \log (x+1) (d-e+f-g)+g x \]
Antiderivative was successfully verified.
[In] Int[((2 + x)*(d + e*x + f*x^2 + g*x^3))/(4 - 5*x^2 + x^4),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+x)*(g*x**3+f*x**2+e*x+d)/(x**4-5*x**2+4),x)
[Out]
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Mathematica [A] time = 0.0519937, size = 55, normalized size = 0.96 \[ \frac{1}{6} (-3 \log (1-x) (d+e+f+g)+2 \log (2-x) (d+2 e+4 f+8 g)+\log (x+1) (d-e+f-g)+6 g x) \]
Antiderivative was successfully verified.
[In] Integrate[((2 + x)*(d + e*x + f*x^2 + g*x^3))/(4 - 5*x^2 + x^4),x]
[Out]
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Maple [A] time = 0.011, size = 89, normalized size = 1.6 \[ gx-{\frac{\ln \left ( -1+x \right ) d}{2}}-{\frac{\ln \left ( -1+x \right ) e}{2}}-{\frac{\ln \left ( -1+x \right ) f}{2}}-{\frac{\ln \left ( -1+x \right ) g}{2}}+{\frac{\ln \left ( 1+x \right ) d}{6}}-{\frac{\ln \left ( 1+x \right ) e}{6}}+{\frac{\ln \left ( 1+x \right ) f}{6}}-{\frac{\ln \left ( 1+x \right ) g}{6}}+{\frac{\ln \left ( x-2 \right ) d}{3}}+{\frac{2\,\ln \left ( x-2 \right ) e}{3}}+{\frac{4\,\ln \left ( x-2 \right ) f}{3}}+{\frac{8\,\ln \left ( x-2 \right ) g}{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+x)*(g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4),x)
[Out]
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Maxima [A] time = 0.700485, size = 63, normalized size = 1.11 \[ g x + \frac{1}{6} \,{\left (d - e + f - g\right )} \log \left (x + 1\right ) - \frac{1}{2} \,{\left (d + e + f + g\right )} \log \left (x - 1\right ) + \frac{1}{3} \,{\left (d + 2 \, e + 4 \, f + 8 \, g\right )} \log \left (x - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x^3 + f*x^2 + e*x + d)*(x + 2)/(x^4 - 5*x^2 + 4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.287316, size = 63, normalized size = 1.11 \[ g x + \frac{1}{6} \,{\left (d - e + f - g\right )} \log \left (x + 1\right ) - \frac{1}{2} \,{\left (d + e + f + g\right )} \log \left (x - 1\right ) + \frac{1}{3} \,{\left (d + 2 \, e + 4 \, f + 8 \, g\right )} \log \left (x - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x^3 + f*x^2 + e*x + d)*(x + 2)/(x^4 - 5*x^2 + 4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 71.2712, size = 1389, normalized size = 24.37 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2+x)*(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 = 0.295258, size = 72, normalized size = 1.26 \[ g x + \frac{1}{6} \,{\left (d + f - g - e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{2} \,{\left (d + f + g + e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{3} \,{\left (d + 4 \, f + 8 \, g + 2 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x^3 + f*x^2 + e*x + d)*(x + 2)/(x^4 - 5*x^2 + 4),x, algorithm="giac")
[Out]