Optimal. Leaf size=68 \[ \frac{1}{12 (1-x)}+\frac{1}{36 (2-x)}-\frac{1}{36 (x+1)}+\frac{1}{18} \log (1-x)-\frac{35}{432} \log (2-x)+\frac{1}{54} \log (x+1)+\frac{1}{144} \log (x+2) \]
[Out]
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Rubi [A] time = 0.109365, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{1}{12 (1-x)}+\frac{1}{36 (2-x)}-\frac{1}{36 (x+1)}+\frac{1}{18} \log (1-x)-\frac{35}{432} \log (2-x)+\frac{1}{54} \log (x+1)+\frac{1}{144} \log (x+2) \]
Antiderivative was successfully verified.
[In] Int[(2 + x)/(4 - 5*x^2 + x^4)^2,x]
[Out]
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Rubi in Sympy [A] time = 21.859, size = 48, normalized size = 0.71 \[ \frac{\log{\left (- x + 1 \right )}}{18} - \frac{35 \log{\left (- x + 2 \right )}}{432} + \frac{\log{\left (x + 1 \right )}}{54} + \frac{\log{\left (x + 2 \right )}}{144} - \frac{1}{36 \left (x + 1\right )} + \frac{1}{36 \left (- x + 2\right )} + \frac{1}{12 \left (- x + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((2+x)/(x**4-5*x**2+4)**2,x)
[Out]
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Mathematica [A] time = 0.0548921, size = 60, normalized size = 0.88 \[ \frac{1}{432} \left (\frac{12 \left (-5 x^2+6 x+5\right )}{x^3-2 x^2-x+2}+24 \log (1-x)-35 \log (2-x)+8 \log (x+1)+3 \log (x+2)\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(2 + x)/(4 - 5*x^2 + x^4)^2,x]
[Out]
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Maple [A] time = 0.021, size = 47, normalized size = 0.7 \[{\frac{\ln \left ( 2+x \right ) }{144}}-{\frac{1}{-12+12\,x}}+{\frac{\ln \left ( -1+x \right ) }{18}}-{\frac{1}{36+36\,x}}+{\frac{\ln \left ( 1+x \right ) }{54}}-{\frac{1}{36\,x-72}}-{\frac{35\,\ln \left ( x-2 \right ) }{432}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((2+x)/(x^4-5*x^2+4)^2,x)
[Out]
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Maxima [A] time = 0.706691, size = 70, normalized size = 1.03 \[ -\frac{5 \, x^{2} - 6 \, x - 5}{36 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )}} + \frac{1}{144} \, \log \left (x + 2\right ) + \frac{1}{54} \, \log \left (x + 1\right ) + \frac{1}{18} \, \log \left (x - 1\right ) - \frac{35}{432} \, \log \left (x - 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 2)/(x^4 - 5*x^2 + 4)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.254751, size = 139, normalized size = 2.04 \[ -\frac{60 \, x^{2} - 3 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )} \log \left (x + 2\right ) - 8 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )} \log \left (x + 1\right ) - 24 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )} \log \left (x - 1\right ) + 35 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )} \log \left (x - 2\right ) - 72 \, x - 60}{432 \,{\left (x^{3} - 2 \, x^{2} - x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 2)/(x^4 - 5*x^2 + 4)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 0.846249, size = 53, normalized size = 0.78 \[ - \frac{5 x^{2} - 6 x - 5}{36 x^{3} - 72 x^{2} - 36 x + 72} - \frac{35 \log{\left (x - 2 \right )}}{432} + \frac{\log{\left (x - 1 \right )}}{18} + \frac{\log{\left (x + 1 \right )}}{54} + \frac{\log{\left (x + 2 \right )}}{144} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((2+x)/(x**4-5*x**2+4)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.287628, size = 76, normalized size = 1.12 \[ -\frac{5 \, x^{2} - 6 \, x - 5}{36 \,{\left (x + 1\right )}{\left (x - 1\right )}{\left (x - 2\right )}} + \frac{1}{144} \,{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{1}{54} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{18} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) - \frac{35}{432} \,{\rm ln}\left ({\left | x - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + 2)/(x^4 - 5*x^2 + 4)^2,x, algorithm="giac")
[Out]