Optimal. Leaf size=106 \[ \frac{d-2 e+4 f-8 g+16 h}{12 (x+2)}-\frac{1}{18} \log (1-x) (d+e+f+g+h)+\frac{1}{48} \log (2-x) (d+2 e+4 f+8 g+16 h)+\frac{1}{6} \log (x+1) (d-e+f-g+h)-\frac{1}{144} \log (x+2) (19 d-26 e+28 f-8 g-80 h) \]
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Rubi [A] time = 0.485613, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{d-2 e+4 f-8 g+16 h}{12 (x+2)}-\frac{1}{18} \log (1-x) (d+e+f+g+h)+\frac{1}{48} \log (2-x) (d+2 e+4 f+8 g+16 h)+\frac{1}{6} \log (x+1) (d-e+f-g+h)-\frac{1}{144} \log (x+2) (19 d-26 e+28 f-8 g-80 h) \]
Antiderivative was successfully verified.
[In] Int[((2 - x - 2*x^2 + x^3)*(d + e*x + f*x^2 + g*x^3 + h*x^4))/(4 - 5*x^2 + x^4)^2,x]
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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((x**3-2*x**2-x+2)*(h*x**4+g*x**3+f*x**2+e*x+d)/(x**4-5*x**2+4)**2,x)
[Out]
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Mathematica [A] time = 0.114195, size = 102, normalized size = 0.96 \[ \frac{1}{144} \left (\frac{12 (d-2 e+4 f-8 g+16 h)}{x+2}+24 \log (-x-1) (d-e+f-g+h)-8 \log (1-x) (d+e+f+g+h)+3 \log (2-x) (d+2 (e+2 f+4 g+8 h))+\log (x+2) (-19 d+26 e-28 f+8 g+80 h)\right ) \]
Antiderivative was successfully verified.
[In] Integrate[((2 - x - 2*x^2 + x^3)*(d + e*x + f*x^2 + g*x^3 + h*x^4))/(4 - 5*x^2 + x^4)^2,x]
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Maple [A] time = 0.017, size = 182, normalized size = 1.7 \[{\frac{4\,h}{6+3\,x}}-{\frac{2\,g}{6+3\,x}}+{\frac{d}{24+12\,x}}-{\frac{e}{12+6\,x}}+{\frac{f}{6+3\,x}}+{\frac{\ln \left ( 1+x \right ) d}{6}}-{\frac{\ln \left ( 1+x \right ) e}{6}}-{\frac{\ln \left ( -1+x \right ) d}{18}}-{\frac{\ln \left ( -1+x \right ) e}{18}}+{\frac{\ln \left ( x-2 \right ) h}{3}}+{\frac{\ln \left ( 1+x \right ) h}{6}}+{\frac{5\,\ln \left ( 2+x \right ) h}{9}}-{\frac{\ln \left ( -1+x \right ) h}{18}}-{\frac{\ln \left ( 1+x \right ) g}{6}}+{\frac{\ln \left ( x-2 \right ) g}{6}}-{\frac{\ln \left ( -1+x \right ) g}{18}}+{\frac{\ln \left ( 2+x \right ) g}{18}}+{\frac{\ln \left ( x-2 \right ) d}{48}}+{\frac{\ln \left ( x-2 \right ) e}{24}}+{\frac{13\,\ln \left ( 2+x \right ) e}{72}}+{\frac{\ln \left ( x-2 \right ) f}{12}}-{\frac{19\,\ln \left ( 2+x \right ) d}{144}}+{\frac{\ln \left ( 1+x \right ) f}{6}}-{\frac{\ln \left ( -1+x \right ) f}{18}}-{\frac{7\,\ln \left ( 2+x \right ) f}{36}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((x^3-2*x^2-x+2)*(h*x^4+g*x^3+f*x^2+e*x+d)/(x^4-5*x^2+4)^2,x)
[Out]
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Maxima [A] time = 0.694834, size = 124, normalized size = 1.17 \[ -\frac{1}{144} \,{\left (19 \, d - 26 \, e + 28 \, f - 8 \, g - 80 \, h\right )} \log \left (x + 2\right ) + \frac{1}{6} \,{\left (d - e + f - g + h\right )} \log \left (x + 1\right ) - \frac{1}{18} \,{\left (d + e + f + g + h\right )} \log \left (x - 1\right ) + \frac{1}{48} \,{\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h\right )} \log \left (x - 2\right ) + \frac{d - 2 \, e + 4 \, f - 8 \, g + 16 \, h}{12 \,{\left (x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((h*x^4 + g*x^3 + f*x^2 + e*x + d)*(x^3 - 2*x^2 - x + 2)/(x^4 - 5*x^2 + 4)^2,x, algorithm="maxima")
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Fricas [A] time = 4.17267, size = 221, normalized size = 2.08 \[ -\frac{{\left ({\left (19 \, d - 26 \, e + 28 \, f - 8 \, g - 80 \, h\right )} x + 38 \, d - 52 \, e + 56 \, f - 16 \, g - 160 \, h\right )} \log \left (x + 2\right ) - 24 \,{\left ({\left (d - e + f - g + h\right )} x + 2 \, d - 2 \, e + 2 \, f - 2 \, g + 2 \, h\right )} \log \left (x + 1\right ) + 8 \,{\left ({\left (d + e + f + g + h\right )} x + 2 \, d + 2 \, e + 2 \, f + 2 \, g + 2 \, h\right )} \log \left (x - 1\right ) - 3 \,{\left ({\left (d + 2 \, e + 4 \, f + 8 \, g + 16 \, h\right )} x + 2 \, d + 4 \, e + 8 \, f + 16 \, g + 32 \, h\right )} \log \left (x - 2\right ) - 12 \, d + 24 \, e - 48 \, f + 96 \, g - 192 \, h}{144 \,{\left (x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((h*x^4 + g*x^3 + f*x^2 + e*x + d)*(x^3 - 2*x^2 - x + 2)/(x^4 - 5*x^2 + 4)^2,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((x**3-2*x**2-x+2)*(h*x**4+g*x**3+f*x**2+e*x+d)/(x**4-5*x**2+4)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.283838, size = 136, normalized size = 1.28 \[ -\frac{1}{144} \,{\left (19 \, d + 28 \, f - 8 \, g - 80 \, h - 26 \, e\right )}{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{1}{6} \,{\left (d + f - g + h - e\right )}{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{18} \,{\left (d + f + g + h + e\right )}{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{48} \,{\left (d + 4 \, f + 8 \, g + 16 \, h + 2 \, e\right )}{\rm ln}\left ({\left | x - 2 \right |}\right ) + \frac{d + 4 \, f - 8 \, g + 16 \, h - 2 \, e}{12 \,{\left (x + 2\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((h*x^4 + g*x^3 + f*x^2 + e*x + d)*(x^3 - 2*x^2 - x + 2)/(x^4 - 5*x^2 + 4)^2,x, algorithm="giac")
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