3.91 \(\int \frac{2-3 x+x^2}{\left (4-5 x^2+x^4\right )^2} \, dx\)

Optimal. Leaf size=56 \[ -\frac{3 x+5}{12 \left (x^2+3 x+2\right )}-\frac{1}{36} \log (1-x)+\frac{1}{144} \log (2-x)-\frac{7}{36} \log (x+1)+\frac{31}{144} \log (x+2) \]

[Out]

-(5 + 3*x)/(12*(2 + 3*x + x^2)) - Log[1 - x]/36 + Log[2 - x]/144 - (7*Log[1 + x]
)/36 + (31*Log[2 + x])/144

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Rubi [A]  time = 0.131818, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ -\frac{3 x+5}{12 \left (x^2+3 x+2\right )}-\frac{1}{36} \log (1-x)+\frac{1}{144} \log (2-x)-\frac{7}{36} \log (x+1)+\frac{31}{144} \log (x+2) \]

Antiderivative was successfully verified.

[In]  Int[(2 - 3*x + x^2)/(4 - 5*x^2 + x^4)^2,x]

[Out]

-(5 + 3*x)/(12*(2 + 3*x + x^2)) - Log[1 - x]/36 + Log[2 - x]/144 - (7*Log[1 + x]
)/36 + (31*Log[2 + x])/144

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Rubi in Sympy [A]  time = 31.265, size = 44, normalized size = 0.79 \[ - \frac{18 x + 30}{72 \left (x^{2} + 3 x + 2\right )} - \frac{\log{\left (- x + 1 \right )}}{36} + \frac{\log{\left (- x + 2 \right )}}{144} - \frac{7 \log{\left (x + 1 \right )}}{36} + \frac{31 \log{\left (x + 2 \right )}}{144} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x**2-3*x+2)/(x**4-5*x**2+4)**2,x)

[Out]

-(18*x + 30)/(72*(x**2 + 3*x + 2)) - log(-x + 1)/36 + log(-x + 2)/144 - 7*log(x
+ 1)/36 + 31*log(x + 2)/144

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Mathematica [A]  time = 0.0464615, size = 48, normalized size = 0.86 \[ \frac{1}{144} \left (-\frac{12 (3 x+5)}{x^2+3 x+2}-4 \log (1-x)+\log (2-x)-28 \log (x+1)+31 \log (x+2)\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(2 - 3*x + x^2)/(4 - 5*x^2 + x^4)^2,x]

[Out]

((-12*(5 + 3*x))/(2 + 3*x + x^2) - 4*Log[1 - x] + Log[2 - x] - 28*Log[1 + x] + 3
1*Log[2 + x])/144

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Maple [A]  time = 0.018, size = 40, normalized size = 0.7 \[ -{\frac{1}{24+12\,x}}+{\frac{31\,\ln \left ( 2+x \right ) }{144}}-{\frac{\ln \left ( -1+x \right ) }{36}}-{\frac{1}{6+6\,x}}-{\frac{7\,\ln \left ( 1+x \right ) }{36}}+{\frac{\ln \left ( x-2 \right ) }{144}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((x^2-3*x+2)/(x^4-5*x^2+4)^2,x)

[Out]

-1/12/(2+x)+31/144*ln(2+x)-1/36*ln(-1+x)-1/6/(1+x)-7/36*ln(1+x)+1/144*ln(x-2)

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Maxima [A]  time = 0.699696, size = 57, normalized size = 1.02 \[ -\frac{3 \, x + 5}{12 \,{\left (x^{2} + 3 \, x + 2\right )}} + \frac{31}{144} \, \log \left (x + 2\right ) - \frac{7}{36} \, \log \left (x + 1\right ) - \frac{1}{36} \, \log \left (x - 1\right ) + \frac{1}{144} \, \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 3*x + 2)/(x^4 - 5*x^2 + 4)^2,x, algorithm="maxima")

[Out]

-1/12*(3*x + 5)/(x^2 + 3*x + 2) + 31/144*log(x + 2) - 7/36*log(x + 1) - 1/36*log
(x - 1) + 1/144*log(x - 2)

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Fricas [A]  time = 0.258591, size = 97, normalized size = 1.73 \[ \frac{31 \,{\left (x^{2} + 3 \, x + 2\right )} \log \left (x + 2\right ) - 28 \,{\left (x^{2} + 3 \, x + 2\right )} \log \left (x + 1\right ) - 4 \,{\left (x^{2} + 3 \, x + 2\right )} \log \left (x - 1\right ) +{\left (x^{2} + 3 \, x + 2\right )} \log \left (x - 2\right ) - 36 \, x - 60}{144 \,{\left (x^{2} + 3 \, x + 2\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 3*x + 2)/(x^4 - 5*x^2 + 4)^2,x, algorithm="fricas")

[Out]

1/144*(31*(x^2 + 3*x + 2)*log(x + 2) - 28*(x^2 + 3*x + 2)*log(x + 1) - 4*(x^2 +
3*x + 2)*log(x - 1) + (x^2 + 3*x + 2)*log(x - 2) - 36*x - 60)/(x^2 + 3*x + 2)

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Sympy [A]  time = 0.78408, size = 44, normalized size = 0.79 \[ - \frac{3 x + 5}{12 x^{2} + 36 x + 24} + \frac{\log{\left (x - 2 \right )}}{144} - \frac{\log{\left (x - 1 \right )}}{36} - \frac{7 \log{\left (x + 1 \right )}}{36} + \frac{31 \log{\left (x + 2 \right )}}{144} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x**2-3*x+2)/(x**4-5*x**2+4)**2,x)

[Out]

-(3*x + 5)/(12*x**2 + 36*x + 24) + log(x - 2)/144 - log(x - 1)/36 - 7*log(x + 1)
/36 + 31*log(x + 2)/144

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GIAC/XCAS [A]  time = 0.283228, size = 62, normalized size = 1.11 \[ -\frac{3 \, x + 5}{12 \,{\left (x + 2\right )}{\left (x + 1\right )}} + \frac{31}{144} \,{\rm ln}\left ({\left | x + 2 \right |}\right ) - \frac{7}{36} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{36} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{144} \,{\rm ln}\left ({\left | x - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^2 - 3*x + 2)/(x^4 - 5*x^2 + 4)^2,x, algorithm="giac")

[Out]

-1/12*(3*x + 5)/((x + 2)*(x + 1)) + 31/144*ln(abs(x + 2)) - 7/36*ln(abs(x + 1))
- 1/36*ln(abs(x - 1)) + 1/144*ln(abs(x - 2))