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

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]

1/(12*(1 - x)) + 1/(36*(2 - x)) - 1/(36*(1 + x)) + Log[1 - x]/18 - (35*Log[2 - x
])/432 + Log[1 + x]/54 + Log[2 + x]/144

_______________________________________________________________________________________

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]

1/(12*(1 - x)) + 1/(36*(2 - x)) - 1/(36*(1 + x)) + Log[1 - x]/18 - (35*Log[2 - x
])/432 + Log[1 + x]/54 + Log[2 + x]/144

_______________________________________________________________________________________

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]

log(-x + 1)/18 - 35*log(-x + 2)/432 + log(x + 1)/54 + log(x + 2)/144 - 1/(36*(x
+ 1)) + 1/(36*(-x + 2)) + 1/(12*(-x + 1))

_______________________________________________________________________________________

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]

((12*(5 + 6*x - 5*x^2))/(2 - x - 2*x^2 + x^3) + 24*Log[1 - x] - 35*Log[2 - x] +
8*Log[1 + x] + 3*Log[2 + x])/432

_______________________________________________________________________________________

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]

1/144*ln(2+x)-1/12/(-1+x)+1/18*ln(-1+x)-1/36/(1+x)+1/54*ln(1+x)-1/36/(x-2)-35/43
2*ln(x-2)

_______________________________________________________________________________________

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]

-1/36*(5*x^2 - 6*x - 5)/(x^3 - 2*x^2 - x + 2) + 1/144*log(x + 2) + 1/54*log(x +
1) + 1/18*log(x - 1) - 35/432*log(x - 2)

_______________________________________________________________________________________

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]

-1/432*(60*x^2 - 3*(x^3 - 2*x^2 - x + 2)*log(x + 2) - 8*(x^3 - 2*x^2 - x + 2)*lo
g(x + 1) - 24*(x^3 - 2*x^2 - x + 2)*log(x - 1) + 35*(x^3 - 2*x^2 - x + 2)*log(x
- 2) - 72*x - 60)/(x^3 - 2*x^2 - x + 2)

_______________________________________________________________________________________

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]

-(5*x**2 - 6*x - 5)/(36*x**3 - 72*x**2 - 36*x + 72) - 35*log(x - 2)/432 + log(x
- 1)/18 + log(x + 1)/54 + log(x + 2)/144

_______________________________________________________________________________________

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]

-1/36*(5*x^2 - 6*x - 5)/((x + 1)*(x - 1)*(x - 2)) + 1/144*ln(abs(x + 2)) + 1/54*
ln(abs(x + 1)) + 1/18*ln(abs(x - 1)) - 35/432*ln(abs(x - 2))