3.11 \(\int \frac{1}{4-5 x^2+x^4} \, dx\)

Optimal. Leaf size=17 \[ \frac{1}{3} \tanh ^{-1}(x)-\frac{1}{6} \tanh ^{-1}\left (\frac{x}{2}\right ) \]

[Out]

-ArcTanh[x/2]/6 + ArcTanh[x]/3

_______________________________________________________________________________________

Rubi [A]  time = 0.0159435, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{1}{3} \tanh ^{-1}(x)-\frac{1}{6} \tanh ^{-1}\left (\frac{x}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(4 - 5*x^2 + x^4)^(-1),x]

[Out]

-ArcTanh[x/2]/6 + ArcTanh[x]/3

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.8714, size = 10, normalized size = 0.59 \[ - \frac{\operatorname{atanh}{\left (\frac{x}{2} \right )}}{6} + \frac{\operatorname{atanh}{\left (x \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**4-5*x**2+4),x)

[Out]

-atanh(x/2)/6 + atanh(x)/3

_______________________________________________________________________________________

Mathematica [B]  time = 0.00898096, size = 37, normalized size = 2.18 \[ -\frac{1}{6} \log (1-x)+\frac{1}{12} \log (2-x)+\frac{1}{6} \log (x+1)-\frac{1}{12} \log (x+2) \]

Antiderivative was successfully verified.

[In]  Integrate[(4 - 5*x^2 + x^4)^(-1),x]

[Out]

-Log[1 - x]/6 + Log[2 - x]/12 + Log[1 + x]/6 - Log[2 + x]/12

_______________________________________________________________________________________

Maple [B]  time = 0.012, size = 26, normalized size = 1.5 \[ -{\frac{\ln \left ( 2+x \right ) }{12}}-{\frac{\ln \left ( -1+x \right ) }{6}}+{\frac{\ln \left ( 1+x \right ) }{6}}+{\frac{\ln \left ( x-2 \right ) }{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^4-5*x^2+4),x)

[Out]

-1/12*ln(2+x)-1/6*ln(-1+x)+1/6*ln(1+x)+1/12*ln(x-2)

_______________________________________________________________________________________

Maxima [A]  time = 0.679632, size = 34, normalized size = 2. \[ -\frac{1}{12} \, \log \left (x + 2\right ) + \frac{1}{6} \, \log \left (x + 1\right ) - \frac{1}{6} \, \log \left (x - 1\right ) + \frac{1}{12} \, \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^4 - 5*x^2 + 4),x, algorithm="maxima")

[Out]

-1/12*log(x + 2) + 1/6*log(x + 1) - 1/6*log(x - 1) + 1/12*log(x - 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.261078, size = 34, normalized size = 2. \[ -\frac{1}{12} \, \log \left (x + 2\right ) + \frac{1}{6} \, \log \left (x + 1\right ) - \frac{1}{6} \, \log \left (x - 1\right ) + \frac{1}{12} \, \log \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^4 - 5*x^2 + 4),x, algorithm="fricas")

[Out]

-1/12*log(x + 2) + 1/6*log(x + 1) - 1/6*log(x - 1) + 1/12*log(x - 2)

_______________________________________________________________________________________

Sympy [A]  time = 0.480528, size = 26, normalized size = 1.53 \[ \frac{\log{\left (x - 2 \right )}}{12} - \frac{\log{\left (x - 1 \right )}}{6} + \frac{\log{\left (x + 1 \right )}}{6} - \frac{\log{\left (x + 2 \right )}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**4-5*x**2+4),x)

[Out]

log(x - 2)/12 - log(x - 1)/6 + log(x + 1)/6 - log(x + 2)/12

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.269179, size = 39, normalized size = 2.29 \[ -\frac{1}{12} \,{\rm ln}\left ({\left | x + 2 \right |}\right ) + \frac{1}{6} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) + \frac{1}{12} \,{\rm ln}\left ({\left | x - 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x^4 - 5*x^2 + 4),x, algorithm="giac")

[Out]

-1/12*ln(abs(x + 2)) + 1/6*ln(abs(x + 1)) - 1/6*ln(abs(x - 1)) + 1/12*ln(abs(x -
2))