Optimal. Leaf size=19 \[ \frac{1}{4} \tanh ^{-1}(x)-\frac{x}{4 \left (x^2+1\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0440524, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{1}{4} \tanh ^{-1}(x)-\frac{x}{4 \left (x^2+1\right )} \]
Antiderivative was successfully verified.
[In] Int[x^2/((1 - x^2)*(1 + x^2)^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 8.29075, size = 12, normalized size = 0.63 \[ - \frac{x}{4 \left (x^{2} + 1\right )} + \frac{\operatorname{atanh}{\left (x \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(-x**2+1)/(x**2+1)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0178874, size = 27, normalized size = 1.42 \[ \frac{1}{8} \left (-\frac{2 x}{x^2+1}-\log (1-x)+\log (x+1)\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^2/((1 - x^2)*(1 + x^2)^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.014, size = 24, normalized size = 1.3 \[ -{\frac{\ln \left ( -1+x \right ) }{8}}+{\frac{\ln \left ( 1+x \right ) }{8}}-{\frac{x}{4\,{x}^{2}+4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(-x^2+1)/(x^2+1)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.804053, size = 31, normalized size = 1.63 \[ -\frac{x}{4 \,{\left (x^{2} + 1\right )}} + \frac{1}{8} \, \log \left (x + 1\right ) - \frac{1}{8} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x^2/((x^2 + 1)^2*(x^2 - 1)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.254631, size = 46, normalized size = 2.42 \[ \frac{{\left (x^{2} + 1\right )} \log \left (x + 1\right ) -{\left (x^{2} + 1\right )} \log \left (x - 1\right ) - 2 \, x}{8 \,{\left (x^{2} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x^2/((x^2 + 1)^2*(x^2 - 1)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.279657, size = 20, normalized size = 1.05 \[ - \frac{x}{4 x^{2} + 4} - \frac{\log{\left (x - 1 \right )}}{8} + \frac{\log{\left (x + 1 \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(-x**2+1)/(x**2+1)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.260747, size = 41, normalized size = 2.16 \[ -\frac{1}{4 \,{\left (x + \frac{1}{x}\right )}} + \frac{1}{16} \,{\rm ln}\left ({\left | x + \frac{1}{x} + 2 \right |}\right ) - \frac{1}{16} \,{\rm ln}\left ({\left | x + \frac{1}{x} - 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-x^2/((x^2 + 1)^2*(x^2 - 1)),x, algorithm="giac")
[Out]