Optimal. Leaf size=24 \[ \frac{1}{3} \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )-\frac{1}{3} \tanh ^{-1}(x) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0252483, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{1}{3} \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )-\frac{1}{3} \tanh ^{-1}(x) \]
Antiderivative was successfully verified.
[In] Int[x^2/(-2 + x^2 + x^4),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 2.96615, size = 20, normalized size = 0.83 \[ \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{3} - \frac{\operatorname{atanh}{\left (x \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2/(x**4+x**2-2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0159883, size = 32, normalized size = 1.33 \[ \frac{1}{6} \left (\log (1-x)-\log (x+1)+2 \sqrt{2} \tan ^{-1}\left (\frac{x}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^2/(-2 + x^2 + x^4),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 26, normalized size = 1.1 \[ -{\frac{\ln \left ( 1+x \right ) }{6}}+{\frac{\ln \left ( -1+x \right ) }{6}}+{\frac{\sqrt{2}}{3}\arctan \left ({\frac{x\sqrt{2}}{2}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2/(x^4+x^2-2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.51545, size = 34, normalized size = 1.42 \[ \frac{1}{3} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{1}{6} \, \log \left (x + 1\right ) + \frac{1}{6} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(x^4 + x^2 - 2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.217796, size = 34, normalized size = 1.42 \[ \frac{1}{3} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{1}{6} \, \log \left (x + 1\right ) + \frac{1}{6} \, \log \left (x - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(x^4 + x^2 - 2),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.197665, size = 29, normalized size = 1.21 \[ \frac{\log{\left (x - 1 \right )}}{6} - \frac{\log{\left (x + 1 \right )}}{6} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x}{2} \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2/(x**4+x**2-2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.20265, size = 36, normalized size = 1.5 \[ \frac{1}{3} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2} x\right ) - \frac{1}{6} \,{\rm ln}\left ({\left | x + 1 \right |}\right ) + \frac{1}{6} \,{\rm ln}\left ({\left | x - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^2/(x^4 + x^2 - 2),x, algorithm="giac")
[Out]