Optimal. Leaf size=14 \[ \frac{\tanh ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0143112, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\tanh ^{-1}\left (\sqrt{2} x\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[(1 - 2*x^2)/(1 - 4*x^2 + 4*x^4),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 4.57746, size = 14, normalized size = 1. \[ \frac{\sqrt{2} \operatorname{atanh}{\left (\sqrt{2} x \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-2*x**2+1)/(4*x**4-4*x**2+1),x)
[Out]
_______________________________________________________________________________________
Mathematica [B] time = 0.0112848, size = 32, normalized size = 2.29 \[ \frac{\log \left (2 x+\sqrt{2}\right )-\log \left (\sqrt{2}-2 x\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - 2*x^2)/(1 - 4*x^2 + 4*x^4),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.003, size = 12, normalized size = 0.9 \[{\frac{{\it Artanh} \left ( \sqrt{2}x \right ) \sqrt{2}}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-2*x^2+1)/(4*x^4-4*x^2+1),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.826188, size = 35, normalized size = 2.5 \[ -\frac{1}{4} \, \sqrt{2} \log \left (\frac{2 \,{\left (2 \, x - \sqrt{2}\right )}}{4 \, x + 2 \, \sqrt{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*x^2 - 1)/(4*x^4 - 4*x^2 + 1),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.280236, size = 42, normalized size = 3. \[ \frac{1}{4} \, \sqrt{2} \log \left (\frac{\sqrt{2}{\left (2 \, x^{2} + 1\right )} + 4 \, x}{2 \, x^{2} - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*x^2 - 1)/(4*x^4 - 4*x^2 + 1),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.16878, size = 32, normalized size = 2.29 \[ - \frac{\sqrt{2} \log{\left (x - \frac{\sqrt{2}}{2} \right )}}{4} + \frac{\sqrt{2} \log{\left (x + \frac{\sqrt{2}}{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x**2+1)/(4*x**4-4*x**2+1),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.27209, size = 39, normalized size = 2.79 \[ \frac{1}{4} \, \sqrt{2}{\rm ln}\left ({\left | x + \frac{1}{2} \, \sqrt{2} \right |}\right ) - \frac{1}{4} \, \sqrt{2}{\rm ln}\left ({\left | x - \frac{1}{2} \, \sqrt{2} \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*x^2 - 1)/(4*x^4 - 4*x^2 + 1),x, algorithm="giac")
[Out]