Optimal. Leaf size=29 \[ \frac{1}{2} \log \left (2 x^2+x+1\right )-\frac{1}{2} \log \left (2 x^2-x+1\right ) \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0326194, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{1}{2} \log \left (2 x^2+x+1\right )-\frac{1}{2} \log \left (2 x^2-x+1\right ) \]
Antiderivative was successfully verified.
[In] Int[(1 - 2*x^2)/(1 + 3*x^2 + 4*x^4),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 11.4104, size = 26, normalized size = 0.9 \[ - \frac{\log{\left (x^{2} - \frac{x}{2} + \frac{1}{2} \right )}}{2} + \frac{\log{\left (x^{2} + \frac{x}{2} + \frac{1}{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-2*x**2+1)/(4*x**4+3*x**2+1),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.00986284, size = 29, normalized size = 1. \[ \frac{1}{2} \log \left (2 x^2+x+1\right )-\frac{1}{2} \log \left (2 x^2-x+1\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(1 - 2*x^2)/(1 + 3*x^2 + 4*x^4),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.005, size = 26, normalized size = 0.9 \[ -{\frac{\ln \left ( 2\,{x}^{2}-x+1 \right ) }{2}}+{\frac{\ln \left ( 2\,{x}^{2}+x+1 \right ) }{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-2*x^2+1)/(4*x^4+3*x^2+1),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 0.744593, size = 34, normalized size = 1.17 \[ \frac{1}{2} \, \log \left (2 \, x^{2} + x + 1\right ) - \frac{1}{2} \, \log \left (2 \, x^{2} - x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*x^2 - 1)/(4*x^4 + 3*x^2 + 1),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.282175, size = 34, normalized size = 1.17 \[ \frac{1}{2} \, \log \left (2 \, x^{2} + x + 1\right ) - \frac{1}{2} \, \log \left (2 \, x^{2} - x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*x^2 - 1)/(4*x^4 + 3*x^2 + 1),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.19189, size = 26, normalized size = 0.9 \[ - \frac{\log{\left (x^{2} - \frac{x}{2} + \frac{1}{2} \right )}}{2} + \frac{\log{\left (x^{2} + \frac{x}{2} + \frac{1}{2} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x**2+1)/(4*x**4+3*x**2+1),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.271278, size = 34, normalized size = 1.17 \[ \frac{1}{2} \,{\rm ln}\left (2 \, x^{2} + x + 1\right ) - \frac{1}{2} \,{\rm ln}\left (2 \, x^{2} - x + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*x^2 - 1)/(4*x^4 + 3*x^2 + 1),x, algorithm="giac")
[Out]