Optimal. Leaf size=50 \[ \frac{\log \left (2 x^2+\sqrt{3} x+1\right )}{2 \sqrt{3}}-\frac{\log \left (2 x^2-\sqrt{3} x+1\right )}{2 \sqrt{3}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0457355, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{\log \left (2 x^2+\sqrt{3} x+1\right )}{2 \sqrt{3}}-\frac{\log \left (2 x^2-\sqrt{3} x+1\right )}{2 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Int[(1 - 2*x^2)/(1 + x^2 + 4*x^4),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 13.8858, size = 46, normalized size = 0.92 \[ - \frac{\sqrt{3} \log{\left (x^{2} - \frac{\sqrt{3} x}{2} + \frac{1}{2} \right )}}{6} + \frac{\sqrt{3} \log{\left (x^{2} + \frac{\sqrt{3} x}{2} + \frac{1}{2} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-2*x**2+1)/(4*x**4+x**2+1),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0221911, size = 42, normalized size = 0.84 \[ \frac{\log \left (2 x^2+\sqrt{3} x+1\right )-\log \left (-2 x^2+\sqrt{3} x-1\right )}{2 \sqrt{3}} \]
Antiderivative was successfully verified.
[In] Integrate[(1 - 2*x^2)/(1 + x^2 + 4*x^4),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.015, size = 39, normalized size = 0.8 \[ -{\frac{\ln \left ( 1+2\,{x}^{2}-x\sqrt{3} \right ) \sqrt{3}}{6}}+{\frac{\ln \left ( 1+2\,{x}^{2}+x\sqrt{3} \right ) \sqrt{3}}{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-2*x^2+1)/(4*x^4+x^2+1),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{2 \, x^{2} - 1}{4 \, x^{4} + x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*x^2 - 1)/(4*x^4 + x^2 + 1),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.278298, size = 59, normalized size = 1.18 \[ \frac{1}{6} \, \sqrt{3} \log \left (\frac{12 \, x^{3} + \sqrt{3}{\left (4 \, x^{4} + 7 \, x^{2} + 1\right )} + 6 \, x}{4 \, x^{4} + x^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*x^2 - 1)/(4*x^4 + x^2 + 1),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.200585, size = 46, normalized size = 0.92 \[ - \frac{\sqrt{3} \log{\left (x^{2} - \frac{\sqrt{3} x}{2} + \frac{1}{2} \right )}}{6} + \frac{\sqrt{3} \log{\left (x^{2} + \frac{\sqrt{3} x}{2} + \frac{1}{2} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-2*x**2+1)/(4*x**4+x**2+1),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{2 \, x^{2} - 1}{4 \, x^{4} + x^{2} + 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-(2*x^2 - 1)/(4*x^4 + x^2 + 1),x, algorithm="giac")
[Out]