Optimal. Leaf size=38 \[ \frac{1}{8} \log \left (4 x^2-4 x+3\right )+x+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{2}}\right )}{4 \sqrt{2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0592986, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{1}{8} \log \left (4 x^2-4 x+3\right )+x+\frac{\tan ^{-1}\left (\frac{1-2 x}{\sqrt{2}}\right )}{4 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[(2 - 3*x + 4*x^2)/(3 - 4*x + 4*x^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 5.67476, size = 32, normalized size = 0.84 \[ x + \frac{\log{\left (4 x^{2} - 4 x + 3 \right )}}{8} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (x - \frac{1}{2}\right ) \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((4*x**2-3*x+2)/(4*x**2-4*x+3),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0168122, size = 38, normalized size = 1. \[ \frac{1}{8} \log \left (4 x^2-4 x+3\right )+x-\frac{\tan ^{-1}\left (\frac{2 x-1}{\sqrt{2}}\right )}{4 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Integrate[(2 - 3*x + 4*x^2)/(3 - 4*x + 4*x^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.009, size = 32, normalized size = 0.8 \[ x+{\frac{\ln \left ( 4\,{x}^{2}-4\,x+3 \right ) }{8}}-{\frac{\sqrt{2}}{8}\arctan \left ({\frac{ \left ( 8\,x-4 \right ) \sqrt{2}}{8}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((4*x^2-3*x+2)/(4*x^2-4*x+3),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.51412, size = 42, normalized size = 1.11 \[ -\frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - 1\right )}\right ) + x + \frac{1}{8} \, \log \left (4 \, x^{2} - 4 \, x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x^2 - 3*x + 2)/(4*x^2 - 4*x + 3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.233421, size = 54, normalized size = 1.42 \[ \frac{1}{16} \, \sqrt{2}{\left (8 \, \sqrt{2} x + \sqrt{2} \log \left (4 \, x^{2} - 4 \, x + 3\right ) - 2 \, \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - 1\right )}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x^2 - 3*x + 2)/(4*x^2 - 4*x + 3),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 0.119971, size = 34, normalized size = 0.89 \[ x + \frac{\log{\left (x^{2} - x + \frac{3}{4} \right )}}{8} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} x - \frac{\sqrt{2}}{2} \right )}}{8} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x**2-3*x+2)/(4*x**2-4*x+3),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.20602, size = 42, normalized size = 1.11 \[ -\frac{1}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (2 \, x - 1\right )}\right ) + x + \frac{1}{8} \,{\rm ln}\left (4 \, x^{2} - 4 \, x + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((4*x^2 - 3*x + 2)/(4*x^2 - 4*x + 3),x, algorithm="giac")
[Out]