3.129 \(\int \frac{1}{\left (4-4 x+x^2\right ) \left (5-4 x+x^2\right )} \, dx\)

Optimal. Leaf size=14 \[ \frac{1}{2-x}+\tan ^{-1}(2-x) \]

[Out]

(2 - x)^(-1) + ArcTan[2 - x]

_______________________________________________________________________________________

Rubi [A]  time = 0.0239587, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ \frac{1}{2-x}+\tan ^{-1}(2-x) \]

Antiderivative was successfully verified.

[In]  Int[1/((4 - 4*x + x^2)*(5 - 4*x + x^2)),x]

[Out]

(2 - x)^(-1) + ArcTan[2 - x]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.55335, size = 10, normalized size = 0.71 \[ - \operatorname{atan}{\left (x - 2 \right )} + \frac{2}{- 2 x + 4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(x**2-4*x+4)/(x**2-4*x+5),x)

[Out]

-atan(x - 2) + 2/(-2*x + 4)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0123414, size = 14, normalized size = 1. \[ \tan ^{-1}(2-x)-\frac{1}{x-2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((4 - 4*x + x^2)*(5 - 4*x + x^2)),x]

[Out]

-(-2 + x)^(-1) + ArcTan[2 - x]

_______________________________________________________________________________________

Maple [A]  time = 0.009, size = 15, normalized size = 1.1 \[ -\arctan \left ( -2+x \right ) - \left ( -2+x \right ) ^{-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(x^2-4*x+4)/(x^2-4*x+5),x)

[Out]

-arctan(-2+x)-1/(-2+x)

_______________________________________________________________________________________

Maxima [A]  time = 1.48672, size = 19, normalized size = 1.36 \[ -\frac{1}{x - 2} - \arctan \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - 4*x + 5)*(x^2 - 4*x + 4)),x, algorithm="maxima")

[Out]

-1/(x - 2) - arctan(x - 2)

_______________________________________________________________________________________

Fricas [A]  time = 0.197063, size = 23, normalized size = 1.64 \[ -\frac{{\left (x - 2\right )} \arctan \left (x - 2\right ) + 1}{x - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - 4*x + 5)*(x^2 - 4*x + 4)),x, algorithm="fricas")

[Out]

-((x - 2)*arctan(x - 2) + 1)/(x - 2)

_______________________________________________________________________________________

Sympy [A]  time = 0.140555, size = 10, normalized size = 0.71 \[ - \operatorname{atan}{\left (x - 2 \right )} - \frac{1}{x - 2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(x**2-4*x+4)/(x**2-4*x+5),x)

[Out]

-atan(x - 2) - 1/(x - 2)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220908, size = 19, normalized size = 1.36 \[ -\frac{1}{x - 2} - \arctan \left (x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((x^2 - 4*x + 5)*(x^2 - 4*x + 4)),x, algorithm="giac")

[Out]

-1/(x - 2) - arctan(x - 2)