3.69 \(\int \frac{1}{\sqrt{4-12 x+9 x^2}} \, dx\)

Optimal. Leaf size=29 \[ -\frac{(2-3 x) \log (2-3 x)}{3 \sqrt{9 x^2-12 x+4}} \]

[Out]

-((2 - 3*x)*Log[2 - 3*x])/(3*Sqrt[4 - 12*x + 9*x^2])

_______________________________________________________________________________________

Rubi [A]  time = 0.014126, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{(2-3 x) \log (2-3 x)}{3 \sqrt{9 x^2-12 x+4}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[4 - 12*x + 9*x^2],x]

[Out]

-((2 - 3*x)*Log[2 - 3*x])/(3*Sqrt[4 - 12*x + 9*x^2])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 1.75393, size = 27, normalized size = 0.93 \[ - \frac{\left (- 9 x + 6\right ) \log{\left (- 3 x + 2 \right )}}{9 \sqrt{9 x^{2} - 12 x + 4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/((-2+3*x)**2)**(1/2),x)

[Out]

-(-9*x + 6)*log(-3*x + 2)/(9*sqrt(9*x**2 - 12*x + 4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0180115, size = 26, normalized size = 0.9 \[ -\frac{(2-3 x) \log (2-3 x)}{3 \sqrt{(2-3 x)^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[4 - 12*x + 9*x^2],x]

[Out]

-((2 - 3*x)*Log[2 - 3*x])/(3*Sqrt[(2 - 3*x)^2])

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 23, normalized size = 0.8 \[{\frac{ \left ( -2+3\,x \right ) \ln \left ( -2+3\,x \right ) }{3}{\frac{1}{\sqrt{ \left ( -2+3\,x \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/((-2+3*x)^2)^(1/2),x)

[Out]

1/3/((-2+3*x)^2)^(1/2)*(-2+3*x)*ln(-2+3*x)

_______________________________________________________________________________________

Maxima [A]  time = 0.804821, size = 8, normalized size = 0.28 \[ \frac{1}{3} \, \log \left (x - \frac{2}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt((3*x - 2)^2),x, algorithm="maxima")

[Out]

1/3*log(x - 2/3)

_______________________________________________________________________________________

Fricas [A]  time = 0.233132, size = 11, normalized size = 0.38 \[ \frac{1}{3} \, \log \left (3 \, x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt((3*x - 2)^2),x, algorithm="fricas")

[Out]

1/3*log(3*x - 2)

_______________________________________________________________________________________

Sympy [A]  time = 0.116029, size = 7, normalized size = 0.24 \[ \frac{\log{\left (3 x - 2 \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-2+3*x)**2)**(1/2),x)

[Out]

log(3*x - 2)/3

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.210258, size = 20, normalized size = 0.69 \[ \frac{1}{3} \,{\rm ln}\left ({\left | 3 \, x - 2 \right |}\right ){\rm sign}\left (3 \, x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/sqrt((3*x - 2)^2),x, algorithm="giac")

[Out]

1/3*ln(abs(3*x - 2))*sign(3*x - 2)