3.2409 \(\int \frac{1}{x \sqrt{4-12 x+9 x^2}} \, dx\)

Optimal. Leaf size=27 \[ -\frac{(2-3 x) \tanh ^{-1}(1-3 x)}{\sqrt{9 x^2-12 x+4}} \]

[Out]

-(((2 - 3*x)*ArcTanh[1 - 3*x])/Sqrt[4 - 12*x + 9*x^2])

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Rubi [B]  time = 0.0455112, antiderivative size = 55, normalized size of antiderivative = 2.04, number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{(2-3 x) \log (x)}{2 \sqrt{9 x^2-12 x+4}}-\frac{(2-3 x) \log (2-3 x)}{2 \sqrt{9 x^2-12 x+4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.38805, size = 49, normalized size = 1.81 \[ \frac{3 \sqrt{9 x^{2} - 12 x + 4} \log{\left (x \right )}}{2 \left (- 9 x + 6\right )} - \frac{3 \sqrt{9 x^{2} - 12 x + 4} \log{\left (- 3 x + 2 \right )}}{2 \left (- 9 x + 6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0286058, size = 31, normalized size = 1.15 \[ \frac{(3 x-2) (\log (2-3 x)-\log (x))}{2 \sqrt{(2-3 x)^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 28, normalized size = 1. \[ -{\frac{ \left ( -2+3\,x \right ) \left ( \ln \left ( x \right ) -\ln \left ( -2+3\,x \right ) \right ) }{2}{\frac{1}{\sqrt{ \left ( -2+3\,x \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/((-2+3*x)^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.746111, size = 32, normalized size = 1.19 \[ -\frac{1}{2} \, \left (-1\right )^{-12 \, x + 8} \log \left (-\frac{12 \, x}{{\left | x \right |}} + \frac{8}{{\left | x \right |}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(-1)^(-12*x + 8)*log(-12*x/abs(x) + 8/abs(x))

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Fricas [A]  time = 0.220857, size = 18, normalized size = 0.67 \[ \frac{1}{2} \, \log \left (3 \, x - 2\right ) - \frac{1}{2} \, \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log(3*x - 2) - 1/2*log(x)

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Sympy [A]  time = 0.2354, size = 12, normalized size = 0.44 \[ - \frac{\log{\left (x \right )}}{2} + \frac{\log{\left (x - \frac{2}{3} \right )}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(x)/2 + log(x - 2/3)/2

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GIAC/XCAS [A]  time = 0.20579, size = 28, normalized size = 1.04 \[ \frac{1}{2} \,{\left ({\rm ln}\left ({\left | 3 \, x - 2 \right |}\right ) -{\rm ln}\left ({\left | x \right |}\right )\right )}{\rm sign}\left (3 \, x - 2\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(ln(abs(3*x - 2)) - ln(abs(x)))*sign(3*x - 2)