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])

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Rubi [A]  time = 0.004519, 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, Rules used = {608, 31} \[ -\frac{(2-3 x) \log (2-3 x)}{3 \sqrt{9 x^2-12 x+4}} \]

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])

Rule 608

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[(b/2 + c*x)/Sqrt[a + b*x + c*x^2], Int[1/(b/2
+ c*x), x], x] /; FreeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{4-12 x+9 x^2}} \, dx &=\frac{(-6+9 x) \int \frac{1}{-6+9 x} \, dx}{\sqrt{4-12 x+9 x^2}}\\ &=-\frac{(2-3 x) \log (2-3 x)}{3 \sqrt{4-12 x+9 x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0107827, 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])

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Maple [A]  time = 0.125, size = 23, normalized size = 0.8 \begin{align*}{\frac{ \left ( -2+3\,x \right ) \ln \left ( -2+3\,x \right ) }{3}{\frac{1}{\sqrt{ \left ( -2+3\,x \right ) ^{2}}}}} \end{align*}

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)

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Maxima [A]  time = 1.67276, size = 8, normalized size = 0.28 \begin{align*} \frac{1}{3} \, \log \left (x - \frac{2}{3}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-2+3*x)^2)^(1/2),x, algorithm="maxima")

[Out]

1/3*log(x - 2/3)

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Fricas [A]  time = 2.06614, size = 24, normalized size = 0.83 \begin{align*} \frac{1}{3} \, \log \left (3 \, x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-2+3*x)^2)^(1/2),x, algorithm="fricas")

[Out]

1/3*log(3*x - 2)

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Sympy [A]  time = 0.09535, size = 7, normalized size = 0.24 \begin{align*} \frac{\log{\left (3 x - 2 \right )}}{3} \end{align*}

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

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Giac [A]  time = 1.62468, size = 20, normalized size = 0.69 \begin{align*} \frac{1}{3} \, \log \left ({\left | 3 \, x - 2 \right |}\right ) \mathrm{sgn}\left (3 \, x - 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/((-2+3*x)^2)^(1/2),x, algorithm="giac")

[Out]

1/3*log(abs(3*x - 2))*sgn(3*x - 2)