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]

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

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Rubi [A]  time = 0.00, antiderivative size = 29, normalized size of antiderivative = 1.00, 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 31

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

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]

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*}

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Mathematica [A]  time = 0.01, size = 26, normalized size = 0.90 \[ -\frac {(2-3 x) \log (2-3 x)}{3 \sqrt {(2-3 x)^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[4 - 12*x + 9*x^2],x]

[Out]

-1/3*((2 - 3*x)*Log[2 - 3*x])/Sqrt[(2 - 3*x)^2]

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fricas [A]  time = 0.87, size = 8, normalized size = 0.28 \[ \frac {1}{3} \, \log \left (3 \, x - 2\right ) \]

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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giac [A]  time = 0.46, size = 15, normalized size = 0.52 \[ \frac {1}{3} \, \log \left ({\left | 3 \, x - 2 \right |}\right ) \mathrm {sgn}\left (3 \, x - 2\right ) \]

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)

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maple [A]  time = 0.05, size = 23, normalized size = 0.79 \[ \frac {\left (3 x -2\right ) \ln \left (3 x -2\right )}{3 \sqrt {\left (3 x -2\right )^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((3*x-2)^2)^(1/2),x)

[Out]

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

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maxima [A]  time = 2.97, size = 6, normalized size = 0.21 \[ \frac {1}{3} \, \log \left (x - \frac {2}{3}\right ) \]

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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mupad [B]  time = 0.35, size = 14, normalized size = 0.48 \[ \frac {\ln \left (3\,x-2\right )\,\mathrm {sign}\left (3\,x-2\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((3*x - 2)^2)^(1/2),x)

[Out]

(log(3*x - 2)*sign(3*x - 2))/3

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sympy [A]  time = 0.08, 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

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