3.1.69 \(\int \frac {1}{\sqrt {4-12 x+9 x^2}} \, dx\) [69]

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

[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 = {622, 31} \begin {gather*} -\frac {(2-3 x) \log (2-3 x)}{3 \sqrt {9 x^2-12 x+4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[4 - 12*x + 9*x^2],x]

[Out]

-1/3*((2 - 3*x)*Log[2 - 3*x])/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 622

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 \begin {gather*} -\frac {(2-3 x) \log (2-3 x)}{3 \sqrt {(2-3 x)^2}} \end {gather*}

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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Maple [A]
time = 0.37, size = 23, normalized size = 0.79

method	result	size
default	\(\frac {\left (-2+3 x \right ) \ln \left (-2+3 x \right )}{3 \sqrt {\left (-2+3 x \right )^{2}}}\)	\(23\)
risch	\(\frac {\sqrt {\left (-2+3 x \right )^{2}}\, \ln \left (-2+3 x \right )}{-6+9 x}\)	\(25\)
meijerg	\(-\frac {2 \ln \left (1-\frac {3 x}{2}\right )}{3 \sqrt {\left (-2+3 x \right )^{2}}}+\frac {x \ln \left (1-\frac {3 x}{2}\right )}{\sqrt {\left (-2+3 x \right )^{2}}}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((-2+3*x)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.48, size = 6, normalized size = 0.21 \begin {gather*} \frac {1}{3} \, \log \left (x - \frac {2}{3}\right ) \end {gather*}

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 = 1.66, size = 8, normalized size = 0.28 \begin {gather*} \frac {1}{3} \, \log \left (3 \, x - 2\right ) \end {gather*}

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 = 0.01, size = 7, normalized size = 0.24 \begin {gather*} \frac {\log {\left (3 x - 2 \right )}}{3} \end {gather*}

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 = 1.50, size = 15, normalized size = 0.52 \begin {gather*} \frac {1}{3} \, \log \left ({\left | 3 \, x - 2 \right |}\right ) \mathrm {sgn}\left (3 \, x - 2\right ) \end {gather*}

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

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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