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

Optimal result

Integrand size = 14, antiderivative size = 29 \[ \int \frac {1}{\sqrt {4-12 x+9 x^2}} \, dx=-\frac {(2-3 x) \log (2-3 x)}{3 \sqrt {4-12 x+9 x^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {622, 31} \[ \int \frac {1}{\sqrt {4-12 x+9 x^2}} \, dx=-\frac {(2-3 x) \log (2-3 x)}{3 \sqrt {9 x^2-12 x+4}} \]

[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*} \text {integral}& = \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] (verified)

Time = 0.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\sqrt {4-12 x+9 x^2}} \, dx=-\frac {(2-3 x) \log (2-3 x)}{3 \sqrt {(2-3 x)^2}} \]

[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]

Maple [A] (verified)

Time = 1.91 (sec) , antiderivative size = 23, normalized size of antiderivative = 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	\(-\frac {2 \ln \left (1-\frac {3 x}{2}\right )}{3 \sqrt {\left (-2+3 x \right )^{2}}}+\frac {x \ln \left (1-\frac {3 x}{2}\right )}{\sqrt {\left (-2+3 x \right )^{2}}}\)	\(36\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.28 \[ \int \frac {1}{\sqrt {4-12 x+9 x^2}} \, dx=\frac {1}{3} \, \log \left (3 \, x - 2\right ) \]

[In]

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

[Out]

1/3*log(3*x - 2)

Sympy [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\sqrt {4-12 x+9 x^2}} \, dx=\frac {\left (x - \frac {2}{3}\right ) \log {\left (x - \frac {2}{3} \right )}}{3 \sqrt {\left (x - \frac {2}{3}\right )^{2}}} \]

[In]

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

[Out]

(x - 2/3)*log(x - 2/3)/(3*sqrt((x - 2/3)**2))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.21 \[ \int \frac {1}{\sqrt {4-12 x+9 x^2}} \, dx=\frac {1}{3} \, \log \left (x - \frac {2}{3}\right ) \]

[In]

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

[Out]

1/3*log(x - 2/3)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.52 \[ \int \frac {1}{\sqrt {4-12 x+9 x^2}} \, dx=\frac {1}{3} \, \log \left ({\left | 3 \, x - 2 \right |}\right ) \mathrm {sgn}\left (3 \, x - 2\right ) \]

[In]

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

[Out]

1/3*log(abs(3*x - 2))*sgn(3*x - 2)

Mupad [B] (verification not implemented)

Time = 9.15 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.48 \[ \int \frac {1}{\sqrt {4-12 x+9 x^2}} \, dx=\frac {\ln \left (3\,x-2\right )\,\mathrm {sign}\left (3\,x-2\right )}{3} \]

[In]

int(1/((3*x - 2)^2)^(1/2),x)

[Out]

(log(3*x - 2)*sign(3*x - 2))/3