3.25.20 \(\int \frac {1}{x \sqrt {4-12 x+9 x^2}} \, dx\) [2420]

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

[Out]

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

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(27)=54\).
time = 0.01, 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, Rules used = {660, 36, 29, 31} \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[4 - 12*x + 9*x^2]),x]

[Out]

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 660

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps

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

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Mathematica [A]
time = 0.01, size = 31, normalized size = 1.15 \begin {gather*} \frac {(-2+3 x) (\log (2-3 x)-\log (x))}{2 \sqrt {(2-3 x)^2}} \end {gather*}

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 = 0.57, size = 28, normalized size = 1.04

method	result	size
default	\(\frac {\left (-2+3 x \right ) \left (\ln \left (-2+3 x \right )-\ln \left (x \right )\right )}{2 \sqrt {\left (-2+3 x \right )^{2}}}\)	\(28\)
risch	\(\frac {\sqrt {\left (-2+3 x \right )^{2}}\, \ln \left (-2+3 x \right )}{6 x -4}-\frac {\sqrt {\left (-2+3 x \right )^{2}}\, \ln \left (x \right )}{2 \left (-2+3 x \right )}\)	\(46\)
meijerg	\(\frac {-\ln \left (1-\frac {3 x}{2}\right )+\ln \left (x \right )-\ln \left (2\right )+\ln \left (3\right )+i \pi }{\sqrt {\left (-2+3 x \right )^{2}}}-\frac {3 x \left (-\ln \left (1-\frac {3 x}{2}\right )+\ln \left (x \right )-\ln \left (2\right )+\ln \left (3\right )+i \pi \right )}{2 \sqrt {\left (-2+3 x \right )^{2}}}\)	\(66\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 24, normalized size = 0.89 \begin {gather*} -\frac {1}{2} \, \left (-1\right )^{-12 \, x + 8} \log \left (-\frac {12 \, x}{{\left | x \right |}} + \frac {8}{{\left | x \right |}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/((-2+3*x)^2)^(1/2),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 = 3.11, size = 13, normalized size = 0.48 \begin {gather*} \frac {1}{2} \, \log \left (3 \, x - 2\right ) - \frac {1}{2} \, \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*log(3*x - 2) - 1/2*log(x)

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Sympy [A]
time = 0.03, size = 12, normalized size = 0.44 \begin {gather*} - \frac {\log {\left (x \right )}}{2} + \frac {\log {\left (x - \frac {2}{3} \right )}}{2} \end {gather*}

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 [A]
time = 2.03, size = 21, normalized size = 0.78 \begin {gather*} \frac {1}{2} \, {\left (\log \left ({\left | 3 \, x - 2 \right |}\right ) - \log \left ({\left | x \right |}\right )\right )} \mathrm {sgn}\left (3 \, x - 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(log(abs(3*x - 2)) - log(abs(x)))*sgn(3*x - 2)

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Mupad [B]
time = 1.23, size = 23, normalized size = 0.85 \begin {gather*} -\frac {\ln \left (\frac {2\,\sqrt {{\left (3\,x-2\right )}^2}-6\,x+4}{x}\right )}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*((3*x - 2)^2)^(1/2)),x)

[Out]

-log((2*((3*x - 2)^2)^(1/2) - 6*x + 4)/x)/2

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