3.25.21 \(\int \frac {1}{x \sqrt {-4+12 x-9 x^2}} \, dx\) [2421]

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, 31, 29} \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}{(6-9 x) 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 = 33, normalized size = 1.22 \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 = 0.65, size = 30, normalized size = 1.11

method	result	size
meijerg	\(\frac {i \left (-\ln \left (1-\frac {3 x}{2}\right )+\ln \left (x \right )-\ln \left (2\right )+\ln \left (3\right )+i \pi \right )}{2}\)	\(25\)
default	\(\frac {\left (-2+3 x \right ) \left (\ln \left (-2+3 x \right )-\ln \left (x \right )\right )}{2 \sqrt {-\left (-2+3 x \right )^{2}}}\)	\(30\)
risch	\(\frac {\left (-2+3 x \right ) \ln \left (-2+3 x \right )}{2 \sqrt {-\left (-2+3 x \right )^{2}}}-\frac {\left (-2+3 x \right ) \ln \left (x \right )}{2 \sqrt {-\left (-2+3 x \right )^{2}}}\)	\(46\)

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 [C] Result contains complex when optimal does not.
time = 0.49, size = 24, normalized size = 0.89 \begin {gather*} -\frac {1}{2} i \, \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*I*(-1)^(-12*x + 8)*log(-12*x/abs(x) + 8/abs(x))

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Fricas [C] Result contains complex when optimal does not.
time = 2.30, size = 11, normalized size = 0.41 \begin {gather*} -\frac {1}{2} i \, \log \left (x - \frac {2}{3}\right ) + \frac {1}{2} i \, \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*I*log(x - 2/3) + 1/2*I*log(x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \sqrt {- \left (3 x - 2\right )^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*sqrt(-(3*x - 2)**2)), x)

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Giac [C] Result contains complex when optimal does not.
time = 0.97, size = 31, normalized size = 1.15 \begin {gather*} \frac {i \, \log \left ({\left | 3 \, x - 2 \right |}\right )}{2 \, \mathrm {sgn}\left (-3 \, x + 2\right )} - \frac {i \, \log \left ({\left | x \right |}\right )}{2 \, \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*I*log(abs(3*x - 2))/sgn(-3*x + 2) - 1/2*I*log(abs(x))/sgn(-3*x + 2)

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Mupad [B]
time = 1.27, size = 27, normalized size = 1.00 \begin {gather*} \frac {\ln \left (\frac {6\,x-4+\sqrt {-{\left (3\,x-2\right )}^2}\,2{}\mathrm {i}}{x}\right )\,1{}\mathrm {i}}{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((6*x + (-(3*x - 2)^2)^(1/2)*2i - 4)/x)*1i)/2

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