Optimal. Leaf size=27 \[ -\frac{(2-3 x) \tanh ^{-1}(1-3 x)}{\sqrt{-9 x^2+12 x-4}} \]
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Rubi [B] time = 0.0129094, 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 = {646, 36, 31, 29} \[ \frac{(2-3 x) \log (x)}{2 \sqrt{-9 x^2+12 x-4}}-\frac{(2-3 x) \log (2-3 x)}{2 \sqrt{-9 x^2+12 x-4}} \]
Antiderivative was successfully verified.
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Rule 646
Rule 36
Rule 31
Rule 29
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*}
Mathematica [A] time = 0.0100771, size = 33, normalized size = 1.22 \[ \frac{(3 x-2) (\log (2-3 x)-\log (x))}{2 \sqrt{-(2-3 x)^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.103, size = 30, normalized size = 1.1 \begin{align*} -{\frac{ \left ( -2+3\,x \right ) \left ( \ln \left ( x \right ) -\ln \left ( -2+3\,x \right ) \right ) }{2}{\frac{1}{\sqrt{- \left ( -2+3\,x \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.46814, size = 32, normalized size = 1.19 \begin{align*} -\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{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 2.09051, size = 49, normalized size = 1.81 \begin{align*} -\frac{1}{2} i \, \log \left (x - \frac{2}{3}\right ) + \frac{1}{2} i \, \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{- \left (3 x - 2\right )^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{undef} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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