Integrand size = 18, antiderivative size = 27 \[ \int \frac {1}{x \sqrt {-4+12 x-9 x^2}} \, dx=-\frac {(2-3 x) \text {arctanh}(1-3 x)}{\sqrt {-4+12 x-9 x^2}} \]
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Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(27)=54\).
Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {660, 36, 31, 29} \[ \int \frac {1}{x \sqrt {-4+12 x-9 x^2}} \, dx=\frac {(2-3 x) \log (x)}{2 \sqrt {-9 x^2+12 x-4}}-\frac {(2-3 x) \log (2-3 x)}{2 \sqrt {-9 x^2+12 x-4}} \]
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Rule 29
Rule 31
Rule 36
Rule 660
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 1.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x \sqrt {-4+12 x-9 x^2}} \, dx=\frac {(-2+3 x) (\log (2-3 x)-\log (x))}{2 \sqrt {-(2-3 x)^2}} \]
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Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
method | result | size |
meijerg | \(\frac {i \left (\ln \left (x \right )+\ln \left (3\right )-\ln \left (2\right )+i \pi -\ln \left (1-\frac {3 x}{2}\right )\right )}{2}\) | \(25\) |
default | \(-\frac {\left (-2+3 x \right ) \left (\ln \left (x \right )-\ln \left (-2+3 x \right )\right )}{2 \sqrt {-\left (-2+3 x \right )^{2}}}\) | \(30\) |
risch | \(-\frac {\left (-2+3 x \right ) \ln \left (x \right )}{2 \sqrt {-\left (-2+3 x \right )^{2}}}+\frac {\left (-2+3 x \right ) \ln \left (-2+3 x \right )}{2 \sqrt {-\left (-2+3 x \right )^{2}}}\) | \(46\) |
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Result contains complex when optimal does not.
Time = 0.42 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.41 \[ \int \frac {1}{x \sqrt {-4+12 x-9 x^2}} \, dx=-\frac {1}{2} i \, \log \left (x - \frac {2}{3}\right ) + \frac {1}{2} i \, \log \left (x\right ) \]
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\[ \int \frac {1}{x \sqrt {-4+12 x-9 x^2}} \, dx=\int \frac {1}{x \sqrt {- \left (3 x - 2\right )^{2}}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \sqrt {-4+12 x-9 x^2}} \, dx=-\frac {1}{2} i \, \left (-1\right )^{-12 \, x + 8} \log \left (-\frac {12 \, x}{{\left | x \right |}} + \frac {8}{{\left | x \right |}}\right ) \]
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x \sqrt {-4+12 x-9 x^2}} \, dx=\frac {i \, \log \left ({\left | 3 \, x - 2 \right |}\right )}{2 \, \mathrm {sgn}\left (-3 \, x + 2\right )} - \frac {i \, \log \left ({\left | x \right |}\right )}{2 \, \mathrm {sgn}\left (-3 \, x + 2\right )} \]
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Time = 10.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {-4+12 x-9 x^2}} \, dx=\frac {\ln \left (\frac {6\,x-4+\sqrt {-{\left (3\,x-2\right )}^2}\,2{}\mathrm {i}}{x}\right )\,1{}\mathrm {i}}{2} \]
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