Integrand size = 34, antiderivative size = 13 \[ \int \frac {-4 x+(-3+4 x) \log \left (\frac {1}{3} (-3+4 x)\right )}{-3 x^2+4 x^3} \, dx=-\frac {\log \left (-1+\frac {4 x}{3}\right )}{x} \]
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Time = 0.08 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.206, Rules used = {1607, 6820, 14, 36, 29, 31, 2442} \[ \int \frac {-4 x+(-3+4 x) \log \left (\frac {1}{3} (-3+4 x)\right )}{-3 x^2+4 x^3} \, dx=-\frac {\log \left (\frac {4 x}{3}-1\right )}{x} \]
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Rule 14
Rule 29
Rule 31
Rule 36
Rule 1607
Rule 2442
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {-4 x+(-3+4 x) \log \left (\frac {1}{3} (-3+4 x)\right )}{x^2 (-3+4 x)} \, dx \\ & = \int \frac {\frac {4 x}{3-4 x}+\log \left (-1+\frac {4 x}{3}\right )}{x^2} \, dx \\ & = \int \left (-\frac {4}{x (-3+4 x)}+\frac {\log \left (-1+\frac {4 x}{3}\right )}{x^2}\right ) \, dx \\ & = -\left (4 \int \frac {1}{x (-3+4 x)} \, dx\right )+\int \frac {\log \left (-1+\frac {4 x}{3}\right )}{x^2} \, dx \\ & = -\frac {\log \left (-1+\frac {4 x}{3}\right )}{x}+\frac {4}{3} \int \frac {1}{x} \, dx+\frac {4}{3} \int \frac {1}{x \left (-1+\frac {4 x}{3}\right )} \, dx-\frac {16}{3} \int \frac {1}{-3+4 x} \, dx \\ & = -\frac {4}{3} \log (3-4 x)+\frac {4 \log (x)}{3}-\frac {\log \left (-1+\frac {4 x}{3}\right )}{x}-\frac {4}{3} \int \frac {1}{x} \, dx+\frac {16}{9} \int \frac {1}{-1+\frac {4 x}{3}} \, dx \\ & = -\frac {\log \left (-1+\frac {4 x}{3}\right )}{x} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {-4 x+(-3+4 x) \log \left (\frac {1}{3} (-3+4 x)\right )}{-3 x^2+4 x^3} \, dx=-\frac {\log \left (-1+\frac {4 x}{3}\right )}{x} \]
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Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92
method | result | size |
norman | \(-\frac {\ln \left (\frac {4 x}{3}-1\right )}{x}\) | \(12\) |
risch | \(-\frac {\ln \left (\frac {4 x}{3}-1\right )}{x}\) | \(12\) |
parallelrisch | \(-\frac {\ln \left (\frac {4 x}{3}-1\right )}{x}\) | \(12\) |
derivativedivides | \(-\frac {4 \ln \left (\frac {4 x}{3}-1\right )}{3}+\frac {\ln \left (\frac {4 x}{3}-1\right ) \left (\frac {4 x}{3}-1\right )}{x}\) | \(25\) |
default | \(-\frac {4 \ln \left (\frac {4 x}{3}-1\right )}{3}+\frac {\ln \left (\frac {4 x}{3}-1\right ) \left (\frac {4 x}{3}-1\right )}{x}\) | \(25\) |
parts | \(-\frac {4 \ln \left (\frac {4 x}{3}\right )}{3}+\frac {\ln \left (\frac {4 x}{3}-1\right ) \left (\frac {4 x}{3}-1\right )}{x}+\frac {4 \ln \left (x \right )}{3}-\frac {4 \ln \left (-3+4 x \right )}{3}\) | \(35\) |
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Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {-4 x+(-3+4 x) \log \left (\frac {1}{3} (-3+4 x)\right )}{-3 x^2+4 x^3} \, dx=-\frac {\log \left (\frac {4}{3} \, x - 1\right )}{x} \]
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Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {-4 x+(-3+4 x) \log \left (\frac {1}{3} (-3+4 x)\right )}{-3 x^2+4 x^3} \, dx=- \frac {\log {\left (\frac {4 x}{3} - 1 \right )}}{x} \]
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Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (11) = 22\).
Time = 0.32 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.38 \[ \int \frac {-4 x+(-3+4 x) \log \left (\frac {1}{3} (-3+4 x)\right )}{-3 x^2+4 x^3} \, dx=\frac {{\left (4 \, x - 3\right )} \log \left (4 \, x - 3\right ) + 3 \, \log \left (3\right )}{3 \, x} - \frac {4}{3} \, \log \left (4 \, x - 3\right ) \]
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Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {-4 x+(-3+4 x) \log \left (\frac {1}{3} (-3+4 x)\right )}{-3 x^2+4 x^3} \, dx=-\frac {\log \left (\frac {4}{3} \, x - 1\right )}{x} \]
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Time = 0.17 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85 \[ \int \frac {-4 x+(-3+4 x) \log \left (\frac {1}{3} (-3+4 x)\right )}{-3 x^2+4 x^3} \, dx=-\frac {\ln \left (\frac {4\,x}{3}-1\right )}{x} \]
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