Integrand size = 27, antiderivative size = 14 \[ \int \frac {80+36 x+(-40+18 x) \log (x)-9 x \log ^2(x)}{9 x} \, dx=\left (\frac {20}{9}+x\right ) (4-\log (x)) \log (x) \]
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Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.93, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {12, 14, 45, 2388, 2338, 2332, 2333} \[ \int \frac {80+36 x+(-40+18 x) \log (x)-9 x \log ^2(x)}{9 x} \, dx=-x \log ^2(x)-\frac {20 \log ^2(x)}{9}+4 x \log (x)+\frac {80 \log (x)}{9} \]
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Rule 12
Rule 14
Rule 45
Rule 2332
Rule 2333
Rule 2338
Rule 2388
Rubi steps \begin{align*} \text {integral}& = \frac {1}{9} \int \frac {80+36 x+(-40+18 x) \log (x)-9 x \log ^2(x)}{x} \, dx \\ & = \frac {1}{9} \int \left (\frac {4 (20+9 x)}{x}+\frac {2 (-20+9 x) \log (x)}{x}-9 \log ^2(x)\right ) \, dx \\ & = \frac {2}{9} \int \frac {(-20+9 x) \log (x)}{x} \, dx+\frac {4}{9} \int \frac {20+9 x}{x} \, dx-\int \log ^2(x) \, dx \\ & = -x \log ^2(x)+\frac {4}{9} \int \left (9+\frac {20}{x}\right ) \, dx+2 (2 \int \log (x) \, dx)-\frac {40}{9} \int \frac {\log (x)}{x} \, dx \\ & = 4 x+\frac {80 \log (x)}{9}-\frac {20 \log ^2(x)}{9}-x \log ^2(x)+2 (-2 x+2 x \log (x)) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.93 \[ \int \frac {80+36 x+(-40+18 x) \log (x)-9 x \log ^2(x)}{9 x} \, dx=\frac {80 \log (x)}{9}+4 x \log (x)-\frac {20 \log ^2(x)}{9}-x \log ^2(x) \]
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Time = 0.82 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57
method | result | size |
risch | \(\frac {\left (-9 x -20\right ) \ln \left (x \right )^{2}}{9}+4 x \ln \left (x \right )+\frac {80 \ln \left (x \right )}{9}\) | \(22\) |
default | \(-x \ln \left (x \right )^{2}+4 x \ln \left (x \right )-\frac {20 \ln \left (x \right )^{2}}{9}+\frac {80 \ln \left (x \right )}{9}\) | \(24\) |
norman | \(-x \ln \left (x \right )^{2}+4 x \ln \left (x \right )-\frac {20 \ln \left (x \right )^{2}}{9}+\frac {80 \ln \left (x \right )}{9}\) | \(24\) |
parallelrisch | \(-x \ln \left (x \right )^{2}+4 x \ln \left (x \right )-\frac {20 \ln \left (x \right )^{2}}{9}+\frac {80 \ln \left (x \right )}{9}\) | \(24\) |
parts | \(-x \ln \left (x \right )^{2}+4 x \ln \left (x \right )-\frac {20 \ln \left (x \right )^{2}}{9}+\frac {80 \ln \left (x \right )}{9}\) | \(24\) |
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Time = 0.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.50 \[ \int \frac {80+36 x+(-40+18 x) \log (x)-9 x \log ^2(x)}{9 x} \, dx=-\frac {1}{9} \, {\left (9 \, x + 20\right )} \log \left (x\right )^{2} + \frac {4}{9} \, {\left (9 \, x + 20\right )} \log \left (x\right ) \]
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Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.71 \[ \int \frac {80+36 x+(-40+18 x) \log (x)-9 x \log ^2(x)}{9 x} \, dx=4 x \log {\left (x \right )} + \left (- x - \frac {20}{9}\right ) \log {\left (x \right )}^{2} + \frac {80 \log {\left (x \right )}}{9} \]
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (13) = 26\).
Time = 0.22 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.29 \[ \int \frac {80+36 x+(-40+18 x) \log (x)-9 x \log ^2(x)}{9 x} \, dx=-{\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x + 2 \, x \log \left (x\right ) - \frac {20}{9} \, \log \left (x\right )^{2} + 2 \, x + \frac {80}{9} \, \log \left (x\right ) \]
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Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.50 \[ \int \frac {80+36 x+(-40+18 x) \log (x)-9 x \log ^2(x)}{9 x} \, dx=-\frac {1}{9} \, {\left (9 \, x + 20\right )} \log \left (x\right )^{2} + 4 \, x \log \left (x\right ) + \frac {80}{9} \, \log \left (x\right ) \]
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Time = 12.37 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {80+36 x+(-40+18 x) \log (x)-9 x \log ^2(x)}{9 x} \, dx=-\frac {\ln \left (x\right )\,\left (9\,x+20\right )\,\left (\ln \left (x\right )-4\right )}{9} \]
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