Integrand size = 16, antiderivative size = 21 \[ \int \frac {(2-\log (x)) (3+\log (x))^2}{x} \, dx=\frac {5}{3} (3+\log (x))^3-\frac {1}{4} (3+\log (x))^4 \]
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Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2412, 45} \[ \int \frac {(2-\log (x)) (3+\log (x))^2}{x} \, dx=\frac {5}{3} (\log (x)+3)^3-\frac {1}{4} (\log (x)+3)^4 \]
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Rule 45
Rule 2412
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int (2-x) (3+x)^2 \, dx,x,\log (x)\right ) \\ & = \text {Subst}\left (\int \left (5 (3+x)^2-(3+x)^3\right ) \, dx,x,\log (x)\right ) \\ & = \frac {5}{3} (3+\log (x))^3-\frac {1}{4} (3+\log (x))^4 \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.38 \[ \int \frac {(2-\log (x)) (3+\log (x))^2}{x} \, dx=18 \log (x)+\frac {3 \log ^2(x)}{2}-\frac {4 \log ^3(x)}{3}-\frac {\log ^4(x)}{4} \]
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Time = 0.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14
method | result | size |
derivativedivides | \(-\frac {\ln \left (x \right )^{4}}{4}-\frac {4 \ln \left (x \right )^{3}}{3}+\frac {3 \ln \left (x \right )^{2}}{2}+18 \ln \left (x \right )\) | \(24\) |
default | \(-\frac {\ln \left (x \right )^{4}}{4}-\frac {4 \ln \left (x \right )^{3}}{3}+\frac {3 \ln \left (x \right )^{2}}{2}+18 \ln \left (x \right )\) | \(24\) |
norman | \(-\frac {\ln \left (x \right )^{4}}{4}-\frac {4 \ln \left (x \right )^{3}}{3}+\frac {3 \ln \left (x \right )^{2}}{2}+18 \ln \left (x \right )\) | \(24\) |
risch | \(-\frac {\ln \left (x \right )^{4}}{4}-\frac {4 \ln \left (x \right )^{3}}{3}+\frac {3 \ln \left (x \right )^{2}}{2}+18 \ln \left (x \right )\) | \(24\) |
parts | \(-\frac {\ln \left (x \right )^{4}}{4}-\frac {4 \ln \left (x \right )^{3}}{3}+\frac {3 \ln \left (x \right )^{2}}{2}+18 \ln \left (x \right )\) | \(24\) |
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none
Time = 0.29 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {(2-\log (x)) (3+\log (x))^2}{x} \, dx=-\frac {1}{4} \, \log \left (x\right )^{4} - \frac {4}{3} \, \log \left (x\right )^{3} + \frac {3}{2} \, \log \left (x\right )^{2} + 18 \, \log \left (x\right ) \]
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Time = 0.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29 \[ \int \frac {(2-\log (x)) (3+\log (x))^2}{x} \, dx=- \frac {\log {\left (x \right )}^{4}}{4} - \frac {4 \log {\left (x \right )}^{3}}{3} + \frac {3 \log {\left (x \right )}^{2}}{2} + 18 \log {\left (x \right )} \]
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none
Time = 0.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {(2-\log (x)) (3+\log (x))^2}{x} \, dx=-\frac {1}{4} \, \log \left (x\right )^{4} - \frac {4}{3} \, \log \left (x\right )^{3} + \frac {3}{2} \, \log \left (x\right )^{2} + 18 \, \log \left (x\right ) \]
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none
Time = 0.30 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {(2-\log (x)) (3+\log (x))^2}{x} \, dx=-\frac {1}{4} \, \log \left (x\right )^{4} - \frac {4}{3} \, \log \left (x\right )^{3} + \frac {3}{2} \, \log \left (x\right )^{2} + 18 \, \log \left (x\right ) \]
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Time = 1.46 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {(2-\log (x)) (3+\log (x))^2}{x} \, dx=\frac {\ln \left (x\right )\,\left (-3\,{\ln \left (x\right )}^3-16\,{\ln \left (x\right )}^2+18\,\ln \left (x\right )+216\right )}{12} \]
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