Integrand size = 32, antiderivative size = 22 \[ \int \frac {-4+40 x \log (x)-2 x \log ^2(x)-x \log ^3(x)}{64 x \log (x)} \, dx=\frac {1}{16} \left (x \left (10-\frac {\log ^2(x)}{4}\right )-\log (\log (x))\right ) \]
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Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {12, 6820, 2339, 29, 2332, 2333} \[ \int \frac {-4+40 x \log (x)-2 x \log ^2(x)-x \log ^3(x)}{64 x \log (x)} \, dx=\frac {5 x}{8}-\frac {1}{64} x \log ^2(x)-\frac {1}{16} \log (\log (x)) \]
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Rule 12
Rule 29
Rule 2332
Rule 2333
Rule 2339
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \frac {1}{64} \int \frac {-4+40 x \log (x)-2 x \log ^2(x)-x \log ^3(x)}{x \log (x)} \, dx \\ & = \frac {1}{64} \int \left (40-\frac {4}{x \log (x)}-2 \log (x)-\log ^2(x)\right ) \, dx \\ & = \frac {5 x}{8}-\frac {1}{64} \int \log ^2(x) \, dx-\frac {1}{32} \int \log (x) \, dx-\frac {1}{16} \int \frac {1}{x \log (x)} \, dx \\ & = \frac {21 x}{32}-\frac {1}{32} x \log (x)-\frac {1}{64} x \log ^2(x)+\frac {1}{32} \int \log (x) \, dx-\frac {1}{16} \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (x)\right ) \\ & = \frac {5 x}{8}-\frac {1}{64} x \log ^2(x)-\frac {1}{16} \log (\log (x)) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {-4+40 x \log (x)-2 x \log ^2(x)-x \log ^3(x)}{64 x \log (x)} \, dx=\frac {5 x}{8}-\frac {1}{64} x \log ^2(x)-\frac {1}{16} \log (\log (x)) \]
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Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77
method | result | size |
default | \(-\frac {x \ln \left (x \right )^{2}}{64}+\frac {5 x}{8}-\frac {\ln \left (\ln \left (x \right )\right )}{16}\) | \(17\) |
norman | \(-\frac {x \ln \left (x \right )^{2}}{64}+\frac {5 x}{8}-\frac {\ln \left (\ln \left (x \right )\right )}{16}\) | \(17\) |
risch | \(-\frac {x \ln \left (x \right )^{2}}{64}+\frac {5 x}{8}-\frac {\ln \left (\ln \left (x \right )\right )}{16}\) | \(17\) |
parallelrisch | \(-\frac {x \ln \left (x \right )^{2}}{64}+\frac {5 x}{8}-\frac {\ln \left (\ln \left (x \right )\right )}{16}\) | \(17\) |
parts | \(-\frac {x \ln \left (x \right )^{2}}{64}+\frac {5 x}{8}-\frac {\ln \left (\ln \left (x \right )\right )}{16}\) | \(17\) |
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Time = 0.23 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {-4+40 x \log (x)-2 x \log ^2(x)-x \log ^3(x)}{64 x \log (x)} \, dx=-\frac {1}{64} \, x \log \left (x\right )^{2} + \frac {5}{8} \, x - \frac {1}{16} \, \log \left (\log \left (x\right )\right ) \]
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Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {-4+40 x \log (x)-2 x \log ^2(x)-x \log ^3(x)}{64 x \log (x)} \, dx=- \frac {x \log {\left (x \right )}^{2}}{64} + \frac {5 x}{8} - \frac {\log {\left (\log {\left (x \right )} \right )}}{16} \]
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Time = 0.20 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {-4+40 x \log (x)-2 x \log ^2(x)-x \log ^3(x)}{64 x \log (x)} \, dx=-\frac {1}{64} \, {\left (\log \left (x\right )^{2} - 2 \, \log \left (x\right ) + 2\right )} x - \frac {1}{32} \, x \log \left (x\right ) + \frac {21}{32} \, x - \frac {1}{16} \, \log \left (\log \left (x\right )\right ) \]
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Time = 0.26 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {-4+40 x \log (x)-2 x \log ^2(x)-x \log ^3(x)}{64 x \log (x)} \, dx=-\frac {1}{64} \, x \log \left (x\right )^{2} + \frac {5}{8} \, x - \frac {1}{16} \, \log \left (\log \left (x\right )\right ) \]
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Time = 7.68 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {-4+40 x \log (x)-2 x \log ^2(x)-x \log ^3(x)}{64 x \log (x)} \, dx=\frac {5\,x}{8}-\frac {\ln \left (\ln \left (x\right )\right )}{16}-\frac {x\,{\ln \left (x\right )}^2}{64} \]
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