Integrand size = 25, antiderivative size = 27 \[ \int \frac {(-3+3 \log (x)) (i \pi +\log (\log (1+5 \log (4))))}{\log ^2(x)} \, dx=\frac {3 x \left (i \pi +\log \left (-\log \left (\frac {5}{5+25 \log (4)}\right )\right )\right )}{\log (x)} \]
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Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {12, 2407, 2334, 2335} \[ \int \frac {(-3+3 \log (x)) (i \pi +\log (\log (1+5 \log (4))))}{\log ^2(x)} \, dx=\frac {3 x (\log (\log (1+5 \log (4)))+i \pi )}{\log (x)} \]
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Rule 12
Rule 2334
Rule 2335
Rule 2407
Rubi steps \begin{align*} \text {integral}& = (i \pi +\log (\log (1+5 \log (4)))) \int \frac {-3+3 \log (x)}{\log ^2(x)} \, dx \\ & = (i \pi +\log (\log (1+5 \log (4)))) \int \left (-\frac {3}{\log ^2(x)}+\frac {3}{\log (x)}\right ) \, dx \\ & = -\left ((3 (i \pi +\log (\log (1+5 \log (4))))) \int \frac {1}{\log ^2(x)} \, dx\right )+(3 (i \pi +\log (\log (1+5 \log (4))))) \int \frac {1}{\log (x)} \, dx \\ & = \frac {3 x (i \pi +\log (\log (1+5 \log (4))))}{\log (x)}+3 (i \pi +\log (\log (1+5 \log (4)))) \text {li}(x)-(3 (i \pi +\log (\log (1+5 \log (4))))) \int \frac {1}{\log (x)} \, dx \\ & = \frac {3 x (i \pi +\log (\log (1+5 \log (4))))}{\log (x)} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78 \[ \int \frac {(-3+3 \log (x)) (i \pi +\log (\log (1+5 \log (4))))}{\log ^2(x)} \, dx=\frac {3 x (i \pi +\log (\log (1+5 \log (4))))}{\log (x)} \]
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Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67
method | result | size |
default | \(\frac {3 \ln \left (-\ln \left (10 \ln \left (2\right )+1\right )\right ) x}{\ln \left (x \right )}\) | \(18\) |
parallelrisch | \(\frac {3 \ln \left (-\ln \left (10 \ln \left (2\right )+1\right )\right ) x}{\ln \left (x \right )}\) | \(18\) |
risch | \(\frac {3 \left (\ln \left (\ln \left (10 \ln \left (2\right )+1\right )\right )+i \pi \right ) x}{\ln \left (x \right )}\) | \(21\) |
norman | \(\frac {\left (3 \ln \left (\ln \left (10 \ln \left (2\right )+1\right )\right )+3 i \pi \right ) x}{\ln \left (x \right )}\) | \(22\) |
parts | \(-3 \ln \left (-\ln \left (10 \ln \left (2\right )+1\right )\right ) \left (-\frac {x}{\ln \left (x \right )}-\operatorname {Ei}_{1}\left (-\ln \left (x \right )\right )\right )-3 \ln \left (-\ln \left (10 \ln \left (2\right )+1\right )\right ) \operatorname {Ei}_{1}\left (-\ln \left (x \right )\right )\) | \(48\) |
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none
Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63 \[ \int \frac {(-3+3 \log (x)) (i \pi +\log (\log (1+5 \log (4))))}{\log ^2(x)} \, dx=\frac {3 \, x \log \left (-\log \left (10 \, \log \left (2\right ) + 1\right )\right )}{\log \left (x\right )} \]
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Time = 0.05 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {(-3+3 \log (x)) (i \pi +\log (\log (1+5 \log (4))))}{\log ^2(x)} \, dx=\frac {3 x \log {\left (\log {\left (1 + 10 \log {\left (2 \right )} \right )} \right )} + 3 i \pi x}{\log {\left (x \right )}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {(-3+3 \log (x)) (i \pi +\log (\log (1+5 \log (4))))}{\log ^2(x)} \, dx=3 \, {\left ({\rm Ei}\left (\log \left (x\right )\right ) - \Gamma \left (-1, -\log \left (x\right )\right )\right )} \log \left (-\log \left (10 \, \log \left (2\right ) + 1\right )\right ) \]
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none
Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63 \[ \int \frac {(-3+3 \log (x)) (i \pi +\log (\log (1+5 \log (4))))}{\log ^2(x)} \, dx=\frac {3 \, x \log \left (-\log \left (10 \, \log \left (2\right ) + 1\right )\right )}{\log \left (x\right )} \]
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Time = 11.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.63 \[ \int \frac {(-3+3 \log (x)) (i \pi +\log (\log (1+5 \log (4))))}{\log ^2(x)} \, dx=\frac {3\,x\,\ln \left (-\ln \left (10\,\ln \left (2\right )+1\right )\right )}{\ln \left (x\right )} \]
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