Integrand size = 44, antiderivative size = 28 \[ \int \frac {20-40 x+20 x^2+\left (20-20 x^2\right ) \log (4 x)+(-40-x) \log ^2(4 x)}{x^2 \log ^2(4 x)} \, dx=\log \left (\frac {3}{x}\right )-\frac {20 \left (-2+\frac {(1-x)^2}{\log (4 x)}\right )}{x} \]
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Time = 0.23 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36, number of steps used = 18, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6874, 45, 2395, 2334, 2335, 2343, 2346, 2209, 2339, 30} \[ \int \frac {20-40 x+20 x^2+\left (20-20 x^2\right ) \log (4 x)+(-40-x) \log ^2(4 x)}{x^2 \log ^2(4 x)} \, dx=\frac {40}{x}-\frac {20 x}{\log (4 x)}-\log (x)+\frac {40}{\log (4 x)}-\frac {20}{x \log (4 x)} \]
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Rule 30
Rule 45
Rule 2209
Rule 2334
Rule 2335
Rule 2339
Rule 2343
Rule 2346
Rule 2395
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-40-x}{x^2}+\frac {20 (-1+x)^2}{x^2 \log ^2(4 x)}-\frac {20 \left (-1+x^2\right )}{x^2 \log (4 x)}\right ) \, dx \\ & = 20 \int \frac {(-1+x)^2}{x^2 \log ^2(4 x)} \, dx-20 \int \frac {-1+x^2}{x^2 \log (4 x)} \, dx+\int \frac {-40-x}{x^2} \, dx \\ & = 20 \int \left (\frac {1}{\log ^2(4 x)}+\frac {1}{x^2 \log ^2(4 x)}-\frac {2}{x \log ^2(4 x)}\right ) \, dx-20 \int \left (\frac {1}{\log (4 x)}-\frac {1}{x^2 \log (4 x)}\right ) \, dx+\int \left (-\frac {40}{x^2}-\frac {1}{x}\right ) \, dx \\ & = \frac {40}{x}-\log (x)+20 \int \frac {1}{\log ^2(4 x)} \, dx+20 \int \frac {1}{x^2 \log ^2(4 x)} \, dx-20 \int \frac {1}{\log (4 x)} \, dx+20 \int \frac {1}{x^2 \log (4 x)} \, dx-40 \int \frac {1}{x \log ^2(4 x)} \, dx \\ & = \frac {40}{x}-\log (x)-\frac {20}{x \log (4 x)}-\frac {20 x}{\log (4 x)}-5 \operatorname {LogIntegral}(4 x)+20 \int \frac {1}{\log (4 x)} \, dx-20 \int \frac {1}{x^2 \log (4 x)} \, dx-40 \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (4 x)\right )+80 \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (4 x)\right ) \\ & = \frac {40}{x}+80 \operatorname {ExpIntegralEi}(-\log (4 x))-\log (x)+\frac {40}{\log (4 x)}-\frac {20}{x \log (4 x)}-\frac {20 x}{\log (4 x)}-80 \text {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (4 x)\right ) \\ & = \frac {40}{x}-\log (x)+\frac {40}{\log (4 x)}-\frac {20}{x \log (4 x)}-\frac {20 x}{\log (4 x)} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.36 \[ \int \frac {20-40 x+20 x^2+\left (20-20 x^2\right ) \log (4 x)+(-40-x) \log ^2(4 x)}{x^2 \log ^2(4 x)} \, dx=\frac {40}{x}-\log (x)+\frac {40}{\log (4 x)}-\frac {20}{x \log (4 x)}-\frac {20 x}{\log (4 x)} \]
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Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14
method | result | size |
risch | \(-\frac {x \ln \left (x \right )-40}{x}-\frac {20 \left (x^{2}-2 x +1\right )}{x \ln \left (4 x \right )}\) | \(32\) |
norman | \(\frac {-20-x \ln \left (4 x \right )^{2}+40 x -20 x^{2}+40 \ln \left (4 x \right )}{x \ln \left (4 x \right )}\) | \(36\) |
parallelrisch | \(\frac {-20-x \ln \left (4 x \right )^{2}+40 x -20 x^{2}+40 \ln \left (4 x \right )}{x \ln \left (4 x \right )}\) | \(36\) |
parts | \(\frac {40}{x}-\ln \left (x \right )-\frac {20 x}{\ln \left (4 x \right )}+\frac {40}{\ln \left (4 x \right )}-\frac {20}{x \ln \left (4 x \right )}\) | \(39\) |
derivativedivides | \(-\ln \left (4 x \right )+\frac {40}{x}-\frac {20 x}{\ln \left (4 x \right )}+\frac {40}{\ln \left (4 x \right )}-\frac {20}{x \ln \left (4 x \right )}\) | \(41\) |
default | \(-\ln \left (4 x \right )+\frac {40}{x}-\frac {20 x}{\ln \left (4 x \right )}+\frac {40}{\ln \left (4 x \right )}-\frac {20}{x \ln \left (4 x \right )}\) | \(41\) |
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Time = 0.24 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.25 \[ \int \frac {20-40 x+20 x^2+\left (20-20 x^2\right ) \log (4 x)+(-40-x) \log ^2(4 x)}{x^2 \log ^2(4 x)} \, dx=-\frac {x \log \left (4 \, x\right )^{2} + 20 \, x^{2} - 40 \, x - 40 \, \log \left (4 \, x\right ) + 20}{x \log \left (4 \, x\right )} \]
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Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.79 \[ \int \frac {20-40 x+20 x^2+\left (20-20 x^2\right ) \log (4 x)+(-40-x) \log ^2(4 x)}{x^2 \log ^2(4 x)} \, dx=- \log {\left (x \right )} + \frac {- 20 x^{2} + 40 x - 20}{x \log {\left (4 x \right )}} + \frac {40}{x} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.86 \[ \int \frac {20-40 x+20 x^2+\left (20-20 x^2\right ) \log (4 x)+(-40-x) \log ^2(4 x)}{x^2 \log ^2(4 x)} \, dx=\frac {40}{x} + \frac {40}{\log \left (4 \, x\right )} + 80 \, {\rm Ei}\left (-\log \left (4 \, x\right )\right ) - 5 \, {\rm Ei}\left (\log \left (4 \, x\right )\right ) + 5 \, \Gamma \left (-1, -\log \left (4 \, x\right )\right ) - 80 \, \Gamma \left (-1, \log \left (4 \, x\right )\right ) - \log \left (x\right ) \]
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Time = 0.25 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {20-40 x+20 x^2+\left (20-20 x^2\right ) \log (4 x)+(-40-x) \log ^2(4 x)}{x^2 \log ^2(4 x)} \, dx=\frac {40}{x} - \frac {20 \, {\left (x^{2} - 2 \, x + 1\right )}}{x \log \left (4 \, x\right )} - \log \left (x\right ) \]
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Time = 8.78 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int \frac {20-40 x+20 x^2+\left (20-20 x^2\right ) \log (4 x)+(-40-x) \log ^2(4 x)}{x^2 \log ^2(4 x)} \, dx=\frac {40}{x}-\ln \left (x\right )-\frac {20\,x^2-40\,x+20}{x\,\ln \left (4\,x\right )} \]
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