Integrand size = 39, antiderivative size = 27 \[ \int \frac {-2-2 x+2 x \log (3 x)+\left (1-3 x+64 x^2\right ) \log ^2(3 x)}{x \log ^2(3 x)} \, dx=-x-2 \left (-2+x-16 x^2\right )+\log (x)+\frac {2 (1+x)}{\log (3 x)} \]
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Time = 0.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {6874, 14, 2395, 2334, 2335, 2339, 30} \[ \int \frac {-2-2 x+2 x \log (3 x)+\left (1-3 x+64 x^2\right ) \log ^2(3 x)}{x \log ^2(3 x)} \, dx=32 x^2-3 x+\frac {2 x}{\log (3 x)}+\log (x)+\frac {2}{\log (3 x)} \]
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Rule 14
Rule 30
Rule 2334
Rule 2335
Rule 2339
Rule 2395
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1-3 x+64 x^2}{x}-\frac {2 (1+x)}{x \log ^2(3 x)}+\frac {2}{\log (3 x)}\right ) \, dx \\ & = -\left (2 \int \frac {1+x}{x \log ^2(3 x)} \, dx\right )+2 \int \frac {1}{\log (3 x)} \, dx+\int \frac {1-3 x+64 x^2}{x} \, dx \\ & = \frac {2 \operatorname {LogIntegral}(3 x)}{3}-2 \int \left (\frac {1}{\log ^2(3 x)}+\frac {1}{x \log ^2(3 x)}\right ) \, dx+\int \left (-3+\frac {1}{x}+64 x\right ) \, dx \\ & = -3 x+32 x^2+\log (x)+\frac {2 \operatorname {LogIntegral}(3 x)}{3}-2 \int \frac {1}{\log ^2(3 x)} \, dx-2 \int \frac {1}{x \log ^2(3 x)} \, dx \\ & = -3 x+32 x^2+\log (x)+\frac {2 x}{\log (3 x)}+\frac {2 \operatorname {LogIntegral}(3 x)}{3}-2 \int \frac {1}{\log (3 x)} \, dx-2 \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (3 x)\right ) \\ & = -3 x+32 x^2+\log (x)+\frac {2}{\log (3 x)}+\frac {2 x}{\log (3 x)} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04 \[ \int \frac {-2-2 x+2 x \log (3 x)+\left (1-3 x+64 x^2\right ) \log ^2(3 x)}{x \log ^2(3 x)} \, dx=-3 x+32 x^2+\log (x)+\frac {2}{\log (3 x)}+\frac {2 x}{\log (3 x)} \]
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Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85
method | result | size |
risch | \(32 x^{2}-3 x +\ln \left (x \right )+\frac {2+2 x}{\ln \left (3 x \right )}\) | \(23\) |
parts | \(32 x^{2}-3 x +\ln \left (x \right )+\frac {2}{\ln \left (3 x \right )}+\frac {2 x}{\ln \left (3 x \right )}\) | \(29\) |
derivativedivides | \(32 x^{2}-3 x +\ln \left (3 x \right )+\frac {2 x}{\ln \left (3 x \right )}+\frac {2}{\ln \left (3 x \right )}\) | \(31\) |
default | \(32 x^{2}-3 x +\ln \left (3 x \right )+\frac {2 x}{\ln \left (3 x \right )}+\frac {2}{\ln \left (3 x \right )}\) | \(31\) |
norman | \(\frac {2+\ln \left (3 x \right )^{2}+2 x -3 x \ln \left (3 x \right )+32 x^{2} \ln \left (3 x \right )}{\ln \left (3 x \right )}\) | \(35\) |
parallelrisch | \(\frac {2+\ln \left (3 x \right )^{2}+2 x -3 x \ln \left (3 x \right )+32 x^{2} \ln \left (3 x \right )}{\ln \left (3 x \right )}\) | \(35\) |
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Time = 0.24 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.19 \[ \int \frac {-2-2 x+2 x \log (3 x)+\left (1-3 x+64 x^2\right ) \log ^2(3 x)}{x \log ^2(3 x)} \, dx=\frac {{\left (32 \, x^{2} - 3 \, x\right )} \log \left (3 \, x\right ) + \log \left (3 \, x\right )^{2} + 2 \, x + 2}{\log \left (3 \, x\right )} \]
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Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.74 \[ \int \frac {-2-2 x+2 x \log (3 x)+\left (1-3 x+64 x^2\right ) \log ^2(3 x)}{x \log ^2(3 x)} \, dx=32 x^{2} - 3 x + \frac {2 x + 2}{\log {\left (3 x \right )}} + \log {\left (x \right )} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.22 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \frac {-2-2 x+2 x \log (3 x)+\left (1-3 x+64 x^2\right ) \log ^2(3 x)}{x \log ^2(3 x)} \, dx=32 \, x^{2} - 3 \, x + \frac {2}{\log \left (3 \, x\right )} + \frac {2}{3} \, {\rm Ei}\left (\log \left (3 \, x\right )\right ) - \frac {2}{3} \, \Gamma \left (-1, -\log \left (3 \, x\right )\right ) + \log \left (x\right ) \]
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Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81 \[ \int \frac {-2-2 x+2 x \log (3 x)+\left (1-3 x+64 x^2\right ) \log ^2(3 x)}{x \log ^2(3 x)} \, dx=32 \, x^{2} - 3 \, x + \frac {2 \, {\left (x + 1\right )}}{\log \left (3 \, x\right )} + \log \left (x\right ) \]
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Time = 8.59 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int \frac {-2-2 x+2 x \log (3 x)+\left (1-3 x+64 x^2\right ) \log ^2(3 x)}{x \log ^2(3 x)} \, dx=\ln \left (x\right )-3\,x+\frac {2\,x+2}{\ln \left (3\,x\right )}+32\,x^2 \]
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