Integrand size = 25, antiderivative size = 20 \[ \int \frac {-4 \log ^2(4)+(2+4 x) \log ^3(x)}{x \log ^3(x)} \, dx=2+2 \left (4+2 x+\frac {\log ^2(4)}{\log ^2(x)}+\log (x)\right ) \]
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Time = 0.09 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6874, 45, 2339, 30} \[ \int \frac {-4 \log ^2(4)+(2+4 x) \log ^3(x)}{x \log ^3(x)} \, dx=4 x+\frac {2 \log ^2(4)}{\log ^2(x)}+2 \log (x) \]
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Rule 30
Rule 45
Rule 2339
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {2 (1+2 x)}{x}-\frac {4 \log ^2(4)}{x \log ^3(x)}\right ) \, dx \\ & = 2 \int \frac {1+2 x}{x} \, dx-\left (4 \log ^2(4)\right ) \int \frac {1}{x \log ^3(x)} \, dx \\ & = 2 \int \left (2+\frac {1}{x}\right ) \, dx-\left (4 \log ^2(4)\right ) \text {Subst}\left (\int \frac {1}{x^3} \, dx,x,\log (x)\right ) \\ & = 4 x+\frac {2 \log ^2(4)}{\log ^2(x)}+2 \log (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {-4 \log ^2(4)+(2+4 x) \log ^3(x)}{x \log ^3(x)} \, dx=4 x+\frac {2 \log ^2(4)}{\log ^2(x)}+2 \log (x) \]
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Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95
method | result | size |
default | \(4 x +2 \ln \left (x \right )+\frac {8 \ln \left (2\right )^{2}}{\ln \left (x \right )^{2}}\) | \(19\) |
risch | \(4 x +2 \ln \left (x \right )+\frac {8 \ln \left (2\right )^{2}}{\ln \left (x \right )^{2}}\) | \(19\) |
parts | \(4 x +2 \ln \left (x \right )+\frac {8 \ln \left (2\right )^{2}}{\ln \left (x \right )^{2}}\) | \(19\) |
norman | \(\frac {2 \ln \left (x \right )^{3}+8 \ln \left (2\right )^{2}+4 x \ln \left (x \right )^{2}}{\ln \left (x \right )^{2}}\) | \(26\) |
parallelrisch | \(\frac {2 \ln \left (x \right )^{3}+8 \ln \left (2\right )^{2}+4 x \ln \left (x \right )^{2}}{\ln \left (x \right )^{2}}\) | \(26\) |
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Time = 0.25 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.20 \[ \int \frac {-4 \log ^2(4)+(2+4 x) \log ^3(x)}{x \log ^3(x)} \, dx=\frac {2 \, {\left (2 \, x \log \left (x\right )^{2} + \log \left (x\right )^{3} + 4 \, \log \left (2\right )^{2}\right )}}{\log \left (x\right )^{2}} \]
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Time = 0.06 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int \frac {-4 \log ^2(4)+(2+4 x) \log ^3(x)}{x \log ^3(x)} \, dx=4 x + 2 \log {\left (x \right )} + \frac {8 \log {\left (2 \right )}^{2}}{\log {\left (x \right )}^{2}} \]
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Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {-4 \log ^2(4)+(2+4 x) \log ^3(x)}{x \log ^3(x)} \, dx=4 \, x + \frac {8 \, \log \left (2\right )^{2}}{\log \left (x\right )^{2}} + 2 \, \log \left (x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {-4 \log ^2(4)+(2+4 x) \log ^3(x)}{x \log ^3(x)} \, dx=4 \, x + \frac {8 \, \log \left (2\right )^{2}}{\log \left (x\right )^{2}} + 2 \, \log \left (x\right ) \]
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Time = 9.40 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {-4 \log ^2(4)+(2+4 x) \log ^3(x)}{x \log ^3(x)} \, dx=4\,x+2\,\ln \left (x\right )+\frac {8\,{\ln \left (2\right )}^2}{{\ln \left (x\right )}^2} \]
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