Integrand size = 32, antiderivative size = 19 \[ \int \frac {-x^2+x \log (4)+\left (3 x^2-2 x \log (4)\right ) \log (x)}{5 \log ^2(x)} \, dx=5+\frac {x^2 (x-\log (4))}{5 \log (x)} \]
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Time = 0.13 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.32, number of steps used = 17, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {12, 6874, 2395, 2343, 2346, 2209} \[ \int \frac {-x^2+x \log (4)+\left (3 x^2-2 x \log (4)\right ) \log (x)}{5 \log ^2(x)} \, dx=\frac {x^3}{5 \log (x)}-\frac {x^2 \log (4)}{5 \log (x)} \]
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Rule 12
Rule 2209
Rule 2343
Rule 2346
Rule 2395
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \int \frac {-x^2+x \log (4)+\left (3 x^2-2 x \log (4)\right ) \log (x)}{\log ^2(x)} \, dx \\ & = \frac {1}{5} \int \left (-\frac {x (x-\log (4))}{\log ^2(x)}+\frac {x (3 x-2 \log (4))}{\log (x)}\right ) \, dx \\ & = -\left (\frac {1}{5} \int \frac {x (x-\log (4))}{\log ^2(x)} \, dx\right )+\frac {1}{5} \int \frac {x (3 x-2 \log (4))}{\log (x)} \, dx \\ & = -\left (\frac {1}{5} \int \left (\frac {x^2}{\log ^2(x)}-\frac {x \log (4)}{\log ^2(x)}\right ) \, dx\right )+\frac {1}{5} \int \left (\frac {3 x^2}{\log (x)}-\frac {2 x \log (4)}{\log (x)}\right ) \, dx \\ & = -\left (\frac {1}{5} \int \frac {x^2}{\log ^2(x)} \, dx\right )+\frac {3}{5} \int \frac {x^2}{\log (x)} \, dx+\frac {1}{5} \log (4) \int \frac {x}{\log ^2(x)} \, dx-\frac {1}{5} (2 \log (4)) \int \frac {x}{\log (x)} \, dx \\ & = \frac {x^3}{5 \log (x)}-\frac {x^2 \log (4)}{5 \log (x)}-\frac {3}{5} \int \frac {x^2}{\log (x)} \, dx+\frac {3}{5} \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )+\frac {1}{5} (2 \log (4)) \int \frac {x}{\log (x)} \, dx-\frac {1}{5} (2 \log (4)) \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right ) \\ & = \frac {3}{5} \operatorname {ExpIntegralEi}(3 \log (x))-\frac {2}{5} \operatorname {ExpIntegralEi}(2 \log (x)) \log (4)+\frac {x^3}{5 \log (x)}-\frac {x^2 \log (4)}{5 \log (x)}-\frac {3}{5} \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )+\frac {1}{5} (2 \log (4)) \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right ) \\ & = \frac {x^3}{5 \log (x)}-\frac {x^2 \log (4)}{5 \log (x)} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.32 \[ \int \frac {-x^2+x \log (4)+\left (3 x^2-2 x \log (4)\right ) \log (x)}{5 \log ^2(x)} \, dx=\frac {x^3}{5 \log (x)}-\frac {x^2 \log (4)}{5 \log (x)} \]
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Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95
method | result | size |
risch | \(-\frac {x^{2} \left (2 \ln \left (2\right )-x \right )}{5 \ln \left (x \right )}\) | \(18\) |
norman | \(\frac {\frac {x^{3}}{5}-\frac {2 x^{2} \ln \left (2\right )}{5}}{\ln \left (x \right )}\) | \(19\) |
parallelrisch | \(-\frac {2 x^{2} \ln \left (2\right )-x^{3}}{5 \ln \left (x \right )}\) | \(20\) |
default | \(\frac {4 \ln \left (2\right ) \operatorname {Ei}_{1}\left (-2 \ln \left (x \right )\right )}{5}+\frac {2 \ln \left (2\right ) \left (-\frac {x^{2}}{\ln \left (x \right )}-2 \,\operatorname {Ei}_{1}\left (-2 \ln \left (x \right )\right )\right )}{5}+\frac {x^{3}}{5 \ln \left (x \right )}\) | \(43\) |
parts | \(\frac {4 \ln \left (2\right ) \operatorname {Ei}_{1}\left (-2 \ln \left (x \right )\right )}{5}+\frac {2 \ln \left (2\right ) \left (-\frac {x^{2}}{\ln \left (x \right )}-2 \,\operatorname {Ei}_{1}\left (-2 \ln \left (x \right )\right )\right )}{5}+\frac {x^{3}}{5 \ln \left (x \right )}\) | \(43\) |
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Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.89 \[ \int \frac {-x^2+x \log (4)+\left (3 x^2-2 x \log (4)\right ) \log (x)}{5 \log ^2(x)} \, dx=\frac {x^{3} - 2 \, x^{2} \log \left (2\right )}{5 \, \log \left (x\right )} \]
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Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-x^2+x \log (4)+\left (3 x^2-2 x \log (4)\right ) \log (x)}{5 \log ^2(x)} \, dx=\frac {x^{3} - 2 x^{2} \log {\left (2 \right )}}{5 \log {\left (x \right )}} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.23 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.84 \[ \int \frac {-x^2+x \log (4)+\left (3 x^2-2 x \log (4)\right ) \log (x)}{5 \log ^2(x)} \, dx=-\frac {4}{5} \, {\rm Ei}\left (2 \, \log \left (x\right )\right ) \log \left (2\right ) + \frac {4}{5} \, \Gamma \left (-1, -2 \, \log \left (x\right )\right ) \log \left (2\right ) + \frac {3}{5} \, {\rm Ei}\left (3 \, \log \left (x\right )\right ) - \frac {3}{5} \, \Gamma \left (-1, -3 \, \log \left (x\right )\right ) \]
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Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {-x^2+x \log (4)+\left (3 x^2-2 x \log (4)\right ) \log (x)}{5 \log ^2(x)} \, dx=\frac {x^{3}}{5 \, \log \left (x\right )} - \frac {2 \, x^{2} \log \left (2\right )}{5 \, \log \left (x\right )} \]
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Time = 12.69 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {-x^2+x \log (4)+\left (3 x^2-2 x \log (4)\right ) \log (x)}{5 \log ^2(x)} \, dx=-\frac {x^3\,\ln \left (4\right )-x^4}{5\,x\,\ln \left (x\right )} \]
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