Integrand size = 53, antiderivative size = 23 \[ \int \frac {e^{-e^5} \left (e^{e^5} x+\left (8 x+e^{e^5} \left (1-3 x^2\right )\right ) \log (x)+2 e^{e^5} x \log (x) \log (\log (x))\right )}{\log (x)} \, dx=x^2 \left (4 e^{-e^5}+\frac {1}{x}-x+\log (\log (x))\right ) \]
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Time = 0.17 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.113, Rules used = {12, 6874, 6820, 2346, 2209, 2602} \[ \int \frac {e^{-e^5} \left (e^{e^5} x+\left (8 x+e^{e^5} \left (1-3 x^2\right )\right ) \log (x)+2 e^{e^5} x \log (x) \log (\log (x))\right )}{\log (x)} \, dx=-x^3+4 e^{-e^5} x^2+x^2 \log (\log (x))+x \]
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Rule 12
Rule 2209
Rule 2346
Rule 2602
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = e^{-e^5} \int \frac {e^{e^5} x+\left (8 x+e^{e^5} \left (1-3 x^2\right )\right ) \log (x)+2 e^{e^5} x \log (x) \log (\log (x))}{\log (x)} \, dx \\ & = e^{-e^5} \int \left (-\frac {-e^{e^5} x-e^{e^5} \log (x)-8 x \log (x)+3 e^{e^5} x^2 \log (x)}{\log (x)}+2 e^{e^5} x \log (\log (x))\right ) \, dx \\ & = 2 \int x \log (\log (x)) \, dx-e^{-e^5} \int \frac {-e^{e^5} x-e^{e^5} \log (x)-8 x \log (x)+3 e^{e^5} x^2 \log (x)}{\log (x)} \, dx \\ & = x^2 \log (\log (x))-e^{-e^5} \int \left (-8 x+e^{e^5} \left (-1+3 x^2\right )-\frac {e^{e^5} x}{\log (x)}\right ) \, dx-\int \frac {x}{\log (x)} \, dx \\ & = 4 e^{-e^5} x^2+x^2 \log (\log (x))-\int \left (-1+3 x^2\right ) \, dx+\int \frac {x}{\log (x)} \, dx-\text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right ) \\ & = x+4 e^{-e^5} x^2-x^3-\text {Ei}(2 \log (x))+x^2 \log (\log (x))+\text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right ) \\ & = x+4 e^{-e^5} x^2-x^3+x^2 \log (\log (x)) \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {e^{-e^5} \left (e^{e^5} x+\left (8 x+e^{e^5} \left (1-3 x^2\right )\right ) \log (x)+2 e^{e^5} x \log (x) \log (\log (x))\right )}{\log (x)} \, dx=x+4 e^{-e^5} x^2-x^3+x^2 \log (\log (x)) \]
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Time = 0.59 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09
method | result | size |
norman | \(x +x^{2} \ln \left (\ln \left (x \right )\right )-x^{3}+4 \,{\mathrm e}^{-{\mathrm e}^{5}} x^{2}\) | \(25\) |
risch | \(x +x^{2} \ln \left (\ln \left (x \right )\right )-x^{3}+4 \,{\mathrm e}^{-{\mathrm e}^{5}} x^{2}\) | \(25\) |
parts | \(x^{2} \ln \left (\ln \left (x \right )\right )-{\mathrm e}^{-{\mathrm e}^{5}} \left ({\mathrm e}^{{\mathrm e}^{5}} x^{3}-x \,{\mathrm e}^{{\mathrm e}^{5}}-4 x^{2}\right )\) | \(35\) |
parallelrisch | \({\mathrm e}^{-{\mathrm e}^{5}} \left (-{\mathrm e}^{{\mathrm e}^{5}} x^{3}+{\mathrm e}^{{\mathrm e}^{5}} x^{2} \ln \left (\ln \left (x \right )\right )+x \,{\mathrm e}^{{\mathrm e}^{5}}+4 x^{2}\right )\) | \(36\) |
default | \({\mathrm e}^{-{\mathrm e}^{5}} \left (-{\mathrm e}^{{\mathrm e}^{5}} \operatorname {Ei}_{1}\left (-2 \ln \left (x \right )\right )+4 x^{2}-{\mathrm e}^{{\mathrm e}^{5}} x^{3}+2 \,{\mathrm e}^{{\mathrm e}^{5}} \left (\frac {x^{2} \ln \left (\ln \left (x \right )\right )}{2}+\frac {\operatorname {Ei}_{1}\left (-2 \ln \left (x \right )\right )}{2}\right )+x \,{\mathrm e}^{{\mathrm e}^{5}}\right )\) | \(59\) |
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Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.48 \[ \int \frac {e^{-e^5} \left (e^{e^5} x+\left (8 x+e^{e^5} \left (1-3 x^2\right )\right ) \log (x)+2 e^{e^5} x \log (x) \log (\log (x))\right )}{\log (x)} \, dx={\left (x^{2} e^{\left (e^{5}\right )} \log \left (\log \left (x\right )\right ) + 4 \, x^{2} - {\left (x^{3} - x\right )} e^{\left (e^{5}\right )}\right )} e^{\left (-e^{5}\right )} \]
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Time = 0.12 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {e^{-e^5} \left (e^{e^5} x+\left (8 x+e^{e^5} \left (1-3 x^2\right )\right ) \log (x)+2 e^{e^5} x \log (x) \log (\log (x))\right )}{\log (x)} \, dx=- x^{3} + x^{2} \log {\left (\log {\left (x \right )} \right )} + \frac {4 x^{2}}{e^{e^{5}}} + x \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.21 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.43 \[ \int \frac {e^{-e^5} \left (e^{e^5} x+\left (8 x+e^{e^5} \left (1-3 x^2\right )\right ) \log (x)+2 e^{e^5} x \log (x) \log (\log (x))\right )}{\log (x)} \, dx=-{\left (x^{3} e^{\left (e^{5}\right )} - 4 \, x^{2} - {\left (x^{2} \log \left (\log \left (x\right )\right ) - {\rm Ei}\left (2 \, \log \left (x\right )\right )\right )} e^{\left (e^{5}\right )} - x e^{\left (e^{5}\right )} - {\rm Ei}\left (2 \, \log \left (x\right )\right ) e^{\left (e^{5}\right )}\right )} e^{\left (-e^{5}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.61 \[ \int \frac {e^{-e^5} \left (e^{e^5} x+\left (8 x+e^{e^5} \left (1-3 x^2\right )\right ) \log (x)+2 e^{e^5} x \log (x) \log (\log (x))\right )}{\log (x)} \, dx=-{\left (x^{3} e^{\left (e^{5}\right )} - x^{2} e^{\left (e^{5}\right )} \log \left (\log \left (x\right )\right ) - 4 \, x^{2} - x e^{\left (e^{5}\right )}\right )} e^{\left (-e^{5}\right )} \]
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Time = 14.24 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-e^5} \left (e^{e^5} x+\left (8 x+e^{e^5} \left (1-3 x^2\right )\right ) \log (x)+2 e^{e^5} x \log (x) \log (\log (x))\right )}{\log (x)} \, dx=x+4\,x^2\,{\mathrm {e}}^{-{\mathrm {e}}^5}+x^2\,\ln \left (\ln \left (x\right )\right )-x^3 \]
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