Integrand size = 49, antiderivative size = 21 \[ \int \frac {-20 x^2+4 e x^2+\left (10 x^2-2 e x^2\right ) \log (x)+\left (-2+e^x (-1+x)\right ) \log ^3(x)}{x^2 \log ^3(x)} \, dx=-1+\frac {2+e^x}{x}-\frac {2 (-5+e) x}{\log ^2(x)} \]
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Time = 0.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {6, 6820, 14, 2228, 2334, 2335} \[ \int \frac {-20 x^2+4 e x^2+\left (10 x^2-2 e x^2\right ) \log (x)+\left (-2+e^x (-1+x)\right ) \log ^3(x)}{x^2 \log ^3(x)} \, dx=\frac {e^x}{x}+\frac {2}{x}+\frac {2 (5-e) x}{\log ^2(x)} \]
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Rule 6
Rule 14
Rule 2228
Rule 2334
Rule 2335
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {(-20+4 e) x^2+\left (10 x^2-2 e x^2\right ) \log (x)+\left (-2+e^x (-1+x)\right ) \log ^3(x)}{x^2 \log ^3(x)} \, dx \\ & = \int \left (\frac {-2+e^x (-1+x)}{x^2}+\frac {4 (-5+e)}{\log ^3(x)}-\frac {2 (-5+e)}{\log ^2(x)}\right ) \, dx \\ & = (2 (5-e)) \int \frac {1}{\log ^2(x)} \, dx-(4 (5-e)) \int \frac {1}{\log ^3(x)} \, dx+\int \frac {-2+e^x (-1+x)}{x^2} \, dx \\ & = \frac {2 (5-e) x}{\log ^2(x)}-\frac {2 (5-e) x}{\log (x)}-(2 (5-e)) \int \frac {1}{\log ^2(x)} \, dx+(2 (5-e)) \int \frac {1}{\log (x)} \, dx+\int \left (-\frac {2}{x^2}+\frac {e^x (-1+x)}{x^2}\right ) \, dx \\ & = \frac {2}{x}+\frac {2 (5-e) x}{\log ^2(x)}+2 (5-e) \text {li}(x)-(2 (5-e)) \int \frac {1}{\log (x)} \, dx+\int \frac {e^x (-1+x)}{x^2} \, dx \\ & = \frac {2}{x}+\frac {e^x}{x}+\frac {2 (5-e) x}{\log ^2(x)} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {-20 x^2+4 e x^2+\left (10 x^2-2 e x^2\right ) \log (x)+\left (-2+e^x (-1+x)\right ) \log ^3(x)}{x^2 \log ^3(x)} \, dx=\frac {2}{x}+\frac {e^x}{x}-\frac {2 (-5+e) x}{\log ^2(x)} \]
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Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {{\mathrm e}^{x}+2}{x}-\frac {2 \left ({\mathrm e}-5\right ) x}{\ln \left (x \right )^{2}}\) | \(21\) |
parallelrisch | \(\frac {{\mathrm e}^{x} \ln \left (x \right )^{2}-2 x^{2} {\mathrm e}+2 \ln \left (x \right )^{2}+10 x^{2}}{\ln \left (x \right )^{2} x}\) | \(35\) |
default | \(\left (-2 \,{\mathrm e}+10\right ) \left (-\frac {x}{\ln \left (x \right )}-\operatorname {Ei}_{1}\left (-\ln \left (x \right )\right )\right )+\left (4 \,{\mathrm e}-20\right ) \left (-\frac {x}{2 \ln \left (x \right )^{2}}-\frac {x}{2 \ln \left (x \right )}-\frac {\operatorname {Ei}_{1}\left (-\ln \left (x \right )\right )}{2}\right )+\frac {{\mathrm e}^{x}}{x}+\frac {2}{x}\) | \(66\) |
parts | \(\left (-2 \,{\mathrm e}+10\right ) \left (-\frac {x}{\ln \left (x \right )}-\operatorname {Ei}_{1}\left (-\ln \left (x \right )\right )\right )+\left (4 \,{\mathrm e}-20\right ) \left (-\frac {x}{2 \ln \left (x \right )^{2}}-\frac {x}{2 \ln \left (x \right )}-\frac {\operatorname {Ei}_{1}\left (-\ln \left (x \right )\right )}{2}\right )+\frac {{\mathrm e}^{x}}{x}+\frac {2}{x}\) | \(66\) |
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Time = 0.41 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52 \[ \int \frac {-20 x^2+4 e x^2+\left (10 x^2-2 e x^2\right ) \log (x)+\left (-2+e^x (-1+x)\right ) \log ^3(x)}{x^2 \log ^3(x)} \, dx=-\frac {2 \, x^{2} e - {\left (e^{x} + 2\right )} \log \left (x\right )^{2} - 10 \, x^{2}}{x \log \left (x\right )^{2}} \]
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Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {-20 x^2+4 e x^2+\left (10 x^2-2 e x^2\right ) \log (x)+\left (-2+e^x (-1+x)\right ) \log ^3(x)}{x^2 \log ^3(x)} \, dx=\frac {- 2 e x + 10 x}{\log {\left (x \right )}^{2}} + \frac {e^{x}}{x} + \frac {2}{x} \]
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Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 2.43 \[ \int \frac {-20 x^2+4 e x^2+\left (10 x^2-2 e x^2\right ) \log (x)+\left (-2+e^x (-1+x)\right ) \log ^3(x)}{x^2 \log ^3(x)} \, dx=-2 \, e \Gamma \left (-1, -\log \left (x\right )\right ) - 4 \, e \Gamma \left (-2, -\log \left (x\right )\right ) + \frac {2}{x} + {\rm Ei}\left (x\right ) - \Gamma \left (-1, -x\right ) + 10 \, \Gamma \left (-1, -\log \left (x\right )\right ) + 20 \, \Gamma \left (-2, -\log \left (x\right )\right ) \]
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Time = 0.30 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {-20 x^2+4 e x^2+\left (10 x^2-2 e x^2\right ) \log (x)+\left (-2+e^x (-1+x)\right ) \log ^3(x)}{x^2 \log ^3(x)} \, dx=-\frac {2 \, x^{2} e - e^{x} \log \left (x\right )^{2} - 10 \, x^{2} - 2 \, \log \left (x\right )^{2}}{x \log \left (x\right )^{2}} \]
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Time = 11.99 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.33 \[ \int \frac {-20 x^2+4 e x^2+\left (10 x^2-2 e x^2\right ) \log (x)+\left (-2+e^x (-1+x)\right ) \log ^3(x)}{x^2 \log ^3(x)} \, dx=\frac {{\mathrm {e}}^x}{x}+\frac {10\,x}{{\ln \left (x\right )}^2}+\frac {2}{x}-\frac {2\,x\,\mathrm {e}}{{\ln \left (x\right )}^2} \]
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