Integrand size = 101, antiderivative size = 29 \[ \int \frac {e^{-x} \left (-40 x-40 x^2+24 x^3+e^2 \left (-10-10 x+x^2\right )+e^x \left (-5 e^2 x^2-20 x^3\right )+\left (20 x^3-20 x^4+e^2 \left (5 x^2-5 x^3\right )\right ) \log \left (e^2+4 x\right )\right )}{5 e^2 x^2+20 x^3} \, dx=-x+e^{-x} \left (-\frac {1}{5}+\frac {2}{x}+x \log \left (e^2+4 x\right )\right ) \]
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Time = 2.30 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.97, number of steps used = 19, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.099, Rules used = {1607, 6874, 6820, 2225, 2208, 2209, 2207, 2634, 12, 2230} \[ \int \frac {e^{-x} \left (-40 x-40 x^2+24 x^3+e^2 \left (-10-10 x+x^2\right )+e^x \left (-5 e^2 x^2-20 x^3\right )+\left (20 x^3-20 x^4+e^2 \left (5 x^2-5 x^3\right )\right ) \log \left (e^2+4 x\right )\right )}{5 e^2 x^2+20 x^3} \, dx=-x-\frac {e^{-x}}{5}+\frac {2 e^{-x}}{x}+e^{-x} \log \left (4 x+e^2\right )-e^{-x} (1-x) \log \left (4 x+e^2\right ) \]
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Rule 12
Rule 1607
Rule 2207
Rule 2208
Rule 2209
Rule 2225
Rule 2230
Rule 2634
Rule 6820
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{-x} \left (-40 x-40 x^2+24 x^3+e^2 \left (-10-10 x+x^2\right )+e^x \left (-5 e^2 x^2-20 x^3\right )+\left (20 x^3-20 x^4+e^2 \left (5 x^2-5 x^3\right )\right ) \log \left (e^2+4 x\right )\right )}{x^2 \left (5 e^2+20 x\right )} \, dx \\ & = \int \left (-1+\frac {e^{-x} \left (-10 e^2-40 \left (1+\frac {e^2}{4}\right ) x-40 \left (1-\frac {e^2}{40}\right ) x^2+24 x^3+5 e^2 x^2 \log \left (e^2+4 x\right )+20 \left (1-\frac {e^2}{4}\right ) x^3 \log \left (e^2+4 x\right )-20 x^4 \log \left (e^2+4 x\right )\right )}{5 x^2 \left (e^2+4 x\right )}\right ) \, dx \\ & = -x+\frac {1}{5} \int \frac {e^{-x} \left (-10 e^2-40 \left (1+\frac {e^2}{4}\right ) x-40 \left (1-\frac {e^2}{40}\right ) x^2+24 x^3+5 e^2 x^2 \log \left (e^2+4 x\right )+20 \left (1-\frac {e^2}{4}\right ) x^3 \log \left (e^2+4 x\right )-20 x^4 \log \left (e^2+4 x\right )\right )}{x^2 \left (e^2+4 x\right )} \, dx \\ & = -x+\frac {1}{5} \int \frac {e^{-x} \left (e^2 \left (-10-10 x+x^2\right )+8 x \left (-5-5 x+3 x^2\right )-5 (-1+x) x^2 \left (e^2+4 x\right ) \log \left (e^2+4 x\right )\right )}{x^2 \left (e^2+4 x\right )} \, dx \\ & = -x+\frac {1}{5} \int \left (\frac {e^{-x} \left (-10 e^2-10 \left (4+e^2\right ) x-\left (40-e^2\right ) x^2+24 x^3\right )}{x^2 \left (e^2+4 x\right )}-5 e^{-x} (-1+x) \log \left (e^2+4 x\right )\right ) \, dx \\ & = -x+\frac {1}{5} \int \frac {e^{-x} \left (-10 e^2-10 \left (4+e^2\right ) x-\left (40-e^2\right ) x^2+24 x^3\right )}{x^2 \left (e^2+4 x\right )} \, dx-\int e^{-x} (-1+x) \log \left (e^2+4 x\right ) \, dx \\ & = -x+e^{-x} \log \left (e^2+4 x\right )-e^{-x} (1-x) \log \left (e^2+4 x\right )+\frac {1}{5} \int \left (6 e^{-x}-\frac {10 e^{-x}}{x^2}-\frac {10 e^{-x}}{x}-\frac {5 e^{2-x}}{e^2+4 x}\right ) \, dx+\int \frac {4 e^{-x} x}{-e^2-4 x} \, dx \\ & = -x+e^{-x} \log \left (e^2+4 x\right )-e^{-x} (1-x) \log \left (e^2+4 x\right )+\frac {6}{5} \int e^{-x} \, dx-2 \int \frac {e^{-x}}{x^2} \, dx-2 \int \frac {e^{-x}}{x} \, dx+4 \int \frac {e^{-x} x}{-e^2-4 x} \, dx-\int \frac {e^{2-x}}{e^2+4 x} \, dx \\ & = -\frac {6 e^{-x}}{5}+\frac {2 e^{-x}}{x}-x-\frac {1}{4} e^{2+\frac {e^2}{4}} \text {Ei}\left (\frac {1}{4} \left (-e^2-4 x\right )\right )-2 \text {Ei}(-x)+e^{-x} \log \left (e^2+4 x\right )-e^{-x} (1-x) \log \left (e^2+4 x\right )+2 \int \frac {e^{-x}}{x} \, dx+4 \int \left (-\frac {e^{-x}}{4}+\frac {e^{2-x}}{4 \left (e^2+4 x\right )}\right ) \, dx \\ & = -\frac {6 e^{-x}}{5}+\frac {2 e^{-x}}{x}-x-\frac {1}{4} e^{2+\frac {e^2}{4}} \text {Ei}\left (\frac {1}{4} \left (-e^2-4 x\right )\right )+e^{-x} \log \left (e^2+4 x\right )-e^{-x} (1-x) \log \left (e^2+4 x\right )-\int e^{-x} \, dx+\int \frac {e^{2-x}}{e^2+4 x} \, dx \\ & = -\frac {e^{-x}}{5}+\frac {2 e^{-x}}{x}-x+e^{-x} \log \left (e^2+4 x\right )-e^{-x} (1-x) \log \left (e^2+4 x\right ) \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.28 \[ \int \frac {e^{-x} \left (-40 x-40 x^2+24 x^3+e^2 \left (-10-10 x+x^2\right )+e^x \left (-5 e^2 x^2-20 x^3\right )+\left (20 x^3-20 x^4+e^2 \left (5 x^2-5 x^3\right )\right ) \log \left (e^2+4 x\right )\right )}{5 e^2 x^2+20 x^3} \, dx=\frac {1}{5} \left (e^{-x} \left (-1+\frac {10}{x}\right )-5 x+5 e^{-x} x \log \left (e^2+4 x\right )\right ) \]
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Time = 0.69 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00
method | result | size |
parts | \(-x +\frac {\left (2+x^{2} \ln \left ({\mathrm e}^{2}+4 x \right )-\frac {x}{5}\right ) {\mathrm e}^{-x}}{x}\) | \(29\) |
default | \(-x +\frac {\left (10-x +5 x^{2} \ln \left ({\mathrm e}^{2}+4 x \right )\right ) {\mathrm e}^{-x}}{5 x}\) | \(31\) |
norman | \(\frac {\left (2+x^{2} \ln \left ({\mathrm e}^{2}+4 x \right )-\frac {x}{5}-{\mathrm e}^{x} x^{2}\right ) {\mathrm e}^{-x}}{x}\) | \(32\) |
risch | \(x \,{\mathrm e}^{-x} \ln \left ({\mathrm e}^{2}+4 x \right )-\frac {\left (5 \,{\mathrm e}^{x} x^{2}+x -10\right ) {\mathrm e}^{-x}}{5 x}\) | \(34\) |
parallelrisch | \(\frac {\left (160+40 x \,{\mathrm e}^{2} {\mathrm e}^{x}-80 \,{\mathrm e}^{x} x^{2}+80 x^{2} \ln \left ({\mathrm e}^{2}+4 x \right )-16 x \right ) {\mathrm e}^{-x}}{80 x}\) | \(41\) |
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Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-x} \left (-40 x-40 x^2+24 x^3+e^2 \left (-10-10 x+x^2\right )+e^x \left (-5 e^2 x^2-20 x^3\right )+\left (20 x^3-20 x^4+e^2 \left (5 x^2-5 x^3\right )\right ) \log \left (e^2+4 x\right )\right )}{5 e^2 x^2+20 x^3} \, dx=-\frac {{\left (5 \, x^{2} e^{x} - 5 \, x^{2} \log \left (4 \, x + e^{2}\right ) + x - 10\right )} e^{\left (-x\right )}}{5 \, x} \]
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Time = 0.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.83 \[ \int \frac {e^{-x} \left (-40 x-40 x^2+24 x^3+e^2 \left (-10-10 x+x^2\right )+e^x \left (-5 e^2 x^2-20 x^3\right )+\left (20 x^3-20 x^4+e^2 \left (5 x^2-5 x^3\right )\right ) \log \left (e^2+4 x\right )\right )}{5 e^2 x^2+20 x^3} \, dx=- x + \frac {\left (5 x^{2} \log {\left (4 x + e^{2} \right )} - x + 10\right ) e^{- x}}{5 x} \]
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Time = 0.24 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.24 \[ \int \frac {e^{-x} \left (-40 x-40 x^2+24 x^3+e^2 \left (-10-10 x+x^2\right )+e^x \left (-5 e^2 x^2-20 x^3\right )+\left (20 x^3-20 x^4+e^2 \left (5 x^2-5 x^3\right )\right ) \log \left (e^2+4 x\right )\right )}{5 e^2 x^2+20 x^3} \, dx=\frac {5 \, x^{2} e^{\left (-x\right )} \log \left (4 \, x + e^{2}\right ) - 5 \, x^{2} - {\left (x - 10\right )} e^{\left (-x\right )}}{5 \, x} \]
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Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {e^{-x} \left (-40 x-40 x^2+24 x^3+e^2 \left (-10-10 x+x^2\right )+e^x \left (-5 e^2 x^2-20 x^3\right )+\left (20 x^3-20 x^4+e^2 \left (5 x^2-5 x^3\right )\right ) \log \left (e^2+4 x\right )\right )}{5 e^2 x^2+20 x^3} \, dx=-\frac {{\left (5 \, x^{2} e^{x} - 5 \, x^{2} \log \left (4 \, x + e^{2}\right ) + x - 10\right )} e^{\left (-x\right )}}{5 \, x} \]
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Timed out. \[ \int \frac {e^{-x} \left (-40 x-40 x^2+24 x^3+e^2 \left (-10-10 x+x^2\right )+e^x \left (-5 e^2 x^2-20 x^3\right )+\left (20 x^3-20 x^4+e^2 \left (5 x^2-5 x^3\right )\right ) \log \left (e^2+4 x\right )\right )}{5 e^2 x^2+20 x^3} \, dx=\int -\frac {{\mathrm {e}}^{-x}\,\left (40\,x+{\mathrm {e}}^2\,\left (-x^2+10\,x+10\right )+{\mathrm {e}}^x\,\left (20\,x^3+5\,{\mathrm {e}}^2\,x^2\right )-\ln \left (4\,x+{\mathrm {e}}^2\right )\,\left ({\mathrm {e}}^2\,\left (5\,x^2-5\,x^3\right )+20\,x^3-20\,x^4\right )+40\,x^2-24\,x^3\right )}{20\,x^3+5\,{\mathrm {e}}^2\,x^2} \,d x \]
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