Integrand size = 216, antiderivative size = 33 \[ \int \frac {e^{\frac {-x^5+e^x \left (-2 x^3+x^4\right )}{-4-20 x^4-5 x^5+e^x \left (-40 x^2+10 x^3+5 x^4\right )}} \left (20 x^4+20 x^8+e^{2 x} \left (80 x^4-80 x^5+20 x^6\right )+e^x \left (24 x^2-8 x^3-4 x^4+80 x^6-40 x^7\right )\right )}{16+160 x^4+40 x^5+400 x^8+200 x^9+25 x^{10}+e^{2 x} \left (1600 x^4-800 x^5-300 x^6+100 x^7+25 x^8\right )+e^x \left (320 x^2-80 x^3-40 x^4+1600 x^6-300 x^8-50 x^9\right )} \, dx=e^{\frac {x}{x+4 (5+x)+\frac {4}{x^2 \left (e^x (2-x)+x^2\right )}}} \]
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\[ \int \frac {e^{\frac {-x^5+e^x \left (-2 x^3+x^4\right )}{-4-20 x^4-5 x^5+e^x \left (-40 x^2+10 x^3+5 x^4\right )}} \left (20 x^4+20 x^8+e^{2 x} \left (80 x^4-80 x^5+20 x^6\right )+e^x \left (24 x^2-8 x^3-4 x^4+80 x^6-40 x^7\right )\right )}{16+160 x^4+40 x^5+400 x^8+200 x^9+25 x^{10}+e^{2 x} \left (1600 x^4-800 x^5-300 x^6+100 x^7+25 x^8\right )+e^x \left (320 x^2-80 x^3-40 x^4+1600 x^6-300 x^8-50 x^9\right )} \, dx=\int \frac {\exp \left (\frac {-x^5+e^x \left (-2 x^3+x^4\right )}{-4-20 x^4-5 x^5+e^x \left (-40 x^2+10 x^3+5 x^4\right )}\right ) \left (20 x^4+20 x^8+e^{2 x} \left (80 x^4-80 x^5+20 x^6\right )+e^x \left (24 x^2-8 x^3-4 x^4+80 x^6-40 x^7\right )\right )}{16+160 x^4+40 x^5+400 x^8+200 x^9+25 x^{10}+e^{2 x} \left (1600 x^4-800 x^5-300 x^6+100 x^7+25 x^8\right )+e^x \left (320 x^2-80 x^3-40 x^4+1600 x^6-300 x^8-50 x^9\right )} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {4 \exp \left (-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}\right ) x^2 \left (5 e^{2 x} (-2+x)^2 x^2-e^x \left (-6+2 x+x^2-20 x^4+10 x^5\right )+5 \left (x^2+x^6\right )\right )}{\left (4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )\right )^2} \, dx \\ & = 4 \int \frac {\exp \left (-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}\right ) x^2 \left (5 e^{2 x} (-2+x)^2 x^2-e^x \left (-6+2 x+x^2-20 x^4+10 x^5\right )+5 \left (x^2+x^6\right )\right )}{\left (4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )\right )^2} \, dx \\ & = 4 \int \left (\frac {\exp \left (-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}\right )}{5 (4+x)^2}+\frac {\exp \left (-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}\right ) \left (-8-6 x+6 x^2+x^3\right )}{5 (-2+x) (4+x)^2 \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )}-\frac {\exp \left (-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}\right ) \left (-64-8 x+24 x^2+4 x^3+320 x^4-80 x^5-20 x^6+25 x^7+5 x^8\right )}{5 (-2+x) (4+x)^2 \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2}\right ) \, dx \\ & = \frac {4}{5} \int \frac {\exp \left (-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}\right )}{(4+x)^2} \, dx+\frac {4}{5} \int \frac {\exp \left (-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}\right ) \left (-8-6 x+6 x^2+x^3\right )}{(-2+x) (4+x)^2 \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )} \, dx-\frac {4}{5} \int \frac {\exp \left (-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}\right ) \left (-64-8 x+24 x^2+4 x^3+320 x^4-80 x^5-20 x^6+25 x^7+5 x^8\right )}{(-2+x) (4+x)^2 \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2} \, dx \\ & = \frac {4}{5} \int \frac {e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}}}{(4+x)^2} \, dx-\frac {4}{5} \int \left (\frac {84 e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}}}{\left (-4-40 e^x x^2+10 e^x x^3-20 x^4+5 e^x x^4-5 x^5\right )^2}+\frac {484 e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}}}{3 (-2+x) \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2}+\frac {40 e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}} x}{\left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2}+\frac {20 e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}} x^2}{\left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2}+\frac {10 e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}} x^3}{\left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2}-\frac {5 e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}} x^4}{\left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2}+\frac {5 e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}} x^5}{\left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2}-\frac {16 e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}}}{(4+x)^2 \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2}-\frac {4 e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}}}{3 (4+x) \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2}\right ) \, dx+\frac {4}{5} \int \left (-\frac {e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}}}{-4-40 e^x x^2+10 e^x x^3-20 x^4+5 e^x x^4-5 x^5}+\frac {e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}}}{3 (-2+x) \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )}-\frac {8 e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}}}{(4+x)^2 \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )}-\frac {e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}}}{3 (4+x) \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )}\right ) \, dx \\ & = \frac {4}{15} \int \frac {e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}}}{(-2+x) \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )} \, dx-\frac {4}{15} \int \frac {e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}}}{(4+x) \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )} \, dx+\frac {4}{5} \int \frac {e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}}}{(4+x)^2} \, dx-\frac {4}{5} \int \frac {e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}}}{-4-40 e^x x^2+10 e^x x^3-20 x^4+5 e^x x^4-5 x^5} \, dx+\frac {16}{15} \int \frac {e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}}}{(4+x) \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2} \, dx+4 \int \frac {e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}} x^4}{\left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2} \, dx-4 \int \frac {e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}} x^5}{\left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2} \, dx-\frac {32}{5} \int \frac {e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}}}{(4+x)^2 \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )} \, dx-8 \int \frac {e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}} x^3}{\left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2} \, dx+\frac {64}{5} \int \frac {e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}}}{(4+x)^2 \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2} \, dx-16 \int \frac {e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}} x^2}{\left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2} \, dx-32 \int \frac {e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}} x}{\left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2} \, dx-\frac {336}{5} \int \frac {e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}}}{\left (-4-40 e^x x^2+10 e^x x^3-20 x^4+5 e^x x^4-5 x^5\right )^2} \, dx-\frac {1936}{15} \int \frac {e^{-\frac {e^x (-2+x) x^3-x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}}}{(-2+x) \left (4+40 e^x x^2-10 e^x x^3+20 x^4-5 e^x x^4+5 x^5\right )^2} \, dx \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45 \[ \int \frac {e^{\frac {-x^5+e^x \left (-2 x^3+x^4\right )}{-4-20 x^4-5 x^5+e^x \left (-40 x^2+10 x^3+5 x^4\right )}} \left (20 x^4+20 x^8+e^{2 x} \left (80 x^4-80 x^5+20 x^6\right )+e^x \left (24 x^2-8 x^3-4 x^4+80 x^6-40 x^7\right )\right )}{16+160 x^4+40 x^5+400 x^8+200 x^9+25 x^{10}+e^{2 x} \left (1600 x^4-800 x^5-300 x^6+100 x^7+25 x^8\right )+e^x \left (320 x^2-80 x^3-40 x^4+1600 x^6-300 x^8-50 x^9\right )} \, dx=e^{\frac {-e^x (-2+x) x^3+x^5}{4+20 x^4+5 x^5-5 e^x x^2 \left (-8+2 x+x^2\right )}} \]
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Time = 31.62 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.67
method | result | size |
risch | \({\mathrm e}^{\frac {x^{3} \left ({\mathrm e}^{x} x -x^{2}-2 \,{\mathrm e}^{x}\right )}{5 \,{\mathrm e}^{x} x^{4}-5 x^{5}+10 \,{\mathrm e}^{x} x^{3}-20 x^{4}-40 \,{\mathrm e}^{x} x^{2}-4}}\) | \(55\) |
parallelrisch | \({\mathrm e}^{\frac {\left (x^{4}-2 x^{3}\right ) {\mathrm e}^{x}-x^{5}}{5 \,{\mathrm e}^{x} x^{4}-5 x^{5}+10 \,{\mathrm e}^{x} x^{3}-20 x^{4}-40 \,{\mathrm e}^{x} x^{2}-4}}\) | \(56\) |
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Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.55 \[ \int \frac {e^{\frac {-x^5+e^x \left (-2 x^3+x^4\right )}{-4-20 x^4-5 x^5+e^x \left (-40 x^2+10 x^3+5 x^4\right )}} \left (20 x^4+20 x^8+e^{2 x} \left (80 x^4-80 x^5+20 x^6\right )+e^x \left (24 x^2-8 x^3-4 x^4+80 x^6-40 x^7\right )\right )}{16+160 x^4+40 x^5+400 x^8+200 x^9+25 x^{10}+e^{2 x} \left (1600 x^4-800 x^5-300 x^6+100 x^7+25 x^8\right )+e^x \left (320 x^2-80 x^3-40 x^4+1600 x^6-300 x^8-50 x^9\right )} \, dx=e^{\left (\frac {x^{5} - {\left (x^{4} - 2 \, x^{3}\right )} e^{x}}{5 \, x^{5} + 20 \, x^{4} - 5 \, {\left (x^{4} + 2 \, x^{3} - 8 \, x^{2}\right )} e^{x} + 4}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (22) = 44\).
Time = 0.71 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.39 \[ \int \frac {e^{\frac {-x^5+e^x \left (-2 x^3+x^4\right )}{-4-20 x^4-5 x^5+e^x \left (-40 x^2+10 x^3+5 x^4\right )}} \left (20 x^4+20 x^8+e^{2 x} \left (80 x^4-80 x^5+20 x^6\right )+e^x \left (24 x^2-8 x^3-4 x^4+80 x^6-40 x^7\right )\right )}{16+160 x^4+40 x^5+400 x^8+200 x^9+25 x^{10}+e^{2 x} \left (1600 x^4-800 x^5-300 x^6+100 x^7+25 x^8\right )+e^x \left (320 x^2-80 x^3-40 x^4+1600 x^6-300 x^8-50 x^9\right )} \, dx=e^{\frac {- x^{5} + \left (x^{4} - 2 x^{3}\right ) e^{x}}{- 5 x^{5} - 20 x^{4} + \left (5 x^{4} + 10 x^{3} - 40 x^{2}\right ) e^{x} - 4}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 152 vs. \(2 (28) = 56\).
Time = 0.79 (sec) , antiderivative size = 152, normalized size of antiderivative = 4.61 \[ \int \frac {e^{\frac {-x^5+e^x \left (-2 x^3+x^4\right )}{-4-20 x^4-5 x^5+e^x \left (-40 x^2+10 x^3+5 x^4\right )}} \left (20 x^4+20 x^8+e^{2 x} \left (80 x^4-80 x^5+20 x^6\right )+e^x \left (24 x^2-8 x^3-4 x^4+80 x^6-40 x^7\right )\right )}{16+160 x^4+40 x^5+400 x^8+200 x^9+25 x^{10}+e^{2 x} \left (1600 x^4-800 x^5-300 x^6+100 x^7+25 x^8\right )+e^x \left (320 x^2-80 x^3-40 x^4+1600 x^6-300 x^8-50 x^9\right )} \, dx=e^{\left (-\frac {4 \, x^{4}}{5 \, x^{5} + 20 \, x^{4} - 5 \, {\left (x^{4} + 2 \, x^{3} - 8 \, x^{2}\right )} e^{x} + 4} + \frac {4 \, x^{3} e^{x}}{5 \, x^{5} + 20 \, x^{4} - 5 \, {\left (x^{4} + 2 \, x^{3} - 8 \, x^{2}\right )} e^{x} + 4} - \frac {8 \, x^{2} e^{x}}{5 \, x^{5} + 20 \, x^{4} - 5 \, {\left (x^{4} + 2 \, x^{3} - 8 \, x^{2}\right )} e^{x} + 4} - \frac {4}{5 \, {\left (5 \, x^{5} + 20 \, x^{4} - 5 \, {\left (x^{4} + 2 \, x^{3} - 8 \, x^{2}\right )} e^{x} + 4\right )}} + \frac {1}{5}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (28) = 56\).
Time = 0.35 (sec) , antiderivative size = 125, normalized size of antiderivative = 3.79 \[ \int \frac {e^{\frac {-x^5+e^x \left (-2 x^3+x^4\right )}{-4-20 x^4-5 x^5+e^x \left (-40 x^2+10 x^3+5 x^4\right )}} \left (20 x^4+20 x^8+e^{2 x} \left (80 x^4-80 x^5+20 x^6\right )+e^x \left (24 x^2-8 x^3-4 x^4+80 x^6-40 x^7\right )\right )}{16+160 x^4+40 x^5+400 x^8+200 x^9+25 x^{10}+e^{2 x} \left (1600 x^4-800 x^5-300 x^6+100 x^7+25 x^8\right )+e^x \left (320 x^2-80 x^3-40 x^4+1600 x^6-300 x^8-50 x^9\right )} \, dx=e^{\left (\frac {x^{5}}{5 \, x^{5} - 5 \, x^{4} e^{x} + 20 \, x^{4} - 10 \, x^{3} e^{x} + 40 \, x^{2} e^{x} + 4} - \frac {x^{4} e^{x}}{5 \, x^{5} - 5 \, x^{4} e^{x} + 20 \, x^{4} - 10 \, x^{3} e^{x} + 40 \, x^{2} e^{x} + 4} + \frac {2 \, x^{3} e^{x}}{5 \, x^{5} - 5 \, x^{4} e^{x} + 20 \, x^{4} - 10 \, x^{3} e^{x} + 40 \, x^{2} e^{x} + 4}\right )} \]
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Time = 9.21 (sec) , antiderivative size = 127, normalized size of antiderivative = 3.85 \[ \int \frac {e^{\frac {-x^5+e^x \left (-2 x^3+x^4\right )}{-4-20 x^4-5 x^5+e^x \left (-40 x^2+10 x^3+5 x^4\right )}} \left (20 x^4+20 x^8+e^{2 x} \left (80 x^4-80 x^5+20 x^6\right )+e^x \left (24 x^2-8 x^3-4 x^4+80 x^6-40 x^7\right )\right )}{16+160 x^4+40 x^5+400 x^8+200 x^9+25 x^{10}+e^{2 x} \left (1600 x^4-800 x^5-300 x^6+100 x^7+25 x^8\right )+e^x \left (320 x^2-80 x^3-40 x^4+1600 x^6-300 x^8-50 x^9\right )} \, dx={\mathrm {e}}^{\frac {x^5}{40\,x^2\,{\mathrm {e}}^x-10\,x^3\,{\mathrm {e}}^x-5\,x^4\,{\mathrm {e}}^x+20\,x^4+5\,x^5+4}}\,{\mathrm {e}}^{\frac {2\,x^3\,{\mathrm {e}}^x}{40\,x^2\,{\mathrm {e}}^x-10\,x^3\,{\mathrm {e}}^x-5\,x^4\,{\mathrm {e}}^x+20\,x^4+5\,x^5+4}}\,{\mathrm {e}}^{-\frac {x^4\,{\mathrm {e}}^x}{40\,x^2\,{\mathrm {e}}^x-10\,x^3\,{\mathrm {e}}^x-5\,x^4\,{\mathrm {e}}^x+20\,x^4+5\,x^5+4}} \]
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