Integrand size = 156, antiderivative size = 24 \[ \int \frac {76800-156160 x+36000 x^2+50200 x^3-31650 x^4+6750 x^5-500 x^6+e^{3 e} \left (512-896 x-384 x^2+256 x^3\right )+e^{2 e} \left (8192-15104 x-2784 x^2+5280 x^3-960 x^4\right )+e^e \left (43520-84352 x+2880 x^2+30200 x^3-11400 x^4+1200 x^5\right )}{8000+64 e^{3 e}+e^{2 e} (960-240 x)-6000 x+1500 x^2-125 x^3+e^e \left (4800-2400 x+300 x^2\right )} \, dx=\left (4+x+\frac {x}{5+e^e-\frac {5 x}{4}}-x^2\right )^2 \]
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Leaf count is larger than twice the leaf count of optimal. \(86\) vs. \(2(24)=48\).
Time = 0.18 (sec) , antiderivative size = 86, normalized size of antiderivative = 3.58, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.006, Rules used = {2099} \[ \int \frac {76800-156160 x+36000 x^2+50200 x^3-31650 x^4+6750 x^5-500 x^6+e^{3 e} \left (512-896 x-384 x^2+256 x^3\right )+e^{2 e} \left (8192-15104 x-2784 x^2+5280 x^3-960 x^4\right )+e^e \left (43520-84352 x+2880 x^2+30200 x^3-11400 x^4+1200 x^5\right )}{8000+64 e^{3 e}+e^{2 e} (960-240 x)-6000 x+1500 x^2-125 x^3+e^e \left (4800-2400 x+300 x^2\right )} \, dx=x^4-2 x^3-\frac {27 x^2}{5}+\frac {32}{25} \left (10+e^e\right ) x-\frac {128 \left (5+e^e\right ) \left (55+35 e^e+4 e^{2 e}\right )}{125 \left (4 \left (5+e^e\right )-5 x\right )}+\frac {256 \left (5+e^e\right )^2}{25 \left (4 \left (5+e^e\right )-5 x\right )^2} \]
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Rule 2099
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {32}{25} \left (10+e^e\right )+\frac {512 \left (5+e^e\right )^2}{5 \left (20+4 e^e-5 x\right )^3}+\frac {128 \left (5+e^e\right ) \left (-55-35 e^e-4 e^{2 e}\right )}{25 \left (20+4 e^e-5 x\right )^2}-\frac {54 x}{5}-6 x^2+4 x^3\right ) \, dx \\ & = \frac {256 \left (5+e^e\right )^2}{25 \left (4 \left (5+e^e\right )-5 x\right )^2}-\frac {128 \left (5+e^e\right ) \left (55+35 e^e+4 e^{2 e}\right )}{125 \left (4 \left (5+e^e\right )-5 x\right )}+\frac {32}{25} \left (10+e^e\right ) x-\frac {27 x^2}{5}-2 x^3+x^4 \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(157\) vs. \(2(24)=48\).
Time = 0.13 (sec) , antiderivative size = 157, normalized size of antiderivative = 6.54 \[ \int \frac {76800-156160 x+36000 x^2+50200 x^3-31650 x^4+6750 x^5-500 x^6+e^{3 e} \left (512-896 x-384 x^2+256 x^3\right )+e^{2 e} \left (8192-15104 x-2784 x^2+5280 x^3-960 x^4\right )+e^e \left (43520-84352 x+2880 x^2+30200 x^3-11400 x^4+1200 x^5\right )}{8000+64 e^{3 e}+e^{2 e} (960-240 x)-6000 x+1500 x^2-125 x^3+e^e \left (4800-2400 x+300 x^2\right )} \, dx=\frac {-4096 e^{6 e}+10240 e^{5 e} (-11+x)-1280 e^{4 e} \left (989-180 x+5 x^2\right )-3200 e^{3 e} \left (2324-629 x+35 x^2\right )+400 e^{2 e} \left (-60016+21520 x-1920 x^2-50 x^3+25 x^4\right )-1000 e^e \left (40064-18256 x+2720 x^2+45 x^3-150 x^4+25 x^5\right )+625 \left (-42496+25088 x-7040 x^2+600 x^3+665 x^4-250 x^5+25 x^6\right )}{625 \left (20+4 e^e-5 x\right )^2} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(104\) vs. \(2(22)=44\).
Time = 0.11 (sec) , antiderivative size = 105, normalized size of antiderivative = 4.38
method | result | size |
risch | \(x^{4}-2 x^{3}+\frac {32 x \,{\mathrm e}^{{\mathrm e}}}{25}-\frac {27 x^{2}}{5}+\frac {64 x}{5}+\frac {\frac {\left (32 \,{\mathrm e}^{3 \,{\mathrm e}}+440 \,{\mathrm e}^{2 \,{\mathrm e}}+1840 \,{\mathrm e}^{{\mathrm e}}+2200\right ) x}{25}-\frac {128 \,{\mathrm e}^{4 \,{\mathrm e}}}{125}-\frac {96 \,{\mathrm e}^{3 \,{\mathrm e}}}{5}-\frac {3216 \,{\mathrm e}^{2 \,{\mathrm e}}}{25}-\frac {1792 \,{\mathrm e}^{{\mathrm e}}}{5}-336}{{\mathrm e}^{2 \,{\mathrm e}}-\frac {5 x \,{\mathrm e}^{{\mathrm e}}}{2}+\frac {25 x^{2}}{16}+10 \,{\mathrm e}^{{\mathrm e}}-\frac {25 x}{2}+25}\) | \(105\) |
norman | \(\frac {\left (-40 \,{\mathrm e}^{{\mathrm e}}-250\right ) x^{5}+\left (-32 \,{\mathrm e}^{2 \,{\mathrm e}}-72 \,{\mathrm e}^{{\mathrm e}}+600\right ) x^{3}+\left (16 \,{\mathrm e}^{2 \,{\mathrm e}}+240 \,{\mathrm e}^{{\mathrm e}}+665\right ) x^{4}+\left (-\frac {896 \,{\mathrm e}^{3 \,{\mathrm e}}}{5}-\frac {15104 \,{\mathrm e}^{2 \,{\mathrm e}}}{5}-31232-\frac {84352 \,{\mathrm e}^{{\mathrm e}}}{5}\right ) x +25 x^{6}+\frac {1792 \,{\mathrm e}^{4 \,{\mathrm e}}}{25}+\frac {40448 \,{\mathrm e}^{3 \,{\mathrm e}}}{25}+\frac {340224 \,{\mathrm e}^{2 \,{\mathrm e}}}{25}+\frac {252928 \,{\mathrm e}^{{\mathrm e}}}{5}+70144}{\left (4 \,{\mathrm e}^{{\mathrm e}}-5 x +20\right )^{2}}\) | \(117\) |
gosper | \(\frac {400 x^{4} {\mathrm e}^{2 \,{\mathrm e}}-1000 x^{5} {\mathrm e}^{{\mathrm e}}+625 x^{6}-800 \,{\mathrm e}^{2 \,{\mathrm e}} x^{3}+6000 x^{4} {\mathrm e}^{{\mathrm e}}-6250 x^{5}+1792 \,{\mathrm e}^{4 \,{\mathrm e}}-4480 \,{\mathrm e}^{3 \,{\mathrm e}} x -1800 x^{3} {\mathrm e}^{{\mathrm e}}+16625 x^{4}+40448 \,{\mathrm e}^{3 \,{\mathrm e}}-75520 \,{\mathrm e}^{2 \,{\mathrm e}} x +15000 x^{3}+340224 \,{\mathrm e}^{2 \,{\mathrm e}}-421760 x \,{\mathrm e}^{{\mathrm e}}+1264640 \,{\mathrm e}^{{\mathrm e}}-780800 x +1753600}{400 \,{\mathrm e}^{2 \,{\mathrm e}}-1000 x \,{\mathrm e}^{{\mathrm e}}+625 x^{2}+4000 \,{\mathrm e}^{{\mathrm e}}-5000 x +10000}\) | \(150\) |
parallelrisch | \(\frac {400 x^{4} {\mathrm e}^{2 \,{\mathrm e}}-1000 x^{5} {\mathrm e}^{{\mathrm e}}+625 x^{6}-800 \,{\mathrm e}^{2 \,{\mathrm e}} x^{3}+6000 x^{4} {\mathrm e}^{{\mathrm e}}-6250 x^{5}+1792 \,{\mathrm e}^{4 \,{\mathrm e}}-4480 \,{\mathrm e}^{3 \,{\mathrm e}} x -1800 x^{3} {\mathrm e}^{{\mathrm e}}+16625 x^{4}+40448 \,{\mathrm e}^{3 \,{\mathrm e}}-75520 \,{\mathrm e}^{2 \,{\mathrm e}} x +15000 x^{3}+340224 \,{\mathrm e}^{2 \,{\mathrm e}}-421760 x \,{\mathrm e}^{{\mathrm e}}+1264640 \,{\mathrm e}^{{\mathrm e}}-780800 x +1753600}{400 \,{\mathrm e}^{2 \,{\mathrm e}}-1000 x \,{\mathrm e}^{{\mathrm e}}+625 x^{2}+4000 \,{\mathrm e}^{{\mathrm e}}-5000 x +10000}\) | \(150\) |
default | \(x^{4}-2 x^{3}-\frac {27 x^{2}}{5}+\frac {32 x \,{\mathrm e}^{{\mathrm e}}}{25}+\frac {64 x}{5}-\frac {128 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (125 \textit {\_Z}^{3}+\left (-300 \,{\mathrm e}^{{\mathrm e}}-1500\right ) \textit {\_Z}^{2}+\left (2400 \,{\mathrm e}^{{\mathrm e}}+240 \,{\mathrm e}^{2 \,{\mathrm e}}+6000\right ) \textit {\_Z} -4800 \,{\mathrm e}^{{\mathrm e}}-960 \,{\mathrm e}^{2 \,{\mathrm e}}-64 \,{\mathrm e}^{3 \,{\mathrm e}}-8000\right )}{\sum }\frac {\left (-5000+1150 \textit {\_R} \,{\mathrm e}^{{\mathrm e}}+275 \,{\mathrm e}^{2 \,{\mathrm e}} \textit {\_R} +20 \,{\mathrm e}^{3 \,{\mathrm e}} \textit {\_R} -5500 \,{\mathrm e}^{{\mathrm e}}-16 \,{\mathrm e}^{4 \,{\mathrm e}}-2000 \,{\mathrm e}^{2 \,{\mathrm e}}-300 \,{\mathrm e}^{3 \,{\mathrm e}}+1375 \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{400+16 \,{\mathrm e}^{2 \,{\mathrm e}}-40 \textit {\_R} \,{\mathrm e}^{{\mathrm e}}+25 \textit {\_R}^{2}+160 \,{\mathrm e}^{{\mathrm e}}-200 \textit {\_R}}\right )}{375}\) | \(173\) |
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Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (25) = 50\).
Time = 0.25 (sec) , antiderivative size = 135, normalized size of antiderivative = 5.62 \[ \int \frac {76800-156160 x+36000 x^2+50200 x^3-31650 x^4+6750 x^5-500 x^6+e^{3 e} \left (512-896 x-384 x^2+256 x^3\right )+e^{2 e} \left (8192-15104 x-2784 x^2+5280 x^3-960 x^4\right )+e^e \left (43520-84352 x+2880 x^2+30200 x^3-11400 x^4+1200 x^5\right )}{8000+64 e^{3 e}+e^{2 e} (960-240 x)-6000 x+1500 x^2-125 x^3+e^e \left (4800-2400 x+300 x^2\right )} \, dx=\frac {3125 \, x^{6} - 31250 \, x^{5} + 83125 \, x^{4} + 75000 \, x^{3} - 590000 \, x^{2} + 2560 \, {\left (2 \, x - 15\right )} e^{\left (3 \, e\right )} + 80 \, {\left (25 \, x^{4} - 50 \, x^{3} - 215 \, x^{2} + 1080 \, x - 3216\right )} e^{\left (2 \, e\right )} - 200 \, {\left (25 \, x^{5} - 150 \, x^{4} + 45 \, x^{3} + 1020 \, x^{2} - 2336 \, x + 3584\right )} e^{e} + 816000 \, x - 2048 \, e^{\left (4 \, e\right )} - 672000}{125 \, {\left (25 \, x^{2} - 40 \, {\left (x - 4\right )} e^{e} - 200 \, x + 16 \, e^{\left (2 \, e\right )} + 400\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 122 vs. \(2 (20) = 40\).
Time = 0.58 (sec) , antiderivative size = 122, normalized size of antiderivative = 5.08 \[ \int \frac {76800-156160 x+36000 x^2+50200 x^3-31650 x^4+6750 x^5-500 x^6+e^{3 e} \left (512-896 x-384 x^2+256 x^3\right )+e^{2 e} \left (8192-15104 x-2784 x^2+5280 x^3-960 x^4\right )+e^e \left (43520-84352 x+2880 x^2+30200 x^3-11400 x^4+1200 x^5\right )}{8000+64 e^{3 e}+e^{2 e} (960-240 x)-6000 x+1500 x^2-125 x^3+e^e \left (4800-2400 x+300 x^2\right )} \, dx=x^{4} - 2 x^{3} - \frac {27 x^{2}}{5} + x \left (\frac {64}{5} + \frac {32 e^{e}}{25}\right ) + \frac {x \left (176000 + 147200 e^{e} + 35200 e^{2 e} + 2560 e^{3 e}\right ) - 38400 e^{3 e} - 2048 e^{4 e} - 257280 e^{2 e} - 716800 e^{e} - 672000}{3125 x^{2} + x \left (- 5000 e^{e} - 25000\right ) + 50000 + 20000 e^{e} + 2000 e^{2 e}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 105 vs. \(2 (25) = 50\).
Time = 0.19 (sec) , antiderivative size = 105, normalized size of antiderivative = 4.38 \[ \int \frac {76800-156160 x+36000 x^2+50200 x^3-31650 x^4+6750 x^5-500 x^6+e^{3 e} \left (512-896 x-384 x^2+256 x^3\right )+e^{2 e} \left (8192-15104 x-2784 x^2+5280 x^3-960 x^4\right )+e^e \left (43520-84352 x+2880 x^2+30200 x^3-11400 x^4+1200 x^5\right )}{8000+64 e^{3 e}+e^{2 e} (960-240 x)-6000 x+1500 x^2-125 x^3+e^e \left (4800-2400 x+300 x^2\right )} \, dx=x^{4} - 2 \, x^{3} - \frac {27}{5} \, x^{2} + \frac {32}{25} \, x {\left (e^{e} + 10\right )} + \frac {128 \, {\left (5 \, x {\left (4 \, e^{\left (3 \, e\right )} + 55 \, e^{\left (2 \, e\right )} + 230 \, e^{e} + 275\right )} - 16 \, e^{\left (4 \, e\right )} - 300 \, e^{\left (3 \, e\right )} - 2010 \, e^{\left (2 \, e\right )} - 5600 \, e^{e} - 5250\right )}}{125 \, {\left (25 \, x^{2} - 40 \, x {\left (e^{e} + 5\right )} + 16 \, e^{\left (2 \, e\right )} + 160 \, e^{e} + 400\right )}} \]
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\[ \int \frac {76800-156160 x+36000 x^2+50200 x^3-31650 x^4+6750 x^5-500 x^6+e^{3 e} \left (512-896 x-384 x^2+256 x^3\right )+e^{2 e} \left (8192-15104 x-2784 x^2+5280 x^3-960 x^4\right )+e^e \left (43520-84352 x+2880 x^2+30200 x^3-11400 x^4+1200 x^5\right )}{8000+64 e^{3 e}+e^{2 e} (960-240 x)-6000 x+1500 x^2-125 x^3+e^e \left (4800-2400 x+300 x^2\right )} \, dx=\int { \frac {2 \, {\left (250 \, x^{6} - 3375 \, x^{5} + 15825 \, x^{4} - 25100 \, x^{3} - 18000 \, x^{2} - 64 \, {\left (2 \, x^{3} - 3 \, x^{2} - 7 \, x + 4\right )} e^{\left (3 \, e\right )} + 16 \, {\left (30 \, x^{4} - 165 \, x^{3} + 87 \, x^{2} + 472 \, x - 256\right )} e^{\left (2 \, e\right )} - 4 \, {\left (150 \, x^{5} - 1425 \, x^{4} + 3775 \, x^{3} + 360 \, x^{2} - 10544 \, x + 5440\right )} e^{e} + 78080 \, x - 38400\right )}}{125 \, x^{3} - 1500 \, x^{2} + 240 \, {\left (x - 4\right )} e^{\left (2 \, e\right )} - 300 \, {\left (x^{2} - 8 \, x + 16\right )} e^{e} + 6000 \, x - 64 \, e^{\left (3 \, e\right )} - 8000} \,d x } \]
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Time = 8.21 (sec) , antiderivative size = 195, normalized size of antiderivative = 8.12 \[ \int \frac {76800-156160 x+36000 x^2+50200 x^3-31650 x^4+6750 x^5-500 x^6+e^{3 e} \left (512-896 x-384 x^2+256 x^3\right )+e^{2 e} \left (8192-15104 x-2784 x^2+5280 x^3-960 x^4\right )+e^e \left (43520-84352 x+2880 x^2+30200 x^3-11400 x^4+1200 x^5\right )}{8000+64 e^{3 e}+e^{2 e} (960-240 x)-6000 x+1500 x^2-125 x^3+e^e \left (4800-2400 x+300 x^2\right )} \, dx=x^2\,\left (\frac {96\,{\mathrm {e}}^{2\,\mathrm {e}}}{25}+\frac {192\,{\mathrm {e}}^{\mathrm {e}}}{5}-\frac {96\,{\left ({\mathrm {e}}^{\mathrm {e}}+5\right )}^2}{25}+\frac {453}{5}\right )-x\,\left (\frac {1056\,{\mathrm {e}}^{2\,\mathrm {e}}}{25}+\frac {256\,{\mathrm {e}}^{3\,\mathrm {e}}}{125}+\frac {1208\,{\mathrm {e}}^{\mathrm {e}}}{5}-\frac {288\,{\left ({\mathrm {e}}^{\mathrm {e}}+5\right )}^2}{25}-\frac {256\,{\left ({\mathrm {e}}^{\mathrm {e}}+5\right )}^3}{125}-\left (\frac {12\,{\mathrm {e}}^{\mathrm {e}}}{5}+12\right )\,\left (\frac {192\,{\mathrm {e}}^{2\,\mathrm {e}}}{25}+\frac {384\,{\mathrm {e}}^{\mathrm {e}}}{5}-\frac {192\,{\left ({\mathrm {e}}^{\mathrm {e}}+5\right )}^2}{25}+\frac {906}{5}\right )+\frac {2008}{5}\right )-\frac {51456\,{\mathrm {e}}^{2\,\mathrm {e}}+7680\,{\mathrm {e}}^{3\,\mathrm {e}}+\frac {2048\,{\mathrm {e}}^{4\,\mathrm {e}}}{5}+143360\,{\mathrm {e}}^{\mathrm {e}}-x\,\left (7040\,{\mathrm {e}}^{2\,\mathrm {e}}+512\,{\mathrm {e}}^{3\,\mathrm {e}}+29440\,{\mathrm {e}}^{\mathrm {e}}+35200\right )+134400}{625\,x^2+\left (-1000\,{\mathrm {e}}^{\mathrm {e}}-5000\right )\,x+400\,{\mathrm {e}}^{2\,\mathrm {e}}+4000\,{\mathrm {e}}^{\mathrm {e}}+10000}-2\,x^3+x^4 \]
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