Integrand size = 267, antiderivative size = 29 \[ \int \frac {20 e^5+20 e^{5+x}+\left (20+20 e^x\right ) \log \left (-e^{e^3}+e^x+x\right )}{e^x \left (-16+40 e^{10}-25 e^{20}\right )+e^{e^3} \left (16-40 e^{10}+25 e^{20}\right )-16 x+40 e^{10} x-25 e^{20} x+\left (e^x \left (80 e^5-100 e^{15}\right )+e^{e^3} \left (-80 e^5+100 e^{15}\right )+80 e^5 x-100 e^{15} x\right ) \log \left (-e^{e^3}+e^x+x\right )+\left (e^x \left (40-150 e^{10}\right )+e^{e^3} \left (-40+150 e^{10}\right )+40 x-150 e^{10} x\right ) \log ^2\left (-e^{e^3}+e^x+x\right )+\left (100 e^{5+e^3}-100 e^{5+x}-100 e^5 x\right ) \log ^3\left (-e^{e^3}+e^x+x\right )+\left (25 e^{e^3}-25 e^x-25 x\right ) \log ^4\left (-e^{e^3}+e^x+x\right )} \, dx=-4+\frac {2}{-4+5 \left (e^5+\log \left (-e^{e^3}+e^x+x\right )\right )^2} \]
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Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.015, Rules used = {6, 6820, 12, 6818} \[ \int \frac {20 e^5+20 e^{5+x}+\left (20+20 e^x\right ) \log \left (-e^{e^3}+e^x+x\right )}{e^x \left (-16+40 e^{10}-25 e^{20}\right )+e^{e^3} \left (16-40 e^{10}+25 e^{20}\right )-16 x+40 e^{10} x-25 e^{20} x+\left (e^x \left (80 e^5-100 e^{15}\right )+e^{e^3} \left (-80 e^5+100 e^{15}\right )+80 e^5 x-100 e^{15} x\right ) \log \left (-e^{e^3}+e^x+x\right )+\left (e^x \left (40-150 e^{10}\right )+e^{e^3} \left (-40+150 e^{10}\right )+40 x-150 e^{10} x\right ) \log ^2\left (-e^{e^3}+e^x+x\right )+\left (100 e^{5+e^3}-100 e^{5+x}-100 e^5 x\right ) \log ^3\left (-e^{e^3}+e^x+x\right )+\left (25 e^{e^3}-25 e^x-25 x\right ) \log ^4\left (-e^{e^3}+e^x+x\right )} \, dx=-\frac {2}{-5 \log ^2\left (x+e^x-e^{e^3}\right )-10 e^5 \log \left (x+e^x-e^{e^3}\right )-5 e^{10}+4} \]
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Rule 6
Rule 12
Rule 6818
Rule 6820
Rubi steps \begin{align*} \text {integral}& = \int \frac {20 e^5+20 e^{5+x}+\left (20+20 e^x\right ) \log \left (-e^{e^3}+e^x+x\right )}{e^x \left (-16+40 e^{10}-25 e^{20}\right )+e^{e^3} \left (16-40 e^{10}+25 e^{20}\right )-25 e^{20} x+\left (-16+40 e^{10}\right ) x+\left (e^x \left (80 e^5-100 e^{15}\right )+e^{e^3} \left (-80 e^5+100 e^{15}\right )+80 e^5 x-100 e^{15} x\right ) \log \left (-e^{e^3}+e^x+x\right )+\left (e^x \left (40-150 e^{10}\right )+e^{e^3} \left (-40+150 e^{10}\right )+40 x-150 e^{10} x\right ) \log ^2\left (-e^{e^3}+e^x+x\right )+\left (100 e^{5+e^3}-100 e^{5+x}-100 e^5 x\right ) \log ^3\left (-e^{e^3}+e^x+x\right )+\left (25 e^{e^3}-25 e^x-25 x\right ) \log ^4\left (-e^{e^3}+e^x+x\right )} \, dx \\ & = \int \frac {20 e^5+20 e^{5+x}+\left (20+20 e^x\right ) \log \left (-e^{e^3}+e^x+x\right )}{e^x \left (-16+40 e^{10}-25 e^{20}\right )+e^{e^3} \left (16-40 e^{10}+25 e^{20}\right )+\left (-16+40 e^{10}-25 e^{20}\right ) x+\left (e^x \left (80 e^5-100 e^{15}\right )+e^{e^3} \left (-80 e^5+100 e^{15}\right )+80 e^5 x-100 e^{15} x\right ) \log \left (-e^{e^3}+e^x+x\right )+\left (e^x \left (40-150 e^{10}\right )+e^{e^3} \left (-40+150 e^{10}\right )+40 x-150 e^{10} x\right ) \log ^2\left (-e^{e^3}+e^x+x\right )+\left (100 e^{5+e^3}-100 e^{5+x}-100 e^5 x\right ) \log ^3\left (-e^{e^3}+e^x+x\right )+\left (25 e^{e^3}-25 e^x-25 x\right ) \log ^4\left (-e^{e^3}+e^x+x\right )} \, dx \\ & = \int \frac {20 \left (1+e^x\right ) \left (e^5+\log \left (-e^{e^3}+e^x+x\right )\right )}{\left (e^{e^3}-e^x-x\right ) \left (4 \left (1-\frac {5 e^{10}}{4}\right )-10 e^5 \log \left (-e^{e^3}+e^x+x\right )-5 \log ^2\left (-e^{e^3}+e^x+x\right )\right )^2} \, dx \\ & = 20 \int \frac {\left (1+e^x\right ) \left (e^5+\log \left (-e^{e^3}+e^x+x\right )\right )}{\left (e^{e^3}-e^x-x\right ) \left (4 \left (1-\frac {5 e^{10}}{4}\right )-10 e^5 \log \left (-e^{e^3}+e^x+x\right )-5 \log ^2\left (-e^{e^3}+e^x+x\right )\right )^2} \, dx \\ & = -\frac {2}{4-5 e^{10}-10 e^5 \log \left (-e^{e^3}+e^x+x\right )-5 \log ^2\left (-e^{e^3}+e^x+x\right )} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.59 \[ \int \frac {20 e^5+20 e^{5+x}+\left (20+20 e^x\right ) \log \left (-e^{e^3}+e^x+x\right )}{e^x \left (-16+40 e^{10}-25 e^{20}\right )+e^{e^3} \left (16-40 e^{10}+25 e^{20}\right )-16 x+40 e^{10} x-25 e^{20} x+\left (e^x \left (80 e^5-100 e^{15}\right )+e^{e^3} \left (-80 e^5+100 e^{15}\right )+80 e^5 x-100 e^{15} x\right ) \log \left (-e^{e^3}+e^x+x\right )+\left (e^x \left (40-150 e^{10}\right )+e^{e^3} \left (-40+150 e^{10}\right )+40 x-150 e^{10} x\right ) \log ^2\left (-e^{e^3}+e^x+x\right )+\left (100 e^{5+e^3}-100 e^{5+x}-100 e^5 x\right ) \log ^3\left (-e^{e^3}+e^x+x\right )+\left (25 e^{e^3}-25 e^x-25 x\right ) \log ^4\left (-e^{e^3}+e^x+x\right )} \, dx=\frac {2}{-4+5 e^{10}+10 e^5 \log \left (-e^{e^3}+e^x+x\right )+5 \log ^2\left (-e^{e^3}+e^x+x\right )} \]
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Time = 1.70 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34
method | result | size |
risch | \(\frac {2}{5 \,{\mathrm e}^{10}+10 \,{\mathrm e}^{5} \ln \left (-{\mathrm e}^{{\mathrm e}^{3}}+{\mathrm e}^{x}+x \right )+5 \ln \left (-{\mathrm e}^{{\mathrm e}^{3}}+{\mathrm e}^{x}+x \right )^{2}-4}\) | \(39\) |
norman | \(\frac {2}{5 \,{\mathrm e}^{10}+10 \,{\mathrm e}^{5} \ln \left (-{\mathrm e}^{{\mathrm e}^{3}}+{\mathrm e}^{x}+x \right )+5 \ln \left (-{\mathrm e}^{{\mathrm e}^{3}}+{\mathrm e}^{x}+x \right )^{2}-4}\) | \(41\) |
parallelrisch | \(\frac {2}{5 \,{\mathrm e}^{10}+10 \,{\mathrm e}^{5} \ln \left (-{\mathrm e}^{{\mathrm e}^{3}}+{\mathrm e}^{x}+x \right )+5 \ln \left (-{\mathrm e}^{{\mathrm e}^{3}}+{\mathrm e}^{x}+x \right )^{2}-4}\) | \(41\) |
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (25) = 50\).
Time = 0.30 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.00 \[ \int \frac {20 e^5+20 e^{5+x}+\left (20+20 e^x\right ) \log \left (-e^{e^3}+e^x+x\right )}{e^x \left (-16+40 e^{10}-25 e^{20}\right )+e^{e^3} \left (16-40 e^{10}+25 e^{20}\right )-16 x+40 e^{10} x-25 e^{20} x+\left (e^x \left (80 e^5-100 e^{15}\right )+e^{e^3} \left (-80 e^5+100 e^{15}\right )+80 e^5 x-100 e^{15} x\right ) \log \left (-e^{e^3}+e^x+x\right )+\left (e^x \left (40-150 e^{10}\right )+e^{e^3} \left (-40+150 e^{10}\right )+40 x-150 e^{10} x\right ) \log ^2\left (-e^{e^3}+e^x+x\right )+\left (100 e^{5+e^3}-100 e^{5+x}-100 e^5 x\right ) \log ^3\left (-e^{e^3}+e^x+x\right )+\left (25 e^{e^3}-25 e^x-25 x\right ) \log ^4\left (-e^{e^3}+e^x+x\right )} \, dx=\frac {2}{10 \, e^{5} \log \left ({\left (x e^{5} + e^{\left (x + 5\right )} - e^{\left (e^{3} + 5\right )}\right )} e^{\left (-5\right )}\right ) + 5 \, \log \left ({\left (x e^{5} + e^{\left (x + 5\right )} - e^{\left (e^{3} + 5\right )}\right )} e^{\left (-5\right )}\right )^{2} + 5 \, e^{10} - 4} \]
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Time = 0.18 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.34 \[ \int \frac {20 e^5+20 e^{5+x}+\left (20+20 e^x\right ) \log \left (-e^{e^3}+e^x+x\right )}{e^x \left (-16+40 e^{10}-25 e^{20}\right )+e^{e^3} \left (16-40 e^{10}+25 e^{20}\right )-16 x+40 e^{10} x-25 e^{20} x+\left (e^x \left (80 e^5-100 e^{15}\right )+e^{e^3} \left (-80 e^5+100 e^{15}\right )+80 e^5 x-100 e^{15} x\right ) \log \left (-e^{e^3}+e^x+x\right )+\left (e^x \left (40-150 e^{10}\right )+e^{e^3} \left (-40+150 e^{10}\right )+40 x-150 e^{10} x\right ) \log ^2\left (-e^{e^3}+e^x+x\right )+\left (100 e^{5+e^3}-100 e^{5+x}-100 e^5 x\right ) \log ^3\left (-e^{e^3}+e^x+x\right )+\left (25 e^{e^3}-25 e^x-25 x\right ) \log ^4\left (-e^{e^3}+e^x+x\right )} \, dx=\frac {2}{5 \log {\left (x + e^{x} - e^{e^{3}} \right )}^{2} + 10 e^{5} \log {\left (x + e^{x} - e^{e^{3}} \right )} - 4 + 5 e^{10}} \]
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Time = 0.60 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.31 \[ \int \frac {20 e^5+20 e^{5+x}+\left (20+20 e^x\right ) \log \left (-e^{e^3}+e^x+x\right )}{e^x \left (-16+40 e^{10}-25 e^{20}\right )+e^{e^3} \left (16-40 e^{10}+25 e^{20}\right )-16 x+40 e^{10} x-25 e^{20} x+\left (e^x \left (80 e^5-100 e^{15}\right )+e^{e^3} \left (-80 e^5+100 e^{15}\right )+80 e^5 x-100 e^{15} x\right ) \log \left (-e^{e^3}+e^x+x\right )+\left (e^x \left (40-150 e^{10}\right )+e^{e^3} \left (-40+150 e^{10}\right )+40 x-150 e^{10} x\right ) \log ^2\left (-e^{e^3}+e^x+x\right )+\left (100 e^{5+e^3}-100 e^{5+x}-100 e^5 x\right ) \log ^3\left (-e^{e^3}+e^x+x\right )+\left (25 e^{e^3}-25 e^x-25 x\right ) \log ^4\left (-e^{e^3}+e^x+x\right )} \, dx=\frac {2}{10 \, e^{5} \log \left (x + e^{x} - e^{\left (e^{3}\right )}\right ) + 5 \, \log \left (x + e^{x} - e^{\left (e^{3}\right )}\right )^{2} + 5 \, e^{10} - 4} \]
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Timed out. \[ \int \frac {20 e^5+20 e^{5+x}+\left (20+20 e^x\right ) \log \left (-e^{e^3}+e^x+x\right )}{e^x \left (-16+40 e^{10}-25 e^{20}\right )+e^{e^3} \left (16-40 e^{10}+25 e^{20}\right )-16 x+40 e^{10} x-25 e^{20} x+\left (e^x \left (80 e^5-100 e^{15}\right )+e^{e^3} \left (-80 e^5+100 e^{15}\right )+80 e^5 x-100 e^{15} x\right ) \log \left (-e^{e^3}+e^x+x\right )+\left (e^x \left (40-150 e^{10}\right )+e^{e^3} \left (-40+150 e^{10}\right )+40 x-150 e^{10} x\right ) \log ^2\left (-e^{e^3}+e^x+x\right )+\left (100 e^{5+e^3}-100 e^{5+x}-100 e^5 x\right ) \log ^3\left (-e^{e^3}+e^x+x\right )+\left (25 e^{e^3}-25 e^x-25 x\right ) \log ^4\left (-e^{e^3}+e^x+x\right )} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {20 e^5+20 e^{5+x}+\left (20+20 e^x\right ) \log \left (-e^{e^3}+e^x+x\right )}{e^x \left (-16+40 e^{10}-25 e^{20}\right )+e^{e^3} \left (16-40 e^{10}+25 e^{20}\right )-16 x+40 e^{10} x-25 e^{20} x+\left (e^x \left (80 e^5-100 e^{15}\right )+e^{e^3} \left (-80 e^5+100 e^{15}\right )+80 e^5 x-100 e^{15} x\right ) \log \left (-e^{e^3}+e^x+x\right )+\left (e^x \left (40-150 e^{10}\right )+e^{e^3} \left (-40+150 e^{10}\right )+40 x-150 e^{10} x\right ) \log ^2\left (-e^{e^3}+e^x+x\right )+\left (100 e^{5+e^3}-100 e^{5+x}-100 e^5 x\right ) \log ^3\left (-e^{e^3}+e^x+x\right )+\left (25 e^{e^3}-25 e^x-25 x\right ) \log ^4\left (-e^{e^3}+e^x+x\right )} \, dx=\int -\frac {20\,{\mathrm {e}}^5+\ln \left (x-{\mathrm {e}}^{{\mathrm {e}}^3}+{\mathrm {e}}^x\right )\,\left (20\,{\mathrm {e}}^x+20\right )+20\,{\mathrm {e}}^5\,{\mathrm {e}}^x}{\left (25\,x-25\,{\mathrm {e}}^{{\mathrm {e}}^3}+25\,{\mathrm {e}}^x\right )\,{\ln \left (x-{\mathrm {e}}^{{\mathrm {e}}^3}+{\mathrm {e}}^x\right )}^4+\left (100\,x\,{\mathrm {e}}^5-100\,{\mathrm {e}}^5\,{\mathrm {e}}^{{\mathrm {e}}^3}+100\,{\mathrm {e}}^5\,{\mathrm {e}}^x\right )\,{\ln \left (x-{\mathrm {e}}^{{\mathrm {e}}^3}+{\mathrm {e}}^x\right )}^3+\left (150\,x\,{\mathrm {e}}^{10}-40\,x-{\mathrm {e}}^{{\mathrm {e}}^3}\,\left (150\,{\mathrm {e}}^{10}-40\right )+{\mathrm {e}}^x\,\left (150\,{\mathrm {e}}^{10}-40\right )\right )\,{\ln \left (x-{\mathrm {e}}^{{\mathrm {e}}^3}+{\mathrm {e}}^x\right )}^2+\left ({\mathrm {e}}^{{\mathrm {e}}^3}\,\left (80\,{\mathrm {e}}^5-100\,{\mathrm {e}}^{15}\right )-{\mathrm {e}}^x\,\left (80\,{\mathrm {e}}^5-100\,{\mathrm {e}}^{15}\right )-80\,x\,{\mathrm {e}}^5+100\,x\,{\mathrm {e}}^{15}\right )\,\ln \left (x-{\mathrm {e}}^{{\mathrm {e}}^3}+{\mathrm {e}}^x\right )+16\,x-{\mathrm {e}}^{{\mathrm {e}}^3}\,\left (25\,{\mathrm {e}}^{20}-40\,{\mathrm {e}}^{10}+16\right )+{\mathrm {e}}^x\,\left (25\,{\mathrm {e}}^{20}-40\,{\mathrm {e}}^{10}+16\right )-40\,x\,{\mathrm {e}}^{10}+25\,x\,{\mathrm {e}}^{20}} \,d x \]
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