Integrand size = 196, antiderivative size = 28 \[ \int \frac {\frac {5 \left (12-6 e^x-18 x\right )}{e^2}+\frac {25 \left (-200+297 x+18 x^2+e^x (109+6 x)\right )}{e^4}+\frac {25 \left (8-4 e^x-12 x\right ) \log \left (e^x+x\right )}{e^4}}{9 e^x+9 x+\frac {5 \left (e^x (-300-18 x)-300 x-18 x^2\right )}{e^2}+\frac {25 \left (2500 x+309 x^2+9 x^3+e^x \left (2500+309 x+9 x^2\right )\right )}{e^4}+\left (\frac {5 \left (12 e^x+12 x\right )}{e^2}+\frac {25 \left (e^x (-200-12 x)-200 x-12 x^2\right )}{e^4}\right ) \log \left (e^x+x\right )+\frac {25 \left (4 e^x+4 x\right ) \log ^2\left (e^x+x\right )}{e^4}} \, dx=\log \left (x+\left (-\frac {e^2}{5}+x+\frac {2}{3} \left (25-\log \left (e^x+x\right )\right )\right )^2\right ) \]
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\[ \int \frac {\frac {5 \left (12-6 e^x-18 x\right )}{e^2}+\frac {25 \left (-200+297 x+18 x^2+e^x (109+6 x)\right )}{e^4}+\frac {25 \left (8-4 e^x-12 x\right ) \log \left (e^x+x\right )}{e^4}}{9 e^x+9 x+\frac {5 \left (e^x (-300-18 x)-300 x-18 x^2\right )}{e^2}+\frac {25 \left (2500 x+309 x^2+9 x^3+e^x \left (2500+309 x+9 x^2\right )\right )}{e^4}+\left (\frac {5 \left (12 e^x+12 x\right )}{e^2}+\frac {25 \left (e^x (-200-12 x)-200 x-12 x^2\right )}{e^4}\right ) \log \left (e^x+x\right )+\frac {25 \left (4 e^x+4 x\right ) \log ^2\left (e^x+x\right )}{e^4}} \, dx=\int \frac {\frac {5 \left (12-6 e^x-18 x\right )}{e^2}+\frac {25 \left (-200+297 x+18 x^2+e^x (109+6 x)\right )}{e^4}+\frac {25 \left (8-4 e^x-12 x\right ) \log \left (e^x+x\right )}{e^4}}{9 e^x+9 x+\frac {5 \left (e^x (-300-18 x)-300 x-18 x^2\right )}{e^2}+\frac {25 \left (2500 x+309 x^2+9 x^3+e^x \left (2500+309 x+9 x^2\right )\right )}{e^4}+\left (\frac {5 \left (12 e^x+12 x\right )}{e^2}+\frac {25 \left (e^x (-200-12 x)-200 x-12 x^2\right )}{e^4}\right ) \log \left (e^x+x\right )+\frac {25 \left (4 e^x+4 x\right ) \log ^2\left (e^x+x\right )}{e^4}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {5 \left (-6 e^{2+x}-6 e^2 (-2+3 x)+5 e^x (109+6 x)+5 \left (-200+297 x+18 x^2\right )-20 \left (-2+e^x+3 x\right ) \log \left (e^x+x\right )\right )}{\left (e^x+x\right ) \left (9 e^4-30 e^2 (50+3 x)+25 \left (2500+309 x+9 x^2\right )+20 \left (3 e^2-5 (50+3 x)\right ) \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )} \, dx \\ & = 5 \int \frac {-6 e^{2+x}-6 e^2 (-2+3 x)+5 e^x (109+6 x)+5 \left (-200+297 x+18 x^2\right )-20 \left (-2+e^x+3 x\right ) \log \left (e^x+x\right )}{\left (e^x+x\right ) \left (9 e^4-30 e^2 (50+3 x)+25 \left (2500+309 x+9 x^2\right )+20 \left (3 e^2-5 (50+3 x)\right ) \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )} \, dx \\ & = 5 \int \left (\frac {545 \left (1-\frac {6 e^2}{545}\right )+30 x-20 \log \left (e^x+x\right )}{62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )}+\frac {4 (1-x) \left (-250 \left (1-\frac {3 e^2}{250}\right )-15 x+10 \log \left (e^x+x\right )\right )}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )}\right ) \, dx \\ & = 5 \int \frac {545 \left (1-\frac {6 e^2}{545}\right )+30 x-20 \log \left (e^x+x\right )}{62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )} \, dx+20 \int \frac {(1-x) \left (-250 \left (1-\frac {3 e^2}{250}\right )-15 x+10 \log \left (e^x+x\right )\right )}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )} \, dx \\ & = 5 \int \left (\frac {20 \log \left (e^x+x\right )}{-62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )-7725 \left (1-\frac {6 e^2}{515}\right ) x-225 x^2+5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )+300 x \log \left (e^x+x\right )-100 \log ^2\left (e^x+x\right )}+\frac {545-6 e^2}{62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )}+\frac {30 x}{62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )}\right ) \, dx+20 \int \left (\frac {x \left (250 \left (1-\frac {3 e^2}{250}\right )+15 x-10 \log \left (e^x+x\right )\right )}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )}+\frac {-250 \left (1-\frac {3 e^2}{250}\right )-15 x+10 \log \left (e^x+x\right )}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )}\right ) \, dx \\ & = 20 \int \frac {x \left (250 \left (1-\frac {3 e^2}{250}\right )+15 x-10 \log \left (e^x+x\right )\right )}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )} \, dx+20 \int \frac {-250 \left (1-\frac {3 e^2}{250}\right )-15 x+10 \log \left (e^x+x\right )}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )} \, dx+100 \int \frac {\log \left (e^x+x\right )}{-62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )-7725 \left (1-\frac {6 e^2}{515}\right ) x-225 x^2+5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )+300 x \log \left (e^x+x\right )-100 \log ^2\left (e^x+x\right )} \, dx+150 \int \frac {x}{62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )} \, dx+\left (5 \left (545-6 e^2\right )\right ) \int \frac {1}{62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )} \, dx \\ & = 20 \int \left (\frac {\left (250-3 e^2\right ) x}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )}+\frac {15 x^2}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )}+\frac {10 x \log \left (e^x+x\right )}{\left (-e^x-x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )}\right ) \, dx+20 \int \left (\frac {15 x}{\left (-e^x-x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )}+\frac {-250+3 e^2}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )}+\frac {10 \log \left (e^x+x\right )}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )}\right ) \, dx+100 \int \frac {\log \left (e^x+x\right )}{-62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )-7725 \left (1-\frac {6 e^2}{515}\right ) x-225 x^2+5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )+300 x \log \left (e^x+x\right )-100 \log ^2\left (e^x+x\right )} \, dx+150 \int \frac {x}{62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )} \, dx+\left (5 \left (545-6 e^2\right )\right ) \int \frac {1}{62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )} \, dx \\ & = 100 \int \frac {\log \left (e^x+x\right )}{-62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )-7725 \left (1-\frac {6 e^2}{515}\right ) x-225 x^2+5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )+300 x \log \left (e^x+x\right )-100 \log ^2\left (e^x+x\right )} \, dx+150 \int \frac {x}{62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )} \, dx+200 \int \frac {x \log \left (e^x+x\right )}{\left (-e^x-x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )} \, dx+200 \int \frac {\log \left (e^x+x\right )}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )} \, dx+300 \int \frac {x}{\left (-e^x-x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )} \, dx+300 \int \frac {x^2}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )} \, dx+\left (5 \left (545-6 e^2\right )\right ) \int \frac {1}{62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )} \, dx-\left (20 \left (250-3 e^2\right )\right ) \int \frac {1}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )} \, dx+\left (20 \left (250-3 e^2\right )\right ) \int \frac {x}{\left (e^x+x\right ) \left (62500 \left (1+\frac {3 e^2 \left (-500+3 e^2\right )}{62500}\right )+7725 \left (1-\frac {6 e^2}{515}\right ) x+225 x^2-5000 \left (1-\frac {3 e^2}{250}\right ) \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right )} \, dx \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(65\) vs. \(2(28)=56\).
Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.32 \[ \int \frac {\frac {5 \left (12-6 e^x-18 x\right )}{e^2}+\frac {25 \left (-200+297 x+18 x^2+e^x (109+6 x)\right )}{e^4}+\frac {25 \left (8-4 e^x-12 x\right ) \log \left (e^x+x\right )}{e^4}}{9 e^x+9 x+\frac {5 \left (e^x (-300-18 x)-300 x-18 x^2\right )}{e^2}+\frac {25 \left (2500 x+309 x^2+9 x^3+e^x \left (2500+309 x+9 x^2\right )\right )}{e^4}+\left (\frac {5 \left (12 e^x+12 x\right )}{e^2}+\frac {25 \left (e^x (-200-12 x)-200 x-12 x^2\right )}{e^4}\right ) \log \left (e^x+x\right )+\frac {25 \left (4 e^x+4 x\right ) \log ^2\left (e^x+x\right )}{e^4}} \, dx=\log \left (62500-1500 e^2+9 e^4+7725 x-90 e^2 x+225 x^2-5000 \log \left (e^x+x\right )+60 e^2 \log \left (e^x+x\right )-300 x \log \left (e^x+x\right )+100 \log ^2\left (e^x+x\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(55\) vs. \(2(24)=48\).
Time = 0.46 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.00
method | result | size |
risch | \(\ln \left (\ln \left ({\mathrm e}^{x}+x \right )^{2}-3 x \ln \left ({\mathrm e}^{x}+x \right )-50 \ln \left ({\mathrm e}^{x}+x \right )+\frac {3 \,{\mathrm e}^{2} \ln \left ({\mathrm e}^{x}+x \right )}{5}+\frac {9 x^{2}}{4}+\frac {309 x}{4}-\frac {9 \,{\mathrm e}^{2} x}{10}+625-15 \,{\mathrm e}^{2}+\frac {9 \,{\mathrm e}^{4}}{100}\right )\) | \(56\) |
parallelrisch | \(\ln \left (\frac {\left (100 \,{\mathrm e}^{-4} \ln \left ({\mathrm e}^{x}+x \right )^{2}-300 \,{\mathrm e}^{-4} \ln \left ({\mathrm e}^{x}+x \right ) x +225 \,{\mathrm e}^{-4} x^{2}-5000 \,{\mathrm e}^{-4} \ln \left ({\mathrm e}^{x}+x \right )+7725 \,{\mathrm e}^{-4} x +62500 \,{\mathrm e}^{-4}+12 \,{\mathrm e}^{\ln \left (5\right )-2} \ln \left ({\mathrm e}^{x}+x \right )-18 \,{\mathrm e}^{\ln \left (5\right )-2} x -300 \,{\mathrm e}^{\ln \left (5\right )-2}+9\right ) {\mathrm e}^{4}}{225}\right )\) | \(116\) |
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Leaf count of result is larger than twice the leaf count of optimal. 77 vs. \(2 (28) = 56\).
Time = 0.24 (sec) , antiderivative size = 77, normalized size of antiderivative = 2.75 \[ \int \frac {\frac {5 \left (12-6 e^x-18 x\right )}{e^2}+\frac {25 \left (-200+297 x+18 x^2+e^x (109+6 x)\right )}{e^4}+\frac {25 \left (8-4 e^x-12 x\right ) \log \left (e^x+x\right )}{e^4}}{9 e^x+9 x+\frac {5 \left (e^x (-300-18 x)-300 x-18 x^2\right )}{e^2}+\frac {25 \left (2500 x+309 x^2+9 x^3+e^x \left (2500+309 x+9 x^2\right )\right )}{e^4}+\left (\frac {5 \left (12 e^x+12 x\right )}{e^2}+\frac {25 \left (e^x (-200-12 x)-200 x-12 x^2\right )}{e^4}\right ) \log \left (e^x+x\right )+\frac {25 \left (4 e^x+4 x\right ) \log ^2\left (e^x+x\right )}{e^4}} \, dx=\log \left (4 \, e^{\left (2 \, \log \left (5\right ) - 4\right )} \log \left (x + e^{x}\right )^{2} + {\left (9 \, x^{2} + 309 \, x + 2500\right )} e^{\left (2 \, \log \left (5\right ) - 4\right )} - 6 \, {\left (3 \, x + 50\right )} e^{\left (\log \left (5\right ) - 2\right )} - 4 \, {\left ({\left (3 \, x + 50\right )} e^{\left (2 \, \log \left (5\right ) - 4\right )} - 3 \, e^{\left (\log \left (5\right ) - 2\right )}\right )} \log \left (x + e^{x}\right ) + 9\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (24) = 48\).
Time = 0.43 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.18 \[ \int \frac {\frac {5 \left (12-6 e^x-18 x\right )}{e^2}+\frac {25 \left (-200+297 x+18 x^2+e^x (109+6 x)\right )}{e^4}+\frac {25 \left (8-4 e^x-12 x\right ) \log \left (e^x+x\right )}{e^4}}{9 e^x+9 x+\frac {5 \left (e^x (-300-18 x)-300 x-18 x^2\right )}{e^2}+\frac {25 \left (2500 x+309 x^2+9 x^3+e^x \left (2500+309 x+9 x^2\right )\right )}{e^4}+\left (\frac {5 \left (12 e^x+12 x\right )}{e^2}+\frac {25 \left (e^x (-200-12 x)-200 x-12 x^2\right )}{e^4}\right ) \log \left (e^x+x\right )+\frac {25 \left (4 e^x+4 x\right ) \log ^2\left (e^x+x\right )}{e^4}} \, dx=\log {\left (\frac {9 x^{2}}{4} - \frac {9 x e^{2}}{10} + \frac {309 x}{4} + \left (- 3 x - 50 + \frac {3 e^{2}}{5}\right ) \log {\left (x + e^{x} \right )} + \log {\left (x + e^{x} \right )}^{2} - 15 e^{2} + \frac {9 e^{4}}{100} + 625 \right )} \]
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Time = 0.27 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71 \[ \int \frac {\frac {5 \left (12-6 e^x-18 x\right )}{e^2}+\frac {25 \left (-200+297 x+18 x^2+e^x (109+6 x)\right )}{e^4}+\frac {25 \left (8-4 e^x-12 x\right ) \log \left (e^x+x\right )}{e^4}}{9 e^x+9 x+\frac {5 \left (e^x (-300-18 x)-300 x-18 x^2\right )}{e^2}+\frac {25 \left (2500 x+309 x^2+9 x^3+e^x \left (2500+309 x+9 x^2\right )\right )}{e^4}+\left (\frac {5 \left (12 e^x+12 x\right )}{e^2}+\frac {25 \left (e^x (-200-12 x)-200 x-12 x^2\right )}{e^4}\right ) \log \left (e^x+x\right )+\frac {25 \left (4 e^x+4 x\right ) \log ^2\left (e^x+x\right )}{e^4}} \, dx=\log \left (\frac {9}{4} \, x^{2} - \frac {3}{20} \, x {\left (6 \, e^{2} - 515\right )} - \frac {1}{5} \, {\left (15 \, x - 3 \, e^{2} + 250\right )} \log \left (x + e^{x}\right ) + \log \left (x + e^{x}\right )^{2} + \frac {9}{100} \, e^{4} - 15 \, e^{2} + 625\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (28) = 56\).
Time = 0.39 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.04 \[ \int \frac {\frac {5 \left (12-6 e^x-18 x\right )}{e^2}+\frac {25 \left (-200+297 x+18 x^2+e^x (109+6 x)\right )}{e^4}+\frac {25 \left (8-4 e^x-12 x\right ) \log \left (e^x+x\right )}{e^4}}{9 e^x+9 x+\frac {5 \left (e^x (-300-18 x)-300 x-18 x^2\right )}{e^2}+\frac {25 \left (2500 x+309 x^2+9 x^3+e^x \left (2500+309 x+9 x^2\right )\right )}{e^4}+\left (\frac {5 \left (12 e^x+12 x\right )}{e^2}+\frac {25 \left (e^x (-200-12 x)-200 x-12 x^2\right )}{e^4}\right ) \log \left (e^x+x\right )+\frac {25 \left (4 e^x+4 x\right ) \log ^2\left (e^x+x\right )}{e^4}} \, dx=\log \left (225 \, x^{2} - 90 \, x e^{2} - 300 \, x \log \left (x + e^{x}\right ) + 60 \, e^{2} \log \left (x + e^{x}\right ) + 100 \, \log \left (x + e^{x}\right )^{2} + 7725 \, x + 9 \, e^{4} - 1500 \, e^{2} - 5000 \, \log \left (x + e^{x}\right ) + 62500\right ) \]
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Time = 12.22 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.96 \[ \int \frac {\frac {5 \left (12-6 e^x-18 x\right )}{e^2}+\frac {25 \left (-200+297 x+18 x^2+e^x (109+6 x)\right )}{e^4}+\frac {25 \left (8-4 e^x-12 x\right ) \log \left (e^x+x\right )}{e^4}}{9 e^x+9 x+\frac {5 \left (e^x (-300-18 x)-300 x-18 x^2\right )}{e^2}+\frac {25 \left (2500 x+309 x^2+9 x^3+e^x \left (2500+309 x+9 x^2\right )\right )}{e^4}+\left (\frac {5 \left (12 e^x+12 x\right )}{e^2}+\frac {25 \left (e^x (-200-12 x)-200 x-12 x^2\right )}{e^4}\right ) \log \left (e^x+x\right )+\frac {25 \left (4 e^x+4 x\right ) \log ^2\left (e^x+x\right )}{e^4}} \, dx=\ln \left (\frac {309\,x}{4}-50\,\ln \left (x+{\mathrm {e}}^x\right )-15\,{\mathrm {e}}^2+\frac {9\,{\mathrm {e}}^4}{100}-\frac {9\,x\,{\mathrm {e}}^2}{10}+{\ln \left (x+{\mathrm {e}}^x\right )}^2+\frac {9\,x^2}{4}+\frac {3\,\ln \left (x+{\mathrm {e}}^x\right )\,{\mathrm {e}}^2}{5}-3\,x\,\ln \left (x+{\mathrm {e}}^x\right )+625\right ) \]
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