Integrand size = 250, antiderivative size = 33 \[ \int \frac {-x^2+\left (e^x \left (2 x^2+2 x^3+75 x^4+65 x^5+15 x^6+x^7\right )+e^{2 x} \left (-40 x-1000 x^3-400 x^4-6290 x^5-5000 x^6-1500 x^7-200 x^8-10 x^9\right )\right ) \log (x)+\left (-x^2+e^x \left (40 x+500 x^3+200 x^4+20 x^5\right )\right ) \log (x) \log (\log (x))-10 x \log (x) \log ^2(\log (x))}{\left (x^2+e^x \left (-20 x-250 x^3-100 x^4-10 x^5\right )+e^{2 x} \left (100+2500 x^2+1000 x^3+15725 x^4+12500 x^5+3750 x^6+500 x^7+25 x^8\right )\right ) \log (x)+\left (10 x+e^x \left (-100-1250 x^2-500 x^3-50 x^4\right )\right ) \log (x) \log (\log (x))+25 \log (x) \log ^2(\log (x))} \, dx=\frac {x^2}{-5+\frac {x}{e^x \left (2+x^2 (5+x)^2\right )-\log (\log (x))}} \]
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\[ \int \frac {-x^2+\left (e^x \left (2 x^2+2 x^3+75 x^4+65 x^5+15 x^6+x^7\right )+e^{2 x} \left (-40 x-1000 x^3-400 x^4-6290 x^5-5000 x^6-1500 x^7-200 x^8-10 x^9\right )\right ) \log (x)+\left (-x^2+e^x \left (40 x+500 x^3+200 x^4+20 x^5\right )\right ) \log (x) \log (\log (x))-10 x \log (x) \log ^2(\log (x))}{\left (x^2+e^x \left (-20 x-250 x^3-100 x^4-10 x^5\right )+e^{2 x} \left (100+2500 x^2+1000 x^3+15725 x^4+12500 x^5+3750 x^6+500 x^7+25 x^8\right )\right ) \log (x)+\left (10 x+e^x \left (-100-1250 x^2-500 x^3-50 x^4\right )\right ) \log (x) \log (\log (x))+25 \log (x) \log ^2(\log (x))} \, dx=\int \frac {-x^2+\left (e^x \left (2 x^2+2 x^3+75 x^4+65 x^5+15 x^6+x^7\right )+e^{2 x} \left (-40 x-1000 x^3-400 x^4-6290 x^5-5000 x^6-1500 x^7-200 x^8-10 x^9\right )\right ) \log (x)+\left (-x^2+e^x \left (40 x+500 x^3+200 x^4+20 x^5\right )\right ) \log (x) \log (\log (x))-10 x \log (x) \log ^2(\log (x))}{\left (x^2+e^x \left (-20 x-250 x^3-100 x^4-10 x^5\right )+e^{2 x} \left (100+2500 x^2+1000 x^3+15725 x^4+12500 x^5+3750 x^6+500 x^7+25 x^8\right )\right ) \log (x)+\left (10 x+e^x \left (-100-1250 x^2-500 x^3-50 x^4\right )\right ) \log (x) \log (\log (x))+25 \log (x) \log ^2(\log (x))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (-x-\log (x) \left (e^x \left (10 e^x \left (2+25 x^2+10 x^3+x^4\right )^2-x \left (2+2 x+75 x^2+65 x^3+15 x^4+x^5\right )\right )+\left (x-20 e^x \left (2+25 x^2+10 x^3+x^4\right )\right ) \log (\log (x))+10 \log ^2(\log (x))\right )\right )}{\log (x) \left (x-5 e^x \left (2+25 x^2+10 x^3+x^4\right )+5 \log (\log (x))\right )^2} \, dx \\ & = \int \left (-\frac {2 x}{5}+\frac {x^2 \left (-6+2 x-25 x^2+25 x^3+11 x^4+x^5\right )}{5 \left (2+25 x^2+10 x^3+x^4\right ) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )}+\frac {x^2 \left (-10-125 x^2-50 x^3-5 x^4-2 x \log (x)+2 x^2 \log (x)+25 x^3 \log (x)+45 x^4 \log (x)+13 x^5 \log (x)+x^6 \log (x)+10 x \log (x) \log (\log (x))+250 x^2 \log (x) \log (\log (x))+275 x^3 \log (x) \log (\log (x))+70 x^4 \log (x) \log (\log (x))+5 x^5 \log (x) \log (\log (x))\right )}{5 \left (2+25 x^2+10 x^3+x^4\right ) \log (x) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )^2}\right ) \, dx \\ & = -\frac {x^2}{5}+\frac {1}{5} \int \frac {x^2 \left (-6+2 x-25 x^2+25 x^3+11 x^4+x^5\right )}{\left (2+25 x^2+10 x^3+x^4\right ) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )} \, dx+\frac {1}{5} \int \frac {x^2 \left (-10-125 x^2-50 x^3-5 x^4-2 x \log (x)+2 x^2 \log (x)+25 x^3 \log (x)+45 x^4 \log (x)+13 x^5 \log (x)+x^6 \log (x)+10 x \log (x) \log (\log (x))+250 x^2 \log (x) \log (\log (x))+275 x^3 \log (x) \log (\log (x))+70 x^4 \log (x) \log (\log (x))+5 x^5 \log (x) \log (\log (x))\right )}{\left (2+25 x^2+10 x^3+x^4\right ) \log (x) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )^2} \, dx \\ & = -\frac {x^2}{5}+\frac {1}{5} \int \left (\frac {50}{10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))}-\frac {10 x}{10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))}+\frac {x^2}{10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))}+\frac {x^3}{10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))}-\frac {2 \left (50-10 x+629 x^2+125 x^3\right )}{\left (2+25 x^2+10 x^3+x^4\right ) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )}\right ) \, dx+\frac {1}{5} \int \frac {x^2 \left (-5 \left (2+25 x^2+10 x^3+x^4\right )+x \log (x) \left (-2+2 x+25 x^2+45 x^3+13 x^4+x^5+5 \left (2+50 x+55 x^2+14 x^3+x^4\right ) \log (\log (x))\right )\right )}{\left (2+25 x^2+10 x^3+x^4\right ) \log (x) \left (x-5 e^x \left (2+25 x^2+10 x^3+x^4\right )+5 \log (\log (x))\right )^2} \, dx \\ & = -\frac {x^2}{5}+\frac {1}{5} \int \frac {x^2}{10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))} \, dx+\frac {1}{5} \int \frac {x^3}{10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))} \, dx+\frac {1}{5} \int \left (-\frac {2 x^3}{\left (2+25 x^2+10 x^3+x^4\right ) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )^2}+\frac {2 x^4}{\left (2+25 x^2+10 x^3+x^4\right ) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )^2}+\frac {25 x^5}{\left (2+25 x^2+10 x^3+x^4\right ) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )^2}+\frac {45 x^6}{\left (2+25 x^2+10 x^3+x^4\right ) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )^2}+\frac {13 x^7}{\left (2+25 x^2+10 x^3+x^4\right ) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )^2}+\frac {x^8}{\left (2+25 x^2+10 x^3+x^4\right ) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )^2}-\frac {10 x^2}{\left (2+25 x^2+10 x^3+x^4\right ) \log (x) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )^2}-\frac {125 x^4}{\left (2+25 x^2+10 x^3+x^4\right ) \log (x) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )^2}-\frac {50 x^5}{\left (2+25 x^2+10 x^3+x^4\right ) \log (x) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )^2}-\frac {5 x^6}{\left (2+25 x^2+10 x^3+x^4\right ) \log (x) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )^2}+\frac {10 x^3 \log (\log (x))}{\left (2+25 x^2+10 x^3+x^4\right ) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )^2}+\frac {250 x^4 \log (\log (x))}{\left (2+25 x^2+10 x^3+x^4\right ) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )^2}+\frac {275 x^5 \log (\log (x))}{\left (2+25 x^2+10 x^3+x^4\right ) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )^2}+\frac {70 x^6 \log (\log (x))}{\left (2+25 x^2+10 x^3+x^4\right ) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )^2}+\frac {5 x^7 \log (\log (x))}{\left (2+25 x^2+10 x^3+x^4\right ) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )^2}\right ) \, dx-\frac {2}{5} \int \frac {50-10 x+629 x^2+125 x^3}{\left (2+25 x^2+10 x^3+x^4\right ) \left (10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))\right )} \, dx-2 \int \frac {x}{10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))} \, dx+10 \int \frac {1}{10 e^x-x+125 e^x x^2+50 e^x x^3+5 e^x x^4-5 \log (\log (x))} \, dx \\ & = \text {Too large to display} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {-x^2+\left (e^x \left (2 x^2+2 x^3+75 x^4+65 x^5+15 x^6+x^7\right )+e^{2 x} \left (-40 x-1000 x^3-400 x^4-6290 x^5-5000 x^6-1500 x^7-200 x^8-10 x^9\right )\right ) \log (x)+\left (-x^2+e^x \left (40 x+500 x^3+200 x^4+20 x^5\right )\right ) \log (x) \log (\log (x))-10 x \log (x) \log ^2(\log (x))}{\left (x^2+e^x \left (-20 x-250 x^3-100 x^4-10 x^5\right )+e^{2 x} \left (100+2500 x^2+1000 x^3+15725 x^4+12500 x^5+3750 x^6+500 x^7+25 x^8\right )\right ) \log (x)+\left (10 x+e^x \left (-100-1250 x^2-500 x^3-50 x^4\right )\right ) \log (x) \log (\log (x))+25 \log (x) \log ^2(\log (x))} \, dx=-\frac {1}{5} x^2 \left (1-\frac {x}{x-5 e^x \left (2+25 x^2+10 x^3+x^4\right )+5 \log (\log (x))}\right ) \]
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Time = 231.83 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.45
method | result | size |
risch | \(-\frac {x^{2}}{5}-\frac {x^{3}}{5 \left (5 \,{\mathrm e}^{x} x^{4}+50 \,{\mathrm e}^{x} x^{3}+125 \,{\mathrm e}^{x} x^{2}+10 \,{\mathrm e}^{x}-5 \ln \left (\ln \left (x \right )\right )-x \right )}\) | \(48\) |
parallelrisch | \(-\frac {50 x^{6} {\mathrm e}^{x}+1250 \,{\mathrm e}^{x} x^{4}+100 \,{\mathrm e}^{x} x^{2}+500 x^{5} {\mathrm e}^{x}-50 x^{2} \ln \left (\ln \left (x \right )\right )}{50 \left (5 \,{\mathrm e}^{x} x^{4}+50 \,{\mathrm e}^{x} x^{3}+125 \,{\mathrm e}^{x} x^{2}+10 \,{\mathrm e}^{x}-5 \ln \left (\ln \left (x \right )\right )-x \right )}\) | \(76\) |
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Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.88 \[ \int \frac {-x^2+\left (e^x \left (2 x^2+2 x^3+75 x^4+65 x^5+15 x^6+x^7\right )+e^{2 x} \left (-40 x-1000 x^3-400 x^4-6290 x^5-5000 x^6-1500 x^7-200 x^8-10 x^9\right )\right ) \log (x)+\left (-x^2+e^x \left (40 x+500 x^3+200 x^4+20 x^5\right )\right ) \log (x) \log (\log (x))-10 x \log (x) \log ^2(\log (x))}{\left (x^2+e^x \left (-20 x-250 x^3-100 x^4-10 x^5\right )+e^{2 x} \left (100+2500 x^2+1000 x^3+15725 x^4+12500 x^5+3750 x^6+500 x^7+25 x^8\right )\right ) \log (x)+\left (10 x+e^x \left (-100-1250 x^2-500 x^3-50 x^4\right )\right ) \log (x) \log (\log (x))+25 \log (x) \log ^2(\log (x))} \, dx=\frac {x^{2} \log \left (\log \left (x\right )\right ) - {\left (x^{6} + 10 \, x^{5} + 25 \, x^{4} + 2 \, x^{2}\right )} e^{x}}{5 \, {\left (x^{4} + 10 \, x^{3} + 25 \, x^{2} + 2\right )} e^{x} - x - 5 \, \log \left (\log \left (x\right )\right )} \]
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Time = 0.46 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {-x^2+\left (e^x \left (2 x^2+2 x^3+75 x^4+65 x^5+15 x^6+x^7\right )+e^{2 x} \left (-40 x-1000 x^3-400 x^4-6290 x^5-5000 x^6-1500 x^7-200 x^8-10 x^9\right )\right ) \log (x)+\left (-x^2+e^x \left (40 x+500 x^3+200 x^4+20 x^5\right )\right ) \log (x) \log (\log (x))-10 x \log (x) \log ^2(\log (x))}{\left (x^2+e^x \left (-20 x-250 x^3-100 x^4-10 x^5\right )+e^{2 x} \left (100+2500 x^2+1000 x^3+15725 x^4+12500 x^5+3750 x^6+500 x^7+25 x^8\right )\right ) \log (x)+\left (10 x+e^x \left (-100-1250 x^2-500 x^3-50 x^4\right )\right ) \log (x) \log (\log (x))+25 \log (x) \log ^2(\log (x))} \, dx=- \frac {x^{3}}{- 5 x + \left (25 x^{4} + 250 x^{3} + 625 x^{2} + 50\right ) e^{x} - 25 \log {\left (\log {\left (x \right )} \right )}} - \frac {x^{2}}{5} \]
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Time = 0.34 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.88 \[ \int \frac {-x^2+\left (e^x \left (2 x^2+2 x^3+75 x^4+65 x^5+15 x^6+x^7\right )+e^{2 x} \left (-40 x-1000 x^3-400 x^4-6290 x^5-5000 x^6-1500 x^7-200 x^8-10 x^9\right )\right ) \log (x)+\left (-x^2+e^x \left (40 x+500 x^3+200 x^4+20 x^5\right )\right ) \log (x) \log (\log (x))-10 x \log (x) \log ^2(\log (x))}{\left (x^2+e^x \left (-20 x-250 x^3-100 x^4-10 x^5\right )+e^{2 x} \left (100+2500 x^2+1000 x^3+15725 x^4+12500 x^5+3750 x^6+500 x^7+25 x^8\right )\right ) \log (x)+\left (10 x+e^x \left (-100-1250 x^2-500 x^3-50 x^4\right )\right ) \log (x) \log (\log (x))+25 \log (x) \log ^2(\log (x))} \, dx=\frac {x^{2} \log \left (\log \left (x\right )\right ) - {\left (x^{6} + 10 \, x^{5} + 25 \, x^{4} + 2 \, x^{2}\right )} e^{x}}{5 \, {\left (x^{4} + 10 \, x^{3} + 25 \, x^{2} + 2\right )} e^{x} - x - 5 \, \log \left (\log \left (x\right )\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (32) = 64\).
Time = 0.42 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.24 \[ \int \frac {-x^2+\left (e^x \left (2 x^2+2 x^3+75 x^4+65 x^5+15 x^6+x^7\right )+e^{2 x} \left (-40 x-1000 x^3-400 x^4-6290 x^5-5000 x^6-1500 x^7-200 x^8-10 x^9\right )\right ) \log (x)+\left (-x^2+e^x \left (40 x+500 x^3+200 x^4+20 x^5\right )\right ) \log (x) \log (\log (x))-10 x \log (x) \log ^2(\log (x))}{\left (x^2+e^x \left (-20 x-250 x^3-100 x^4-10 x^5\right )+e^{2 x} \left (100+2500 x^2+1000 x^3+15725 x^4+12500 x^5+3750 x^6+500 x^7+25 x^8\right )\right ) \log (x)+\left (10 x+e^x \left (-100-1250 x^2-500 x^3-50 x^4\right )\right ) \log (x) \log (\log (x))+25 \log (x) \log ^2(\log (x))} \, dx=-\frac {x^{6} e^{x} + 10 \, x^{5} e^{x} + 25 \, x^{4} e^{x} + 2 \, x^{2} e^{x} - x^{2} \log \left (\log \left (x\right )\right )}{5 \, x^{4} e^{x} + 50 \, x^{3} e^{x} + 125 \, x^{2} e^{x} - x + 10 \, e^{x} - 5 \, \log \left (\log \left (x\right )\right )} \]
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Timed out. \[ \int \frac {-x^2+\left (e^x \left (2 x^2+2 x^3+75 x^4+65 x^5+15 x^6+x^7\right )+e^{2 x} \left (-40 x-1000 x^3-400 x^4-6290 x^5-5000 x^6-1500 x^7-200 x^8-10 x^9\right )\right ) \log (x)+\left (-x^2+e^x \left (40 x+500 x^3+200 x^4+20 x^5\right )\right ) \log (x) \log (\log (x))-10 x \log (x) \log ^2(\log (x))}{\left (x^2+e^x \left (-20 x-250 x^3-100 x^4-10 x^5\right )+e^{2 x} \left (100+2500 x^2+1000 x^3+15725 x^4+12500 x^5+3750 x^6+500 x^7+25 x^8\right )\right ) \log (x)+\left (10 x+e^x \left (-100-1250 x^2-500 x^3-50 x^4\right )\right ) \log (x) \log (\log (x))+25 \log (x) \log ^2(\log (x))} \, dx=\int -\frac {\ln \left (x\right )\,\left ({\mathrm {e}}^{2\,x}\,\left (10\,x^9+200\,x^8+1500\,x^7+5000\,x^6+6290\,x^5+400\,x^4+1000\,x^3+40\,x\right )-{\mathrm {e}}^x\,\left (x^7+15\,x^6+65\,x^5+75\,x^4+2\,x^3+2\,x^2\right )\right )+x^2-\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left ({\mathrm {e}}^x\,\left (20\,x^5+200\,x^4+500\,x^3+40\,x\right )-x^2\right )+10\,x\,{\ln \left (\ln \left (x\right )\right )}^2\,\ln \left (x\right )}{25\,\ln \left (x\right )\,{\ln \left (\ln \left (x\right )\right )}^2+\ln \left (x\right )\,\left (10\,x-{\mathrm {e}}^x\,\left (50\,x^4+500\,x^3+1250\,x^2+100\right )\right )\,\ln \left (\ln \left (x\right )\right )+\ln \left (x\right )\,\left ({\mathrm {e}}^{2\,x}\,\left (25\,x^8+500\,x^7+3750\,x^6+12500\,x^5+15725\,x^4+1000\,x^3+2500\,x^2+100\right )-{\mathrm {e}}^x\,\left (10\,x^5+100\,x^4+250\,x^3+20\,x\right )+x^2\right )} \,d x \]
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