Integrand size = 250, antiderivative size = 33 \[ \int \frac {-x^2+\left (e^x \left (2 x^2+2 x^3+75 x^4+65 x^5+15 x^6+x^7\right )+e^{2 x} \left (-40 x-1000 x^3-400 x^4-6290 x^5-5000 x^6-1500 x^7-200 x^8-10 x^9\right )\right ) \log (x)+\left (-x^2+e^x \left (40 x+500 x^3+200 x^4+20 x^5\right )\right ) \log (x) \log (\log (x))-10 x \log (x) \log ^2(\log (x))}{\left (x^2+e^x \left (-20 x-250 x^3-100 x^4-10 x^5\right )+e^{2 x} \left (100+2500 x^2+1000 x^3+15725 x^4+12500 x^5+3750 x^6+500 x^7+25 x^8\right )\right ) \log (x)+\left (10 x+e^x \left (-100-1250 x^2-500 x^3-50 x^4\right )\right ) \log (x) \log (\log (x))+25 \log (x) \log ^2(\log (x))} \, dx=\frac {x^2}{-5+\frac {x}{e^x \left (2+x^2 (5+x)^2\right )-\log (\log (x))}} \]
Time = 0.28 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {-x^2+\left (e^x \left (2 x^2+2 x^3+75 x^4+65 x^5+15 x^6+x^7\right )+e^{2 x} \left (-40 x-1000 x^3-400 x^4-6290 x^5-5000 x^6-1500 x^7-200 x^8-10 x^9\right )\right ) \log (x)+\left (-x^2+e^x \left (40 x+500 x^3+200 x^4+20 x^5\right )\right ) \log (x) \log (\log (x))-10 x \log (x) \log ^2(\log (x))}{\left (x^2+e^x \left (-20 x-250 x^3-100 x^4-10 x^5\right )+e^{2 x} \left (100+2500 x^2+1000 x^3+15725 x^4+12500 x^5+3750 x^6+500 x^7+25 x^8\right )\right ) \log (x)+\left (10 x+e^x \left (-100-1250 x^2-500 x^3-50 x^4\right )\right ) \log (x) \log (\log (x))+25 \log (x) \log ^2(\log (x))} \, dx=-\frac {1}{5} x^2 \left (1-\frac {x}{x-5 e^x \left (2+25 x^2+10 x^3+x^4\right )+5 \log (\log (x))}\right ) \]
Integrate[(-x^2 + (E^x*(2*x^2 + 2*x^3 + 75*x^4 + 65*x^5 + 15*x^6 + x^7) + E^(2*x)*(-40*x - 1000*x^3 - 400*x^4 - 6290*x^5 - 5000*x^6 - 1500*x^7 - 200 *x^8 - 10*x^9))*Log[x] + (-x^2 + E^x*(40*x + 500*x^3 + 200*x^4 + 20*x^5))* Log[x]*Log[Log[x]] - 10*x*Log[x]*Log[Log[x]]^2)/((x^2 + E^x*(-20*x - 250*x ^3 - 100*x^4 - 10*x^5) + E^(2*x)*(100 + 2500*x^2 + 1000*x^3 + 15725*x^4 + 12500*x^5 + 3750*x^6 + 500*x^7 + 25*x^8))*Log[x] + (10*x + E^x*(-100 - 125 0*x^2 - 500*x^3 - 50*x^4))*Log[x]*Log[Log[x]] + 25*Log[x]*Log[Log[x]]^2),x ]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-x^2+\left (e^x \left (20 x^5+200 x^4+500 x^3+40 x\right )-x^2\right ) \log (x) \log (\log (x))+\left (e^x \left (x^7+15 x^6+65 x^5+75 x^4+2 x^3+2 x^2\right )+e^{2 x} \left (-10 x^9-200 x^8-1500 x^7-5000 x^6-6290 x^5-400 x^4-1000 x^3-40 x\right )\right ) \log (x)-10 x \log (x) \log ^2(\log (x))}{\left (e^x \left (-50 x^4-500 x^3-1250 x^2-100\right )+10 x\right ) \log (x) \log (\log (x))+\left (x^2+e^x \left (-10 x^5-100 x^4-250 x^3-20 x\right )+e^{2 x} \left (25 x^8+500 x^7+3750 x^6+12500 x^5+15725 x^4+1000 x^3+2500 x^2+100\right )\right ) \log (x)+25 \log (x) \log ^2(\log (x))} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (-\log (x) \left (\left (x-20 e^x \left (x^4+10 x^3+25 x^2+2\right )\right ) \log (\log (x))+e^x \left (10 e^x \left (x^4+10 x^3+25 x^2+2\right )^2-x \left (x^5+15 x^4+65 x^3+75 x^2+2 x+2\right )\right )+10 \log ^2(\log (x))\right )-x\right )}{\log (x) \left (-5 e^x \left (x^4+10 x^3+25 x^2+2\right )+x+5 \log (\log (x))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (x^5+11 x^4+25 x^3-25 x^2+2 x-6\right ) x^2}{5 \left (x^4+10 x^3+25 x^2+2\right ) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )}+\frac {x^2 \left (x^6 \log (x)+13 x^5 \log (x)+5 x^5 \log (x) \log (\log (x))-5 x^4+45 x^4 \log (x)+70 x^4 \log (x) \log (\log (x))-50 x^3+25 x^3 \log (x)+275 x^3 \log (x) \log (\log (x))-125 x^2+2 x^2 \log (x)+250 x^2 \log (x) \log (\log (x))-2 x \log (x)+10 x \log (x) \log (\log (x))-10\right )}{5 \left (x^4+10 x^3+25 x^2+2\right ) \log (x) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )^2}-\frac {2 x}{5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (-\log (x) \left (\left (x-20 e^x \left (x^4+10 x^3+25 x^2+2\right )\right ) \log (\log (x))+e^x \left (10 e^x \left (x^4+10 x^3+25 x^2+2\right )^2-x \left (x^5+15 x^4+65 x^3+75 x^2+2 x+2\right )\right )+10 \log ^2(\log (x))\right )-x\right )}{\log (x) \left (-5 e^x \left (x^4+10 x^3+25 x^2+2\right )+x+5 \log (\log (x))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (x^5+11 x^4+25 x^3-25 x^2+2 x-6\right ) x^2}{5 \left (x^4+10 x^3+25 x^2+2\right ) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )}+\frac {x^2 \left (x^6 \log (x)+13 x^5 \log (x)+5 x^5 \log (x) \log (\log (x))-5 x^4+45 x^4 \log (x)+70 x^4 \log (x) \log (\log (x))-50 x^3+25 x^3 \log (x)+275 x^3 \log (x) \log (\log (x))-125 x^2+2 x^2 \log (x)+250 x^2 \log (x) \log (\log (x))-2 x \log (x)+10 x \log (x) \log (\log (x))-10\right )}{5 \left (x^4+10 x^3+25 x^2+2\right ) \log (x) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )^2}-\frac {2 x}{5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (-\log (x) \left (\left (x-20 e^x \left (x^4+10 x^3+25 x^2+2\right )\right ) \log (\log (x))+e^x \left (10 e^x \left (x^4+10 x^3+25 x^2+2\right )^2-x \left (x^5+15 x^4+65 x^3+75 x^2+2 x+2\right )\right )+10 \log ^2(\log (x))\right )-x\right )}{\log (x) \left (-5 e^x \left (x^4+10 x^3+25 x^2+2\right )+x+5 \log (\log (x))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (x^5+11 x^4+25 x^3-25 x^2+2 x-6\right ) x^2}{5 \left (x^4+10 x^3+25 x^2+2\right ) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )}+\frac {x^2 \left (x^6 \log (x)+13 x^5 \log (x)+5 x^5 \log (x) \log (\log (x))-5 x^4+45 x^4 \log (x)+70 x^4 \log (x) \log (\log (x))-50 x^3+25 x^3 \log (x)+275 x^3 \log (x) \log (\log (x))-125 x^2+2 x^2 \log (x)+250 x^2 \log (x) \log (\log (x))-2 x \log (x)+10 x \log (x) \log (\log (x))-10\right )}{5 \left (x^4+10 x^3+25 x^2+2\right ) \log (x) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )^2}-\frac {2 x}{5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (-\log (x) \left (\left (x-20 e^x \left (x^4+10 x^3+25 x^2+2\right )\right ) \log (\log (x))+e^x \left (10 e^x \left (x^4+10 x^3+25 x^2+2\right )^2-x \left (x^5+15 x^4+65 x^3+75 x^2+2 x+2\right )\right )+10 \log ^2(\log (x))\right )-x\right )}{\log (x) \left (-5 e^x \left (x^4+10 x^3+25 x^2+2\right )+x+5 \log (\log (x))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (x^5+11 x^4+25 x^3-25 x^2+2 x-6\right ) x^2}{5 \left (x^4+10 x^3+25 x^2+2\right ) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )}+\frac {x^2 \left (x^6 \log (x)+13 x^5 \log (x)+5 x^5 \log (x) \log (\log (x))-5 x^4+45 x^4 \log (x)+70 x^4 \log (x) \log (\log (x))-50 x^3+25 x^3 \log (x)+275 x^3 \log (x) \log (\log (x))-125 x^2+2 x^2 \log (x)+250 x^2 \log (x) \log (\log (x))-2 x \log (x)+10 x \log (x) \log (\log (x))-10\right )}{5 \left (x^4+10 x^3+25 x^2+2\right ) \log (x) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )^2}-\frac {2 x}{5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (-\log (x) \left (\left (x-20 e^x \left (x^4+10 x^3+25 x^2+2\right )\right ) \log (\log (x))+e^x \left (10 e^x \left (x^4+10 x^3+25 x^2+2\right )^2-x \left (x^5+15 x^4+65 x^3+75 x^2+2 x+2\right )\right )+10 \log ^2(\log (x))\right )-x\right )}{\log (x) \left (-5 e^x \left (x^4+10 x^3+25 x^2+2\right )+x+5 \log (\log (x))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (x^5+11 x^4+25 x^3-25 x^2+2 x-6\right ) x^2}{5 \left (x^4+10 x^3+25 x^2+2\right ) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )}+\frac {x^2 \left (x^6 \log (x)+13 x^5 \log (x)+5 x^5 \log (x) \log (\log (x))-5 x^4+45 x^4 \log (x)+70 x^4 \log (x) \log (\log (x))-50 x^3+25 x^3 \log (x)+275 x^3 \log (x) \log (\log (x))-125 x^2+2 x^2 \log (x)+250 x^2 \log (x) \log (\log (x))-2 x \log (x)+10 x \log (x) \log (\log (x))-10\right )}{5 \left (x^4+10 x^3+25 x^2+2\right ) \log (x) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )^2}-\frac {2 x}{5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (-\log (x) \left (\left (x-20 e^x \left (x^4+10 x^3+25 x^2+2\right )\right ) \log (\log (x))+e^x \left (10 e^x \left (x^4+10 x^3+25 x^2+2\right )^2-x \left (x^5+15 x^4+65 x^3+75 x^2+2 x+2\right )\right )+10 \log ^2(\log (x))\right )-x\right )}{\log (x) \left (-5 e^x \left (x^4+10 x^3+25 x^2+2\right )+x+5 \log (\log (x))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (x^5+11 x^4+25 x^3-25 x^2+2 x-6\right ) x^2}{5 \left (x^4+10 x^3+25 x^2+2\right ) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )}+\frac {x^2 \left (x^6 \log (x)+13 x^5 \log (x)+5 x^5 \log (x) \log (\log (x))-5 x^4+45 x^4 \log (x)+70 x^4 \log (x) \log (\log (x))-50 x^3+25 x^3 \log (x)+275 x^3 \log (x) \log (\log (x))-125 x^2+2 x^2 \log (x)+250 x^2 \log (x) \log (\log (x))-2 x \log (x)+10 x \log (x) \log (\log (x))-10\right )}{5 \left (x^4+10 x^3+25 x^2+2\right ) \log (x) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )^2}-\frac {2 x}{5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (-\log (x) \left (\left (x-20 e^x \left (x^4+10 x^3+25 x^2+2\right )\right ) \log (\log (x))+e^x \left (10 e^x \left (x^4+10 x^3+25 x^2+2\right )^2-x \left (x^5+15 x^4+65 x^3+75 x^2+2 x+2\right )\right )+10 \log ^2(\log (x))\right )-x\right )}{\log (x) \left (-5 e^x \left (x^4+10 x^3+25 x^2+2\right )+x+5 \log (\log (x))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (x^5+11 x^4+25 x^3-25 x^2+2 x-6\right ) x^2}{5 \left (x^4+10 x^3+25 x^2+2\right ) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )}+\frac {x^2 \left (x^6 \log (x)+13 x^5 \log (x)+5 x^5 \log (x) \log (\log (x))-5 x^4+45 x^4 \log (x)+70 x^4 \log (x) \log (\log (x))-50 x^3+25 x^3 \log (x)+275 x^3 \log (x) \log (\log (x))-125 x^2+2 x^2 \log (x)+250 x^2 \log (x) \log (\log (x))-2 x \log (x)+10 x \log (x) \log (\log (x))-10\right )}{5 \left (x^4+10 x^3+25 x^2+2\right ) \log (x) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )^2}-\frac {2 x}{5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (-\log (x) \left (\left (x-20 e^x \left (x^4+10 x^3+25 x^2+2\right )\right ) \log (\log (x))+e^x \left (10 e^x \left (x^4+10 x^3+25 x^2+2\right )^2-x \left (x^5+15 x^4+65 x^3+75 x^2+2 x+2\right )\right )+10 \log ^2(\log (x))\right )-x\right )}{\log (x) \left (-5 e^x \left (x^4+10 x^3+25 x^2+2\right )+x+5 \log (\log (x))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (x^5+11 x^4+25 x^3-25 x^2+2 x-6\right ) x^2}{5 \left (x^4+10 x^3+25 x^2+2\right ) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )}+\frac {x^2 \left (x^6 \log (x)+13 x^5 \log (x)+5 x^5 \log (x) \log (\log (x))-5 x^4+45 x^4 \log (x)+70 x^4 \log (x) \log (\log (x))-50 x^3+25 x^3 \log (x)+275 x^3 \log (x) \log (\log (x))-125 x^2+2 x^2 \log (x)+250 x^2 \log (x) \log (\log (x))-2 x \log (x)+10 x \log (x) \log (\log (x))-10\right )}{5 \left (x^4+10 x^3+25 x^2+2\right ) \log (x) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )^2}-\frac {2 x}{5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (-\log (x) \left (\left (x-20 e^x \left (x^4+10 x^3+25 x^2+2\right )\right ) \log (\log (x))+e^x \left (10 e^x \left (x^4+10 x^3+25 x^2+2\right )^2-x \left (x^5+15 x^4+65 x^3+75 x^2+2 x+2\right )\right )+10 \log ^2(\log (x))\right )-x\right )}{\log (x) \left (-5 e^x \left (x^4+10 x^3+25 x^2+2\right )+x+5 \log (\log (x))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (x^5+11 x^4+25 x^3-25 x^2+2 x-6\right ) x^2}{5 \left (x^4+10 x^3+25 x^2+2\right ) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )}+\frac {x^2 \left (x^6 \log (x)+13 x^5 \log (x)+5 x^5 \log (x) \log (\log (x))-5 x^4+45 x^4 \log (x)+70 x^4 \log (x) \log (\log (x))-50 x^3+25 x^3 \log (x)+275 x^3 \log (x) \log (\log (x))-125 x^2+2 x^2 \log (x)+250 x^2 \log (x) \log (\log (x))-2 x \log (x)+10 x \log (x) \log (\log (x))-10\right )}{5 \left (x^4+10 x^3+25 x^2+2\right ) \log (x) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )^2}-\frac {2 x}{5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (-\log (x) \left (\left (x-20 e^x \left (x^4+10 x^3+25 x^2+2\right )\right ) \log (\log (x))+e^x \left (10 e^x \left (x^4+10 x^3+25 x^2+2\right )^2-x \left (x^5+15 x^4+65 x^3+75 x^2+2 x+2\right )\right )+10 \log ^2(\log (x))\right )-x\right )}{\log (x) \left (-5 e^x \left (x^4+10 x^3+25 x^2+2\right )+x+5 \log (\log (x))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (x^5+11 x^4+25 x^3-25 x^2+2 x-6\right ) x^2}{5 \left (x^4+10 x^3+25 x^2+2\right ) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )}+\frac {x^2 \left (x^6 \log (x)+13 x^5 \log (x)+5 x^5 \log (x) \log (\log (x))-5 x^4+45 x^4 \log (x)+70 x^4 \log (x) \log (\log (x))-50 x^3+25 x^3 \log (x)+275 x^3 \log (x) \log (\log (x))-125 x^2+2 x^2 \log (x)+250 x^2 \log (x) \log (\log (x))-2 x \log (x)+10 x \log (x) \log (\log (x))-10\right )}{5 \left (x^4+10 x^3+25 x^2+2\right ) \log (x) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )^2}-\frac {2 x}{5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (-\log (x) \left (\left (x-20 e^x \left (x^4+10 x^3+25 x^2+2\right )\right ) \log (\log (x))+e^x \left (10 e^x \left (x^4+10 x^3+25 x^2+2\right )^2-x \left (x^5+15 x^4+65 x^3+75 x^2+2 x+2\right )\right )+10 \log ^2(\log (x))\right )-x\right )}{\log (x) \left (-5 e^x \left (x^4+10 x^3+25 x^2+2\right )+x+5 \log (\log (x))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (x^5+11 x^4+25 x^3-25 x^2+2 x-6\right ) x^2}{5 \left (x^4+10 x^3+25 x^2+2\right ) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )}+\frac {x^2 \left (x^6 \log (x)+13 x^5 \log (x)+5 x^5 \log (x) \log (\log (x))-5 x^4+45 x^4 \log (x)+70 x^4 \log (x) \log (\log (x))-50 x^3+25 x^3 \log (x)+275 x^3 \log (x) \log (\log (x))-125 x^2+2 x^2 \log (x)+250 x^2 \log (x) \log (\log (x))-2 x \log (x)+10 x \log (x) \log (\log (x))-10\right )}{5 \left (x^4+10 x^3+25 x^2+2\right ) \log (x) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )^2}-\frac {2 x}{5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (-\log (x) \left (\left (x-20 e^x \left (x^4+10 x^3+25 x^2+2\right )\right ) \log (\log (x))+e^x \left (10 e^x \left (x^4+10 x^3+25 x^2+2\right )^2-x \left (x^5+15 x^4+65 x^3+75 x^2+2 x+2\right )\right )+10 \log ^2(\log (x))\right )-x\right )}{\log (x) \left (-5 e^x \left (x^4+10 x^3+25 x^2+2\right )+x+5 \log (\log (x))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (x^5+11 x^4+25 x^3-25 x^2+2 x-6\right ) x^2}{5 \left (x^4+10 x^3+25 x^2+2\right ) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )}+\frac {x^2 \left (x^6 \log (x)+13 x^5 \log (x)+5 x^5 \log (x) \log (\log (x))-5 x^4+45 x^4 \log (x)+70 x^4 \log (x) \log (\log (x))-50 x^3+25 x^3 \log (x)+275 x^3 \log (x) \log (\log (x))-125 x^2+2 x^2 \log (x)+250 x^2 \log (x) \log (\log (x))-2 x \log (x)+10 x \log (x) \log (\log (x))-10\right )}{5 \left (x^4+10 x^3+25 x^2+2\right ) \log (x) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )^2}-\frac {2 x}{5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (-\log (x) \left (\left (x-20 e^x \left (x^4+10 x^3+25 x^2+2\right )\right ) \log (\log (x))+e^x \left (10 e^x \left (x^4+10 x^3+25 x^2+2\right )^2-x \left (x^5+15 x^4+65 x^3+75 x^2+2 x+2\right )\right )+10 \log ^2(\log (x))\right )-x\right )}{\log (x) \left (-5 e^x \left (x^4+10 x^3+25 x^2+2\right )+x+5 \log (\log (x))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (x^5+11 x^4+25 x^3-25 x^2+2 x-6\right ) x^2}{5 \left (x^4+10 x^3+25 x^2+2\right ) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )}+\frac {x^2 \left (x^6 \log (x)+13 x^5 \log (x)+5 x^5 \log (x) \log (\log (x))-5 x^4+45 x^4 \log (x)+70 x^4 \log (x) \log (\log (x))-50 x^3+25 x^3 \log (x)+275 x^3 \log (x) \log (\log (x))-125 x^2+2 x^2 \log (x)+250 x^2 \log (x) \log (\log (x))-2 x \log (x)+10 x \log (x) \log (\log (x))-10\right )}{5 \left (x^4+10 x^3+25 x^2+2\right ) \log (x) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )^2}-\frac {2 x}{5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (-\log (x) \left (\left (x-20 e^x \left (x^4+10 x^3+25 x^2+2\right )\right ) \log (\log (x))+e^x \left (10 e^x \left (x^4+10 x^3+25 x^2+2\right )^2-x \left (x^5+15 x^4+65 x^3+75 x^2+2 x+2\right )\right )+10 \log ^2(\log (x))\right )-x\right )}{\log (x) \left (-5 e^x \left (x^4+10 x^3+25 x^2+2\right )+x+5 \log (\log (x))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (x^5+11 x^4+25 x^3-25 x^2+2 x-6\right ) x^2}{5 \left (x^4+10 x^3+25 x^2+2\right ) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )}+\frac {x^2 \left (x^6 \log (x)+13 x^5 \log (x)+5 x^5 \log (x) \log (\log (x))-5 x^4+45 x^4 \log (x)+70 x^4 \log (x) \log (\log (x))-50 x^3+25 x^3 \log (x)+275 x^3 \log (x) \log (\log (x))-125 x^2+2 x^2 \log (x)+250 x^2 \log (x) \log (\log (x))-2 x \log (x)+10 x \log (x) \log (\log (x))-10\right )}{5 \left (x^4+10 x^3+25 x^2+2\right ) \log (x) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )^2}-\frac {2 x}{5}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x \left (-\log (x) \left (\left (x-20 e^x \left (x^4+10 x^3+25 x^2+2\right )\right ) \log (\log (x))+e^x \left (10 e^x \left (x^4+10 x^3+25 x^2+2\right )^2-x \left (x^5+15 x^4+65 x^3+75 x^2+2 x+2\right )\right )+10 \log ^2(\log (x))\right )-x\right )}{\log (x) \left (-5 e^x \left (x^4+10 x^3+25 x^2+2\right )+x+5 \log (\log (x))\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (x^5+11 x^4+25 x^3-25 x^2+2 x-6\right ) x^2}{5 \left (x^4+10 x^3+25 x^2+2\right ) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )}+\frac {x^2 \left (x^6 \log (x)+13 x^5 \log (x)+5 x^5 \log (x) \log (\log (x))-5 x^4+45 x^4 \log (x)+70 x^4 \log (x) \log (\log (x))-50 x^3+25 x^3 \log (x)+275 x^3 \log (x) \log (\log (x))-125 x^2+2 x^2 \log (x)+250 x^2 \log (x) \log (\log (x))-2 x \log (x)+10 x \log (x) \log (\log (x))-10\right )}{5 \left (x^4+10 x^3+25 x^2+2\right ) \log (x) \left (5 e^x x^4+50 e^x x^3+125 e^x x^2-x+10 e^x-5 \log (\log (x))\right )^2}-\frac {2 x}{5}\right )dx\) |
Int[(-x^2 + (E^x*(2*x^2 + 2*x^3 + 75*x^4 + 65*x^5 + 15*x^6 + x^7) + E^(2*x )*(-40*x - 1000*x^3 - 400*x^4 - 6290*x^5 - 5000*x^6 - 1500*x^7 - 200*x^8 - 10*x^9))*Log[x] + (-x^2 + E^x*(40*x + 500*x^3 + 200*x^4 + 20*x^5))*Log[x] *Log[Log[x]] - 10*x*Log[x]*Log[Log[x]]^2)/((x^2 + E^x*(-20*x - 250*x^3 - 1 00*x^4 - 10*x^5) + E^(2*x)*(100 + 2500*x^2 + 1000*x^3 + 15725*x^4 + 12500* x^5 + 3750*x^6 + 500*x^7 + 25*x^8))*Log[x] + (10*x + E^x*(-100 - 1250*x^2 - 500*x^3 - 50*x^4))*Log[x]*Log[Log[x]] + 25*Log[x]*Log[Log[x]]^2),x]
3.6.41.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
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 | \(-\frac {50 x^{6} {\mathrm e}^{x}+1250 \,{\mathrm e}^{x} x^{4}+100 \,{\mathrm e}^{x} x^{2}+500 x^{5} {\mathrm e}^{x}-50 x^{2} \ln \left (\ln \left (x \right )\right )}{50 \left (5 \,{\mathrm e}^{x} x^{4}+50 \,{\mathrm e}^{x} x^{3}+125 \,{\mathrm e}^{x} x^{2}+10 \,{\mathrm e}^{x}-5 \ln \left (\ln \left (x \right )\right )-x \right )}\) | \(76\) |
int((-10*x*ln(x)*ln(ln(x))^2+((20*x^5+200*x^4+500*x^3+40*x)*exp(x)-x^2)*ln (x)*ln(ln(x))+((-10*x^9-200*x^8-1500*x^7-5000*x^6-6290*x^5-400*x^4-1000*x^ 3-40*x)*exp(x)^2+(x^7+15*x^6+65*x^5+75*x^4+2*x^3+2*x^2)*exp(x))*ln(x)-x^2) /(25*ln(x)*ln(ln(x))^2+((-50*x^4-500*x^3-1250*x^2-100)*exp(x)+10*x)*ln(x)* ln(ln(x))+((25*x^8+500*x^7+3750*x^6+12500*x^5+15725*x^4+1000*x^3+2500*x^2+ 100)*exp(x)^2+(-10*x^5-100*x^4-250*x^3-20*x)*exp(x)+x^2)*ln(x)),x,method=_ RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.88 \[ \int \frac {-x^2+\left (e^x \left (2 x^2+2 x^3+75 x^4+65 x^5+15 x^6+x^7\right )+e^{2 x} \left (-40 x-1000 x^3-400 x^4-6290 x^5-5000 x^6-1500 x^7-200 x^8-10 x^9\right )\right ) \log (x)+\left (-x^2+e^x \left (40 x+500 x^3+200 x^4+20 x^5\right )\right ) \log (x) \log (\log (x))-10 x \log (x) \log ^2(\log (x))}{\left (x^2+e^x \left (-20 x-250 x^3-100 x^4-10 x^5\right )+e^{2 x} \left (100+2500 x^2+1000 x^3+15725 x^4+12500 x^5+3750 x^6+500 x^7+25 x^8\right )\right ) \log (x)+\left (10 x+e^x \left (-100-1250 x^2-500 x^3-50 x^4\right )\right ) \log (x) \log (\log (x))+25 \log (x) \log ^2(\log (x))} \, dx=\frac {x^{2} \log \left (\log \left (x\right )\right ) - {\left (x^{6} + 10 \, x^{5} + 25 \, x^{4} + 2 \, x^{2}\right )} e^{x}}{5 \, {\left (x^{4} + 10 \, x^{3} + 25 \, x^{2} + 2\right )} e^{x} - x - 5 \, \log \left (\log \left (x\right )\right )} \]
integrate((-10*x*log(x)*log(log(x))^2+((20*x^5+200*x^4+500*x^3+40*x)*exp(x )-x^2)*log(x)*log(log(x))+((-10*x^9-200*x^8-1500*x^7-5000*x^6-6290*x^5-400 *x^4-1000*x^3-40*x)*exp(x)^2+(x^7+15*x^6+65*x^5+75*x^4+2*x^3+2*x^2)*exp(x) )*log(x)-x^2)/(25*log(x)*log(log(x))^2+((-50*x^4-500*x^3-1250*x^2-100)*exp (x)+10*x)*log(x)*log(log(x))+((25*x^8+500*x^7+3750*x^6+12500*x^5+15725*x^4 +1000*x^3+2500*x^2+100)*exp(x)^2+(-10*x^5-100*x^4-250*x^3-20*x)*exp(x)+x^2 )*log(x)),x, algorithm=\
(x^2*log(log(x)) - (x^6 + 10*x^5 + 25*x^4 + 2*x^2)*e^x)/(5*(x^4 + 10*x^3 + 25*x^2 + 2)*e^x - x - 5*log(log(x)))
Time = 0.46 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.18 \[ \int \frac {-x^2+\left (e^x \left (2 x^2+2 x^3+75 x^4+65 x^5+15 x^6+x^7\right )+e^{2 x} \left (-40 x-1000 x^3-400 x^4-6290 x^5-5000 x^6-1500 x^7-200 x^8-10 x^9\right )\right ) \log (x)+\left (-x^2+e^x \left (40 x+500 x^3+200 x^4+20 x^5\right )\right ) \log (x) \log (\log (x))-10 x \log (x) \log ^2(\log (x))}{\left (x^2+e^x \left (-20 x-250 x^3-100 x^4-10 x^5\right )+e^{2 x} \left (100+2500 x^2+1000 x^3+15725 x^4+12500 x^5+3750 x^6+500 x^7+25 x^8\right )\right ) \log (x)+\left (10 x+e^x \left (-100-1250 x^2-500 x^3-50 x^4\right )\right ) \log (x) \log (\log (x))+25 \log (x) \log ^2(\log (x))} \, dx=- \frac {x^{3}}{- 5 x + \left (25 x^{4} + 250 x^{3} + 625 x^{2} + 50\right ) e^{x} - 25 \log {\left (\log {\left (x \right )} \right )}} - \frac {x^{2}}{5} \]
integrate((-10*x*ln(x)*ln(ln(x))**2+((20*x**5+200*x**4+500*x**3+40*x)*exp( x)-x**2)*ln(x)*ln(ln(x))+((-10*x**9-200*x**8-1500*x**7-5000*x**6-6290*x**5 -400*x**4-1000*x**3-40*x)*exp(x)**2+(x**7+15*x**6+65*x**5+75*x**4+2*x**3+2 *x**2)*exp(x))*ln(x)-x**2)/(25*ln(x)*ln(ln(x))**2+((-50*x**4-500*x**3-1250 *x**2-100)*exp(x)+10*x)*ln(x)*ln(ln(x))+((25*x**8+500*x**7+3750*x**6+12500 *x**5+15725*x**4+1000*x**3+2500*x**2+100)*exp(x)**2+(-10*x**5-100*x**4-250 *x**3-20*x)*exp(x)+x**2)*ln(x)),x)
Time = 0.34 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.88 \[ \int \frac {-x^2+\left (e^x \left (2 x^2+2 x^3+75 x^4+65 x^5+15 x^6+x^7\right )+e^{2 x} \left (-40 x-1000 x^3-400 x^4-6290 x^5-5000 x^6-1500 x^7-200 x^8-10 x^9\right )\right ) \log (x)+\left (-x^2+e^x \left (40 x+500 x^3+200 x^4+20 x^5\right )\right ) \log (x) \log (\log (x))-10 x \log (x) \log ^2(\log (x))}{\left (x^2+e^x \left (-20 x-250 x^3-100 x^4-10 x^5\right )+e^{2 x} \left (100+2500 x^2+1000 x^3+15725 x^4+12500 x^5+3750 x^6+500 x^7+25 x^8\right )\right ) \log (x)+\left (10 x+e^x \left (-100-1250 x^2-500 x^3-50 x^4\right )\right ) \log (x) \log (\log (x))+25 \log (x) \log ^2(\log (x))} \, dx=\frac {x^{2} \log \left (\log \left (x\right )\right ) - {\left (x^{6} + 10 \, x^{5} + 25 \, x^{4} + 2 \, x^{2}\right )} e^{x}}{5 \, {\left (x^{4} + 10 \, x^{3} + 25 \, x^{2} + 2\right )} e^{x} - x - 5 \, \log \left (\log \left (x\right )\right )} \]
integrate((-10*x*log(x)*log(log(x))^2+((20*x^5+200*x^4+500*x^3+40*x)*exp(x )-x^2)*log(x)*log(log(x))+((-10*x^9-200*x^8-1500*x^7-5000*x^6-6290*x^5-400 *x^4-1000*x^3-40*x)*exp(x)^2+(x^7+15*x^6+65*x^5+75*x^4+2*x^3+2*x^2)*exp(x) )*log(x)-x^2)/(25*log(x)*log(log(x))^2+((-50*x^4-500*x^3-1250*x^2-100)*exp (x)+10*x)*log(x)*log(log(x))+((25*x^8+500*x^7+3750*x^6+12500*x^5+15725*x^4 +1000*x^3+2500*x^2+100)*exp(x)^2+(-10*x^5-100*x^4-250*x^3-20*x)*exp(x)+x^2 )*log(x)),x, algorithm=\
(x^2*log(log(x)) - (x^6 + 10*x^5 + 25*x^4 + 2*x^2)*e^x)/(5*(x^4 + 10*x^3 + 25*x^2 + 2)*e^x - x - 5*log(log(x)))
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 \[ \int \frac {-x^2+\left (e^x \left (2 x^2+2 x^3+75 x^4+65 x^5+15 x^6+x^7\right )+e^{2 x} \left (-40 x-1000 x^3-400 x^4-6290 x^5-5000 x^6-1500 x^7-200 x^8-10 x^9\right )\right ) \log (x)+\left (-x^2+e^x \left (40 x+500 x^3+200 x^4+20 x^5\right )\right ) \log (x) \log (\log (x))-10 x \log (x) \log ^2(\log (x))}{\left (x^2+e^x \left (-20 x-250 x^3-100 x^4-10 x^5\right )+e^{2 x} \left (100+2500 x^2+1000 x^3+15725 x^4+12500 x^5+3750 x^6+500 x^7+25 x^8\right )\right ) \log (x)+\left (10 x+e^x \left (-100-1250 x^2-500 x^3-50 x^4\right )\right ) \log (x) \log (\log (x))+25 \log (x) \log ^2(\log (x))} \, dx=-\frac {x^{6} e^{x} + 10 \, x^{5} e^{x} + 25 \, x^{4} e^{x} + 2 \, x^{2} e^{x} - x^{2} \log \left (\log \left (x\right )\right )}{5 \, x^{4} e^{x} + 50 \, x^{3} e^{x} + 125 \, x^{2} e^{x} - x + 10 \, e^{x} - 5 \, \log \left (\log \left (x\right )\right )} \]
integrate((-10*x*log(x)*log(log(x))^2+((20*x^5+200*x^4+500*x^3+40*x)*exp(x )-x^2)*log(x)*log(log(x))+((-10*x^9-200*x^8-1500*x^7-5000*x^6-6290*x^5-400 *x^4-1000*x^3-40*x)*exp(x)^2+(x^7+15*x^6+65*x^5+75*x^4+2*x^3+2*x^2)*exp(x) )*log(x)-x^2)/(25*log(x)*log(log(x))^2+((-50*x^4-500*x^3-1250*x^2-100)*exp (x)+10*x)*log(x)*log(log(x))+((25*x^8+500*x^7+3750*x^6+12500*x^5+15725*x^4 +1000*x^3+2500*x^2+100)*exp(x)^2+(-10*x^5-100*x^4-250*x^3-20*x)*exp(x)+x^2 )*log(x)),x, algorithm=\
-(x^6*e^x + 10*x^5*e^x + 25*x^4*e^x + 2*x^2*e^x - x^2*log(log(x)))/(5*x^4* e^x + 50*x^3*e^x + 125*x^2*e^x - x + 10*e^x - 5*log(log(x)))
Timed out. \[ \int \frac {-x^2+\left (e^x \left (2 x^2+2 x^3+75 x^4+65 x^5+15 x^6+x^7\right )+e^{2 x} \left (-40 x-1000 x^3-400 x^4-6290 x^5-5000 x^6-1500 x^7-200 x^8-10 x^9\right )\right ) \log (x)+\left (-x^2+e^x \left (40 x+500 x^3+200 x^4+20 x^5\right )\right ) \log (x) \log (\log (x))-10 x \log (x) \log ^2(\log (x))}{\left (x^2+e^x \left (-20 x-250 x^3-100 x^4-10 x^5\right )+e^{2 x} \left (100+2500 x^2+1000 x^3+15725 x^4+12500 x^5+3750 x^6+500 x^7+25 x^8\right )\right ) \log (x)+\left (10 x+e^x \left (-100-1250 x^2-500 x^3-50 x^4\right )\right ) \log (x) \log (\log (x))+25 \log (x) \log ^2(\log (x))} \, dx=\int -\frac {\ln \left (x\right )\,\left ({\mathrm {e}}^{2\,x}\,\left (10\,x^9+200\,x^8+1500\,x^7+5000\,x^6+6290\,x^5+400\,x^4+1000\,x^3+40\,x\right )-{\mathrm {e}}^x\,\left (x^7+15\,x^6+65\,x^5+75\,x^4+2\,x^3+2\,x^2\right )\right )+x^2-\ln \left (\ln \left (x\right )\right )\,\ln \left (x\right )\,\left ({\mathrm {e}}^x\,\left (20\,x^5+200\,x^4+500\,x^3+40\,x\right )-x^2\right )+10\,x\,{\ln \left (\ln \left (x\right )\right )}^2\,\ln \left (x\right )}{25\,\ln \left (x\right )\,{\ln \left (\ln \left (x\right )\right )}^2+\ln \left (x\right )\,\left (10\,x-{\mathrm {e}}^x\,\left (50\,x^4+500\,x^3+1250\,x^2+100\right )\right )\,\ln \left (\ln \left (x\right )\right )+\ln \left (x\right )\,\left ({\mathrm {e}}^{2\,x}\,\left (25\,x^8+500\,x^7+3750\,x^6+12500\,x^5+15725\,x^4+1000\,x^3+2500\,x^2+100\right )-{\mathrm {e}}^x\,\left (10\,x^5+100\,x^4+250\,x^3+20\,x\right )+x^2\right )} \,d x \]
int(-(log(x)*(exp(2*x)*(40*x + 1000*x^3 + 400*x^4 + 6290*x^5 + 5000*x^6 + 1500*x^7 + 200*x^8 + 10*x^9) - exp(x)*(2*x^2 + 2*x^3 + 75*x^4 + 65*x^5 + 1 5*x^6 + x^7)) + x^2 - log(log(x))*log(x)*(exp(x)*(40*x + 500*x^3 + 200*x^4 + 20*x^5) - x^2) + 10*x*log(log(x))^2*log(x))/(log(x)*(exp(2*x)*(2500*x^2 + 1000*x^3 + 15725*x^4 + 12500*x^5 + 3750*x^6 + 500*x^7 + 25*x^8 + 100) - exp(x)*(20*x + 250*x^3 + 100*x^4 + 10*x^5) + x^2) + 25*log(log(x))^2*log( x) + log(log(x))*log(x)*(10*x - exp(x)*(1250*x^2 + 500*x^3 + 50*x^4 + 100) )),x)
int(-(log(x)*(exp(2*x)*(40*x + 1000*x^3 + 400*x^4 + 6290*x^5 + 5000*x^6 + 1500*x^7 + 200*x^8 + 10*x^9) - exp(x)*(2*x^2 + 2*x^3 + 75*x^4 + 65*x^5 + 1 5*x^6 + x^7)) + x^2 - log(log(x))*log(x)*(exp(x)*(40*x + 500*x^3 + 200*x^4 + 20*x^5) - x^2) + 10*x*log(log(x))^2*log(x))/(log(x)*(exp(2*x)*(2500*x^2 + 1000*x^3 + 15725*x^4 + 12500*x^5 + 3750*x^6 + 500*x^7 + 25*x^8 + 100) - exp(x)*(20*x + 250*x^3 + 100*x^4 + 10*x^5) + x^2) + 25*log(log(x))^2*log( x) + log(log(x))*log(x)*(10*x - exp(x)*(1250*x^2 + 500*x^3 + 50*x^4 + 100) )), x)