Integrand size = 246, antiderivative size = 28 \[ \int \frac {e^{\log ^2\left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )} \left (e^{2+2 x} (16+4 x)+e^{2+x} \left (-320-144 x-16 x^2\right )+e^2 \left (-250-150 x-30 x^2-2 x^3\right )\right ) \log \left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )}{-125-75 x-15 x^2-x^3+e^{2+2 x} (5+x)+e^{2+x} \left (-200-80 x-8 x^2\right )+e^2 \left (2625+1450 x+240 x^2+6 x^3-x^4\right )} \, dx=e^{\log ^2\left (5-\frac {1}{e^2}-x+\left (-4+\frac {e^x}{5+x}\right )^2\right )} \] Output:
exp(ln(5+(exp(x)/(5+x)-4)^2-x-1/exp(2))^2)
Time = 0.11 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {e^{\log ^2\left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )} \left (e^{2+2 x} (16+4 x)+e^{2+x} \left (-320-144 x-16 x^2\right )+e^2 \left (-250-150 x-30 x^2-2 x^3\right )\right ) \log \left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )}{-125-75 x-15 x^2-x^3+e^{2+2 x} (5+x)+e^{2+x} \left (-200-80 x-8 x^2\right )+e^2 \left (2625+1450 x+240 x^2+6 x^3-x^4\right )} \, dx=e^{\log ^2\left (21-\frac {1}{e^2}-x+\frac {e^{2 x}}{(5+x)^2}-\frac {8 e^x}{5+x}\right )} \] Input:
Integrate[(E^Log[(-25 + E^(2 + 2*x) + E^(2 + x)*(-40 - 8*x) - 10*x - x^2 + E^2*(525 + 185*x + 11*x^2 - x^3))/(E^2*(25 + 10*x + x^2))]^2*(E^(2 + 2*x) *(16 + 4*x) + E^(2 + x)*(-320 - 144*x - 16*x^2) + E^2*(-250 - 150*x - 30*x ^2 - 2*x^3))*Log[(-25 + E^(2 + 2*x) + E^(2 + x)*(-40 - 8*x) - 10*x - x^2 + E^2*(525 + 185*x + 11*x^2 - x^3))/(E^2*(25 + 10*x + x^2))])/(-125 - 75*x - 15*x^2 - x^3 + E^(2 + 2*x)*(5 + x) + E^(2 + x)*(-200 - 80*x - 8*x^2) + E ^2*(2625 + 1450*x + 240*x^2 + 6*x^3 - x^4)),x]
Output:
E^Log[21 - E^(-2) - x + E^(2*x)/(5 + x)^2 - (8*E^x)/(5 + x)]^2
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.
| \(\displaystyle \int \frac {\left (e^{x+2} \left (-16 x^2-144 x-320\right )+e^2 \left (-2 x^3-30 x^2-150 x-250\right )+e^{2 x+2} (4 x+16)\right ) \log \left (\frac {-x^2+e^2 \left (-x^3+11 x^2+185 x+525\right )-10 x+e^{2 x+2}+e^{x+2} (-8 x-40)-25}{e^2 \left (x^2+10 x+25\right )}\right ) \exp \left (\log ^2\left (\frac {-x^2+e^2 \left (-x^3+11 x^2+185 x+525\right )-10 x+e^{2 x+2}+e^{x+2} (-8 x-40)-25}{e^2 \left (x^2+10 x+25\right )}\right )\right )}{-x^3-15 x^2+e^{x+2} \left (-8 x^2-80 x-200\right )+e^2 \left (-x^4+6 x^3+240 x^2+1450 x+2625\right )-75 x+e^{2 x+2} (x+5)-125} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-e^{x+2} \left (-16 x^2-144 x-320\right )-e^2 \left (-2 x^3-30 x^2-150 x-250\right )-e^{2 x+2} (4 x+16)\right ) \log \left (\frac {-x^2+e^2 \left (-x^3+11 x^2+185 x+525\right )-10 x+e^{2 x+2}+e^{x+2} (-8 x-40)-25}{e^2 \left (x^2+10 x+25\right )}\right ) \exp \left (\log ^2\left (\frac {-x^2+e^2 \left (-x^3+11 x^2+185 x+525\right )-10 x+e^{2 x+2}+e^{x+2} (-8 x-40)-25}{e^2 \left (x^2+10 x+25\right )}\right )\right )}{(x+5) \left (e^2 x^3+\left (1-11 e^2\right ) x^2+8 e^{x+2} x+10 \left (1-\frac {37 e^2}{2}\right ) x+40 e^{x+2}-e^{2 x+2}+25 \left (1-21 e^2\right )\right )}dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {2 \left (-8 e^x \left (x^2+9 x+20\right )-(x+5)^3+2 e^{2 x} (x+4)\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right ) \exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )+2\right )}{(x+5) \left (-e^2 (x-21) (x+5)^2-(x+5)^2-8 e^{x+2} (x+5)+e^{2 x+2}\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 \int \frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )+2\right ) \left (-(x+5)^3+2 e^{2 x} (x+4)-8 e^x \left (x^2+9 x+20\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )}{(x+5) \left (e^2 (21-x) (x+5)^2-(x+5)^2-8 e^{x+2} (x+5)+e^{2 x+2}\right )}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle 2 \int \frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )+2\right ) \left (-(x+5)^3+2 e^{2 x} (x+4)-8 e^x \left (x^2+9 x+20\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{(x+5) \left (e^2 (21-x) (x+5)^2-(x+5)^2-8 e^{x+2} (x+5)+e^{2 x+2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 \exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) (x+4) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{x+5}+\frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) \left (-2 e^2 x^3-2 \left (1-\frac {25 e^2}{2}\right ) x^2-8 e^{x+2} x-18 \left (1-\frac {58 e^2}{3}\right ) x-32 e^{x+2}-40 \left (1-\frac {173 e^2}{8}\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{e^2 x^3+\left (1-11 e^2\right ) x^2+8 e^{x+2} x+10 \left (1-\frac {37 e^2}{2}\right ) x+40 e^{x+2}-e^{2 x+2}+25 \left (1-21 e^2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )+2\right ) \left (-(x+5)^3+2 e^{2 x} (x+4)-8 e^x \left (x^2+9 x+20\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{(x+5) \left (-e^2 (x-21) (x+5)^2-(x+5)^2-8 e^{x+2} (x+5)+e^{2 x+2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 \exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) (x+4) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{x+5}+\frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) \left (-2 e^2 x^3-2 \left (1-\frac {25 e^2}{2}\right ) x^2-8 e^{x+2} x-18 \left (1-\frac {58 e^2}{3}\right ) x-32 e^{x+2}-40 \left (1-\frac {173 e^2}{8}\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{e^2 x^3+\left (1-11 e^2\right ) x^2+8 e^{x+2} x+10 \left (1-\frac {37 e^2}{2}\right ) x+40 e^{x+2}-e^{2 x+2}+25 \left (1-21 e^2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )+2\right ) \left (-(x+5)^3+2 e^{2 x} (x+4)-8 e^x \left (x^2+9 x+20\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{(x+5) \left (-e^2 (x-21) (x+5)^2-(x+5)^2-8 e^{x+2} (x+5)+e^{2 x+2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 \exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) (x+4) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{x+5}+\frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) \left (-2 e^2 x^3-2 \left (1-\frac {25 e^2}{2}\right ) x^2-8 e^{x+2} x-18 \left (1-\frac {58 e^2}{3}\right ) x-32 e^{x+2}-40 \left (1-\frac {173 e^2}{8}\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{e^2 x^3+\left (1-11 e^2\right ) x^2+8 e^{x+2} x+10 \left (1-\frac {37 e^2}{2}\right ) x+40 e^{x+2}-e^{2 x+2}+25 \left (1-21 e^2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )+2\right ) \left (-(x+5)^3+2 e^{2 x} (x+4)-8 e^x \left (x^2+9 x+20\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{(x+5) \left (-e^2 (x-21) (x+5)^2-(x+5)^2-8 e^{x+2} (x+5)+e^{2 x+2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 \exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) (x+4) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{x+5}+\frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) \left (-2 e^2 x^3-2 \left (1-\frac {25 e^2}{2}\right ) x^2-8 e^{x+2} x-18 \left (1-\frac {58 e^2}{3}\right ) x-32 e^{x+2}-40 \left (1-\frac {173 e^2}{8}\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{e^2 x^3+\left (1-11 e^2\right ) x^2+8 e^{x+2} x+10 \left (1-\frac {37 e^2}{2}\right ) x+40 e^{x+2}-e^{2 x+2}+25 \left (1-21 e^2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )+2\right ) \left (-(x+5)^3+2 e^{2 x} (x+4)-8 e^x \left (x^2+9 x+20\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{(x+5) \left (-e^2 (x-21) (x+5)^2-(x+5)^2-8 e^{x+2} (x+5)+e^{2 x+2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 \exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) (x+4) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{x+5}+\frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) \left (-2 e^2 x^3-2 \left (1-\frac {25 e^2}{2}\right ) x^2-8 e^{x+2} x-18 \left (1-\frac {58 e^2}{3}\right ) x-32 e^{x+2}-40 \left (1-\frac {173 e^2}{8}\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{e^2 x^3+\left (1-11 e^2\right ) x^2+8 e^{x+2} x+10 \left (1-\frac {37 e^2}{2}\right ) x+40 e^{x+2}-e^{2 x+2}+25 \left (1-21 e^2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )+2\right ) \left (-(x+5)^3+2 e^{2 x} (x+4)-8 e^x \left (x^2+9 x+20\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{(x+5) \left (-e^2 (x-21) (x+5)^2-(x+5)^2-8 e^{x+2} (x+5)+e^{2 x+2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 \exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) (x+4) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{x+5}+\frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) \left (-2 e^2 x^3-2 \left (1-\frac {25 e^2}{2}\right ) x^2-8 e^{x+2} x-18 \left (1-\frac {58 e^2}{3}\right ) x-32 e^{x+2}-40 \left (1-\frac {173 e^2}{8}\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{e^2 x^3+\left (1-11 e^2\right ) x^2+8 e^{x+2} x+10 \left (1-\frac {37 e^2}{2}\right ) x+40 e^{x+2}-e^{2 x+2}+25 \left (1-21 e^2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )+2\right ) \left (-(x+5)^3+2 e^{2 x} (x+4)-8 e^x \left (x^2+9 x+20\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{(x+5) \left (-e^2 (x-21) (x+5)^2-(x+5)^2-8 e^{x+2} (x+5)+e^{2 x+2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 \exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) (x+4) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{x+5}+\frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) \left (-2 e^2 x^3-2 \left (1-\frac {25 e^2}{2}\right ) x^2-8 e^{x+2} x-18 \left (1-\frac {58 e^2}{3}\right ) x-32 e^{x+2}-40 \left (1-\frac {173 e^2}{8}\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{e^2 x^3+\left (1-11 e^2\right ) x^2+8 e^{x+2} x+10 \left (1-\frac {37 e^2}{2}\right ) x+40 e^{x+2}-e^{2 x+2}+25 \left (1-21 e^2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )+2\right ) \left (-(x+5)^3+2 e^{2 x} (x+4)-8 e^x \left (x^2+9 x+20\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{(x+5) \left (-e^2 (x-21) (x+5)^2-(x+5)^2-8 e^{x+2} (x+5)+e^{2 x+2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 \exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) (x+4) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{x+5}+\frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) \left (-2 e^2 x^3-2 \left (1-\frac {25 e^2}{2}\right ) x^2-8 e^{x+2} x-18 \left (1-\frac {58 e^2}{3}\right ) x-32 e^{x+2}-40 \left (1-\frac {173 e^2}{8}\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{e^2 x^3+\left (1-11 e^2\right ) x^2+8 e^{x+2} x+10 \left (1-\frac {37 e^2}{2}\right ) x+40 e^{x+2}-e^{2 x+2}+25 \left (1-21 e^2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )+2\right ) \left (-(x+5)^3+2 e^{2 x} (x+4)-8 e^x \left (x^2+9 x+20\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{(x+5) \left (-e^2 (x-21) (x+5)^2-(x+5)^2-8 e^{x+2} (x+5)+e^{2 x+2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 \exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) (x+4) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{x+5}+\frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) \left (-2 e^2 x^3-2 \left (1-\frac {25 e^2}{2}\right ) x^2-8 e^{x+2} x-18 \left (1-\frac {58 e^2}{3}\right ) x-32 e^{x+2}-40 \left (1-\frac {173 e^2}{8}\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{e^2 x^3+\left (1-11 e^2\right ) x^2+8 e^{x+2} x+10 \left (1-\frac {37 e^2}{2}\right ) x+40 e^{x+2}-e^{2 x+2}+25 \left (1-21 e^2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )+2\right ) \left (-(x+5)^3+2 e^{2 x} (x+4)-8 e^x \left (x^2+9 x+20\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{(x+5) \left (-e^2 (x-21) (x+5)^2-(x+5)^2-8 e^{x+2} (x+5)+e^{2 x+2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 \exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) (x+4) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{x+5}+\frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) \left (-2 e^2 x^3-2 \left (1-\frac {25 e^2}{2}\right ) x^2-8 e^{x+2} x-18 \left (1-\frac {58 e^2}{3}\right ) x-32 e^{x+2}-40 \left (1-\frac {173 e^2}{8}\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{e^2 x^3+\left (1-11 e^2\right ) x^2+8 e^{x+2} x+10 \left (1-\frac {37 e^2}{2}\right ) x+40 e^{x+2}-e^{2 x+2}+25 \left (1-21 e^2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )+2\right ) \left (-(x+5)^3+2 e^{2 x} (x+4)-8 e^x \left (x^2+9 x+20\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{(x+5) \left (-e^2 (x-21) (x+5)^2-(x+5)^2-8 e^{x+2} (x+5)+e^{2 x+2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 \exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) (x+4) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{x+5}+\frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) \left (-2 e^2 x^3-2 \left (1-\frac {25 e^2}{2}\right ) x^2-8 e^{x+2} x-18 \left (1-\frac {58 e^2}{3}\right ) x-32 e^{x+2}-40 \left (1-\frac {173 e^2}{8}\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{e^2 x^3+\left (1-11 e^2\right ) x^2+8 e^{x+2} x+10 \left (1-\frac {37 e^2}{2}\right ) x+40 e^{x+2}-e^{2 x+2}+25 \left (1-21 e^2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )+2\right ) \left (-(x+5)^3+2 e^{2 x} (x+4)-8 e^x \left (x^2+9 x+20\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{(x+5) \left (-e^2 (x-21) (x+5)^2-(x+5)^2-8 e^{x+2} (x+5)+e^{2 x+2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 \exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) (x+4) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{x+5}+\frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) \left (-2 e^2 x^3-2 \left (1-\frac {25 e^2}{2}\right ) x^2-8 e^{x+2} x-18 \left (1-\frac {58 e^2}{3}\right ) x-32 e^{x+2}-40 \left (1-\frac {173 e^2}{8}\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{e^2 x^3+\left (1-11 e^2\right ) x^2+8 e^{x+2} x+10 \left (1-\frac {37 e^2}{2}\right ) x+40 e^{x+2}-e^{2 x+2}+25 \left (1-21 e^2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )+2\right ) \left (-(x+5)^3+2 e^{2 x} (x+4)-8 e^x \left (x^2+9 x+20\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{(x+5) \left (-e^2 (x-21) (x+5)^2-(x+5)^2-8 e^{x+2} (x+5)+e^{2 x+2}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle 2 \int \left (\frac {2 \exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) (x+4) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{x+5}+\frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )\right ) \left (-2 e^2 x^3-2 \left (1-\frac {25 e^2}{2}\right ) x^2-8 e^{x+2} x-18 \left (1-\frac {58 e^2}{3}\right ) x-32 e^{x+2}-40 \left (1-\frac {173 e^2}{8}\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{e^2 x^3+\left (1-11 e^2\right ) x^2+8 e^{x+2} x+10 \left (1-\frac {37 e^2}{2}\right ) x+40 e^{x+2}-e^{2 x+2}+25 \left (1-21 e^2\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle 2 \int \frac {\exp \left (\log ^2\left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}-\frac {1}{e^2}+21\right )+2\right ) \left (-(x+5)^3+2 e^{2 x} (x+4)-8 e^x \left (x^2+9 x+20\right )\right ) \log \left (-x-\frac {8 e^x}{x+5}+\frac {e^{2 x}}{(x+5)^2}+21 \left (1-\frac {1}{21 e^2}\right )\right )}{(x+5) \left (-e^2 (x-21) (x+5)^2-(x+5)^2-8 e^{x+2} (x+5)+e^{2 x+2}\right )}dx\) |
Input:
Int[(E^Log[(-25 + E^(2 + 2*x) + E^(2 + x)*(-40 - 8*x) - 10*x - x^2 + E^2*( 525 + 185*x + 11*x^2 - x^3))/(E^2*(25 + 10*x + x^2))]^2*(E^(2 + 2*x)*(16 + 4*x) + E^(2 + x)*(-320 - 144*x - 16*x^2) + E^2*(-250 - 150*x - 30*x^2 - 2 *x^3))*Log[(-25 + E^(2 + 2*x) + E^(2 + x)*(-40 - 8*x) - 10*x - x^2 + E^2*( 525 + 185*x + 11*x^2 - x^3))/(E^2*(25 + 10*x + x^2))])/(-125 - 75*x - 15*x ^2 - x^3 + E^(2 + 2*x)*(5 + x) + E^(2 + x)*(-200 - 80*x - 8*x^2) + E^2*(26 25 + 1450*x + 240*x^2 + 6*x^3 - x^4)),x]
Output:
$Aborted
Leaf count of result is larger than twice the leaf count of optimal. \(64\) vs. \(2(27)=54\).
Time = 54.50 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.32
| method | result | size |
| parallelrisch | \({\mathrm e}^{{\ln \left (\frac {\left ({\mathrm e}^{2} {\mathrm e}^{2 x}+\left (-8 x -40\right ) {\mathrm e}^{2} {\mathrm e}^{x}+\left (-x^{3}+11 x^{2}+185 x +525\right ) {\mathrm e}^{2}-x^{2}-10 x -25\right ) {\mathrm e}^{-2}}{x^{2}+10 x +25}\right )}^{2}}\) | \(65\) |
| risch | \({\mathrm e}^{\frac {{\left (-i \pi {\operatorname {csgn}\left (\frac {i \left (\left (x^{3}-11 x^{2}+\left (8 \,{\mathrm e}^{x}-185\right ) x -{\mathrm e}^{2 x}+40 \,{\mathrm e}^{x}-525\right ) {\mathrm e}^{2}+x^{2}+10 x +25\right )}{\left (5+x \right )^{2}}\right )}^{3}-i {\operatorname {csgn}\left (\frac {i \left (\left (x^{3}-11 x^{2}+\left (8 \,{\mathrm e}^{x}-185\right ) x -{\mathrm e}^{2 x}+40 \,{\mathrm e}^{x}-525\right ) {\mathrm e}^{2}+x^{2}+10 x +25\right )}{\left (5+x \right )^{2}}\right )}^{2} \operatorname {csgn}\left (\frac {i}{\left (5+x \right )^{2}}\right ) \pi -i {\operatorname {csgn}\left (\frac {i \left (\left (x^{3}-11 x^{2}+\left (8 \,{\mathrm e}^{x}-185\right ) x -{\mathrm e}^{2 x}+40 \,{\mathrm e}^{x}-525\right ) {\mathrm e}^{2}+x^{2}+10 x +25\right )}{\left (5+x \right )^{2}}\right )}^{2} \operatorname {csgn}\left (i \left (\left (x^{3}-11 x^{2}+\left (8 \,{\mathrm e}^{x}-185\right ) x -{\mathrm e}^{2 x}+40 \,{\mathrm e}^{x}-525\right ) {\mathrm e}^{2}+x^{2}+10 x +25\right )\right ) \pi +i \operatorname {csgn}\left (\frac {i \left (\left (x^{3}-11 x^{2}+\left (8 \,{\mathrm e}^{x}-185\right ) x -{\mathrm e}^{2 x}+40 \,{\mathrm e}^{x}-525\right ) {\mathrm e}^{2}+x^{2}+10 x +25\right )}{\left (5+x \right )^{2}}\right ) \operatorname {csgn}\left (\frac {i}{\left (5+x \right )^{2}}\right ) \operatorname {csgn}\left (i \left (\left (x^{3}-11 x^{2}+\left (8 \,{\mathrm e}^{x}-185\right ) x -{\mathrm e}^{2 x}+40 \,{\mathrm e}^{x}-525\right ) {\mathrm e}^{2}+x^{2}+10 x +25\right )\right ) \pi -i \operatorname {csgn}\left (i \left (5+x \right )^{2}\right )^{3} \pi +2 i \operatorname {csgn}\left (i \left (5+x \right )^{2}\right )^{2} \operatorname {csgn}\left (i \left (5+x \right )\right ) \pi -i \operatorname {csgn}\left (i \left (5+x \right )^{2}\right ) \operatorname {csgn}\left (i \left (5+x \right )\right )^{2} \pi +2 i {\operatorname {csgn}\left (\frac {i \left (\left (x^{3}-11 x^{2}+\left (8 \,{\mathrm e}^{x}-185\right ) x -{\mathrm e}^{2 x}+40 \,{\mathrm e}^{x}-525\right ) {\mathrm e}^{2}+x^{2}+10 x +25\right )}{\left (5+x \right )^{2}}\right )}^{2} \pi -2 i \pi +4 \ln \left (5+x \right )-2 \ln \left (\left (x^{3}-11 x^{2}+\left (8 \,{\mathrm e}^{x}-185\right ) x -{\mathrm e}^{2 x}+40 \,{\mathrm e}^{x}-525\right ) {\mathrm e}^{2}+x^{2}+10 x +25\right )+4\right )}^{2}}{4}}\) | \(491\) |
Input:
int(((4*x+16)*exp(2)*exp(x)^2+(-16*x^2-144*x-320)*exp(2)*exp(x)+(-2*x^3-30 *x^2-150*x-250)*exp(2))*ln((exp(2)*exp(x)^2+(-8*x-40)*exp(2)*exp(x)+(-x^3+ 11*x^2+185*x+525)*exp(2)-x^2-10*x-25)/(x^2+10*x+25)/exp(2))*exp(ln((exp(2) *exp(x)^2+(-8*x-40)*exp(2)*exp(x)+(-x^3+11*x^2+185*x+525)*exp(2)-x^2-10*x- 25)/(x^2+10*x+25)/exp(2))^2)/((5+x)*exp(2)*exp(x)^2+(-8*x^2-80*x-200)*exp( 2)*exp(x)+(-x^4+6*x^3+240*x^2+1450*x+2625)*exp(2)-x^3-15*x^2-75*x-125),x,m ethod=_RETURNVERBOSE)
Output:
exp(ln((exp(2)*exp(x)^2+(-8*x-40)*exp(2)*exp(x)+(-x^3+11*x^2+185*x+525)*ex p(2)-x^2-10*x-25)/(x^2+10*x+25)/exp(2))^2)
Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (25) = 50\).
Time = 0.08 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.25 \[ \int \frac {e^{\log ^2\left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )} \left (e^{2+2 x} (16+4 x)+e^{2+x} \left (-320-144 x-16 x^2\right )+e^2 \left (-250-150 x-30 x^2-2 x^3\right )\right ) \log \left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )}{-125-75 x-15 x^2-x^3+e^{2+2 x} (5+x)+e^{2+x} \left (-200-80 x-8 x^2\right )+e^2 \left (2625+1450 x+240 x^2+6 x^3-x^4\right )} \, dx=e^{\left (\log \left (-\frac {{\left ({\left (x^{3} - 11 \, x^{2} - 185 \, x - 525\right )} e^{4} + {\left (x^{2} + 10 \, x + 25\right )} e^{2} + 8 \, {\left (x + 5\right )} e^{\left (x + 4\right )} - e^{\left (2 \, x + 4\right )}\right )} e^{\left (-4\right )}}{x^{2} + 10 \, x + 25}\right )^{2}\right )} \] Input:
integrate(((4*x+16)*exp(2)*exp(x)^2+(-16*x^2-144*x-320)*exp(2)*exp(x)+(-2* x^3-30*x^2-150*x-250)*exp(2))*log((exp(2)*exp(x)^2+(-8*x-40)*exp(2)*exp(x) +(-x^3+11*x^2+185*x+525)*exp(2)-x^2-10*x-25)/(x^2+10*x+25)/exp(2))*exp(log ((exp(2)*exp(x)^2+(-8*x-40)*exp(2)*exp(x)+(-x^3+11*x^2+185*x+525)*exp(2)-x ^2-10*x-25)/(x^2+10*x+25)/exp(2))^2)/((5+x)*exp(2)*exp(x)^2+(-8*x^2-80*x-2 00)*exp(2)*exp(x)+(-x^4+6*x^3+240*x^2+1450*x+2625)*exp(2)-x^3-15*x^2-75*x- 125),x, algorithm="fricas")
Output:
e^(log(-((x^3 - 11*x^2 - 185*x - 525)*e^4 + (x^2 + 10*x + 25)*e^2 + 8*(x + 5)*e^(x + 4) - e^(2*x + 4))*e^(-4)/(x^2 + 10*x + 25))^2)
Timed out. \[ \int \frac {e^{\log ^2\left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )} \left (e^{2+2 x} (16+4 x)+e^{2+x} \left (-320-144 x-16 x^2\right )+e^2 \left (-250-150 x-30 x^2-2 x^3\right )\right ) \log \left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )}{-125-75 x-15 x^2-x^3+e^{2+2 x} (5+x)+e^{2+x} \left (-200-80 x-8 x^2\right )+e^2 \left (2625+1450 x+240 x^2+6 x^3-x^4\right )} \, dx=\text {Timed out} \] Input:
integrate(((4*x+16)*exp(2)*exp(x)**2+(-16*x**2-144*x-320)*exp(2)*exp(x)+(- 2*x**3-30*x**2-150*x-250)*exp(2))*ln((exp(2)*exp(x)**2+(-8*x-40)*exp(2)*ex p(x)+(-x**3+11*x**2+185*x+525)*exp(2)-x**2-10*x-25)/(x**2+10*x+25)/exp(2)) *exp(ln((exp(2)*exp(x)**2+(-8*x-40)*exp(2)*exp(x)+(-x**3+11*x**2+185*x+525 )*exp(2)-x**2-10*x-25)/(x**2+10*x+25)/exp(2))**2)/((5+x)*exp(2)*exp(x)**2+ (-8*x**2-80*x-200)*exp(2)*exp(x)+(-x**4+6*x**3+240*x**2+1450*x+2625)*exp(2 )-x**3-15*x**2-75*x-125),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 1199 vs. \(2 (25) = 50\).
Time = 0.90 (sec) , antiderivative size = 1199, normalized size of antiderivative = 42.82 \[ \int \frac {e^{\log ^2\left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )} \left (e^{2+2 x} (16+4 x)+e^{2+x} \left (-320-144 x-16 x^2\right )+e^2 \left (-250-150 x-30 x^2-2 x^3\right )\right ) \log \left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )}{-125-75 x-15 x^2-x^3+e^{2+2 x} (5+x)+e^{2+x} \left (-200-80 x-8 x^2\right )+e^2 \left (2625+1450 x+240 x^2+6 x^3-x^4\right )} \, dx=\text {Too large to display} \] Input:
integrate(((4*x+16)*exp(2)*exp(x)^2+(-16*x^2-144*x-320)*exp(2)*exp(x)+(-2* x^3-30*x^2-150*x-250)*exp(2))*log((exp(2)*exp(x)^2+(-8*x-40)*exp(2)*exp(x) +(-x^3+11*x^2+185*x+525)*exp(2)-x^2-10*x-25)/(x^2+10*x+25)/exp(2))*exp(log ((exp(2)*exp(x)^2+(-8*x-40)*exp(2)*exp(x)+(-x^3+11*x^2+185*x+525)*exp(2)-x ^2-10*x-25)/(x^2+10*x+25)/exp(2))^2)/((5+x)*exp(2)*exp(x)^2+(-8*x^2-80*x-2 00)*exp(2)*exp(x)+(-x^4+6*x^3+240*x^2+1450*x+2625)*exp(2)-x^3-15*x^2-75*x- 125),x, algorithm="maxima")
Output:
(x^8*e^4 + 40*x^7*e^4 + 700*x^6*e^4 + 7000*x^5*e^4 + 43750*x^4*e^4 + 17500 0*x^3*e^4 + 437500*x^2*e^4 + 625000*x*e^4 + 390625*e^4)*e^(log(-x^3*e^2 + x^2*(11*e^2 - 1) + 5*x*(37*e^2 - 2) - 8*(x*e^2 + 5*e^2)*e^x + 525*e^2 + e^ (2*x + 2) - 25)^2 - 4*log(-x^3*e^2 + x^2*(11*e^2 - 1) + 5*x*(37*e^2 - 2) - 8*(x*e^2 + 5*e^2)*e^x + 525*e^2 + e^(2*x + 2) - 25)*log(x + 5) + 4*log(x + 5)^2)/(x^12*e^8 - 4*x^11*(11*e^8 - e^6) - 2*x^10*(7*e^8 + 46*e^6 - 3*e^4 ) + 4*x^9*(4249*e^8 - 497*e^6 - 3*e^4 + e^2) + x^8*(20671*e^8 + 26236*e^6 - 3234*e^4 + 76*e^2 + 1) - 40*x^7*(78239*e^8 - 15841*e^6 + 714*e^4 + 14*e^ 2 - 1) - 700*x^6*(30959*e^8 - 1126*e^6 - 501*e^4 + 44*e^2 - 1) + 1000*x^5* (167617*e^8 - 70133*e^6 + 8547*e^4 - 413*e^2 + 7) + 3125*x^4*(1002035*e^8 - 256744*e^6 + 23772*e^4 - 952*e^2 + 14) + 12500*x^3*(1555869*e^8 - 345871 *e^6 + 28524*e^4 - 1036*e^2 + 14) + 31250*x^2*(2014929*e^8 - 415926*e^6 + 32079*e^4 - 1096*e^2 + 14) + 312500*x*(342657*e^8 - 67473*e^6 + 4977*e^4 - 163*e^2 + 2) - 32*(x*e^8 + 5*e^8)*e^(7*x) - 4*(x^3*e^8 - x^2*(107*e^8 - e ^6) - 5*x*(229*e^8 - 2*e^6) - 2925*e^8 + 25*e^6)*e^(6*x) + 32*(3*x^4*e^8 - x^3*(82*e^8 - 3*e^6) - 15*x^2*(112*e^8 - 3*e^6) - 75*x*(122*e^8 - 3*e^6) - 15875*e^8 + 375*e^6)*e^(5*x) + 2*(3*x^6*e^8 - 6*x^5*(75*e^8 - e^6) + x^4 *(1685*e^8 - 390*e^6 + 3*e^4) + 20*x^3*(7685*e^8 - 465*e^6 + 3*e^4) + 75*x ^2*(19495*e^8 - 980*e^6 + 6*e^4) + 250*x*(21595*e^8 - 1005*e^6 + 6*e^4) + 7146875*e^8 - 318750*e^6 + 1875*e^4)*e^(4*x) - 32*(3*x^7*e^8 - x^6*(115...
\[ \int \frac {e^{\log ^2\left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )} \left (e^{2+2 x} (16+4 x)+e^{2+x} \left (-320-144 x-16 x^2\right )+e^2 \left (-250-150 x-30 x^2-2 x^3\right )\right ) \log \left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )}{-125-75 x-15 x^2-x^3+e^{2+2 x} (5+x)+e^{2+x} \left (-200-80 x-8 x^2\right )+e^2 \left (2625+1450 x+240 x^2+6 x^3-x^4\right )} \, dx=\int { \frac {2 \, {\left ({\left (x^{3} + 15 \, x^{2} + 75 \, x + 125\right )} e^{2} - 2 \, {\left (x + 4\right )} e^{\left (2 \, x + 2\right )} + 8 \, {\left (x^{2} + 9 \, x + 20\right )} e^{\left (x + 2\right )}\right )} e^{\left (\log \left (-\frac {{\left (x^{2} + {\left (x^{3} - 11 \, x^{2} - 185 \, x - 525\right )} e^{2} + 8 \, {\left (x + 5\right )} e^{\left (x + 2\right )} + 10 \, x - e^{\left (2 \, x + 2\right )} + 25\right )} e^{\left (-2\right )}}{x^{2} + 10 \, x + 25}\right )^{2}\right )} \log \left (-\frac {{\left (x^{2} + {\left (x^{3} - 11 \, x^{2} - 185 \, x - 525\right )} e^{2} + 8 \, {\left (x + 5\right )} e^{\left (x + 2\right )} + 10 \, x - e^{\left (2 \, x + 2\right )} + 25\right )} e^{\left (-2\right )}}{x^{2} + 10 \, x + 25}\right )}{x^{3} + 15 \, x^{2} + {\left (x^{4} - 6 \, x^{3} - 240 \, x^{2} - 1450 \, x - 2625\right )} e^{2} - {\left (x + 5\right )} e^{\left (2 \, x + 2\right )} + 8 \, {\left (x^{2} + 10 \, x + 25\right )} e^{\left (x + 2\right )} + 75 \, x + 125} \,d x } \] Input:
integrate(((4*x+16)*exp(2)*exp(x)^2+(-16*x^2-144*x-320)*exp(2)*exp(x)+(-2* x^3-30*x^2-150*x-250)*exp(2))*log((exp(2)*exp(x)^2+(-8*x-40)*exp(2)*exp(x) +(-x^3+11*x^2+185*x+525)*exp(2)-x^2-10*x-25)/(x^2+10*x+25)/exp(2))*exp(log ((exp(2)*exp(x)^2+(-8*x-40)*exp(2)*exp(x)+(-x^3+11*x^2+185*x+525)*exp(2)-x ^2-10*x-25)/(x^2+10*x+25)/exp(2))^2)/((5+x)*exp(2)*exp(x)^2+(-8*x^2-80*x-2 00)*exp(2)*exp(x)+(-x^4+6*x^3+240*x^2+1450*x+2625)*exp(2)-x^3-15*x^2-75*x- 125),x, algorithm="giac")
Output:
integrate(2*((x^3 + 15*x^2 + 75*x + 125)*e^2 - 2*(x + 4)*e^(2*x + 2) + 8*( x^2 + 9*x + 20)*e^(x + 2))*e^(log(-(x^2 + (x^3 - 11*x^2 - 185*x - 525)*e^2 + 8*(x + 5)*e^(x + 2) + 10*x - e^(2*x + 2) + 25)*e^(-2)/(x^2 + 10*x + 25) )^2)*log(-(x^2 + (x^3 - 11*x^2 - 185*x - 525)*e^2 + 8*(x + 5)*e^(x + 2) + 10*x - e^(2*x + 2) + 25)*e^(-2)/(x^2 + 10*x + 25))/(x^3 + 15*x^2 + (x^4 - 6*x^3 - 240*x^2 - 1450*x - 2625)*e^2 - (x + 5)*e^(2*x + 2) + 8*(x^2 + 10*x + 25)*e^(x + 2) + 75*x + 125), x)
Time = 5.46 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.11 \[ \int \frac {e^{\log ^2\left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )} \left (e^{2+2 x} (16+4 x)+e^{2+x} \left (-320-144 x-16 x^2\right )+e^2 \left (-250-150 x-30 x^2-2 x^3\right )\right ) \log \left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )}{-125-75 x-15 x^2-x^3+e^{2+2 x} (5+x)+e^{2+x} \left (-200-80 x-8 x^2\right )+e^2 \left (2625+1450 x+240 x^2+6 x^3-x^4\right )} \, dx={\mathrm {e}}^{{\ln \left ({\mathrm {e}}^{-2}\,\left (21\,{\mathrm {e}}^2-1\right )-\frac {40\,{\mathrm {e}}^2\,{\mathrm {e}}^x-{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^2+8\,x\,{\mathrm {e}}^2\,{\mathrm {e}}^x}{{\mathrm {e}}^2\,x^2+10\,{\mathrm {e}}^2\,x+25\,{\mathrm {e}}^2}-x\right )}^2} \] Input:
int((exp(log(-(exp(-2)*(10*x - exp(2*x)*exp(2) - exp(2)*(185*x + 11*x^2 - x^3 + 525) + x^2 + exp(2)*exp(x)*(8*x + 40) + 25))/(10*x + x^2 + 25))^2)*l og(-(exp(-2)*(10*x - exp(2*x)*exp(2) - exp(2)*(185*x + 11*x^2 - x^3 + 525) + x^2 + exp(2)*exp(x)*(8*x + 40) + 25))/(10*x + x^2 + 25))*(exp(2)*(150*x + 30*x^2 + 2*x^3 + 250) + exp(2)*exp(x)*(144*x + 16*x^2 + 320) - exp(2*x) *exp(2)*(4*x + 16)))/(75*x - exp(2)*(1450*x + 240*x^2 + 6*x^3 - x^4 + 2625 ) + 15*x^2 + x^3 + exp(2)*exp(x)*(80*x + 8*x^2 + 200) - exp(2*x)*exp(2)*(x + 5) + 125),x)
Output:
exp(log(exp(-2)*(21*exp(2) - 1) - (40*exp(2)*exp(x) - exp(2*x)*exp(2) + 8* x*exp(2)*exp(x))/(25*exp(2) + 10*x*exp(2) + x^2*exp(2)) - x)^2)
Time = 4.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 3.21 \[ \int \frac {e^{\log ^2\left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )} \left (e^{2+2 x} (16+4 x)+e^{2+x} \left (-320-144 x-16 x^2\right )+e^2 \left (-250-150 x-30 x^2-2 x^3\right )\right ) \log \left (\frac {-25+e^{2+2 x}+e^{2+x} (-40-8 x)-10 x-x^2+e^2 \left (525+185 x+11 x^2-x^3\right )}{e^2 \left (25+10 x+x^2\right )}\right )}{-125-75 x-15 x^2-x^3+e^{2+2 x} (5+x)+e^{2+x} \left (-200-80 x-8 x^2\right )+e^2 \left (2625+1450 x+240 x^2+6 x^3-x^4\right )} \, dx=e^{\mathrm {log}\left (\frac {e^{2 x} e^{2}-8 e^{x} e^{2} x -40 e^{x} e^{2}-e^{2} x^{3}+11 e^{2} x^{2}+185 e^{2} x +525 e^{2}-x^{2}-10 x -25}{e^{2} x^{2}+10 e^{2} x +25 e^{2}}\right )^{2}} \] Input:
int(((4*x+16)*exp(2)*exp(x)^2+(-16*x^2-144*x-320)*exp(2)*exp(x)+(-2*x^3-30 *x^2-150*x-250)*exp(2))*log((exp(2)*exp(x)^2+(-8*x-40)*exp(2)*exp(x)+(-x^3 +11*x^2+185*x+525)*exp(2)-x^2-10*x-25)/(x^2+10*x+25)/exp(2))*exp(log((exp( 2)*exp(x)^2+(-8*x-40)*exp(2)*exp(x)+(-x^3+11*x^2+185*x+525)*exp(2)-x^2-10* x-25)/(x^2+10*x+25)/exp(2))^2)/((5+x)*exp(2)*exp(x)^2+(-8*x^2-80*x-200)*ex p(2)*exp(x)+(-x^4+6*x^3+240*x^2+1450*x+2625)*exp(2)-x^3-15*x^2-75*x-125),x )
Output:
e**(log((e**(2*x)*e**2 - 8*e**x*e**2*x - 40*e**x*e**2 - e**2*x**3 + 11*e** 2*x**2 + 185*e**2*x + 525*e**2 - x**2 - 10*x - 25)/(e**2*x**2 + 10*e**2*x + 25*e**2))**2)