Integrand size = 78, antiderivative size = 28 \[ \int \frac {-1+e^{e^{e^{4 x}}} \left (-20 x+4 e^x x+e^{e^{4 x}+4 x} \left (16 x-4 e^x x\right )\right )+5 x \log (x)}{e^{e^{e^{4 x}}} \left (-20 x+5 e^x x\right )+5 x \log (x)} \, dx=x-\frac {1}{5} \log \left (-e^{e^{e^{4 x}}} \left (4-e^x\right )+\log (x)\right ) \]
Time = 1.31 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.29 \[ \int \frac {-1+e^{e^{e^{4 x}}} \left (-20 x+4 e^x x+e^{e^{4 x}+4 x} \left (16 x-4 e^x x\right )\right )+5 x \log (x)}{e^{e^{e^{4 x}}} \left (-20 x+5 e^x x\right )+5 x \log (x)} \, dx=\frac {1}{5} \left (5 x-\log \left (-4 e^{e^{e^{4 x}}}+e^{e^{e^{4 x}}+x}+\log (x)\right )\right ) \]
Integrate[(-1 + E^E^E^(4*x)*(-20*x + 4*E^x*x + E^(E^(4*x) + 4*x)*(16*x - 4 *E^x*x)) + 5*x*Log[x])/(E^E^E^(4*x)*(-20*x + 5*E^x*x) + 5*x*Log[x]),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.
| \(\displaystyle \int \frac {e^{e^{e^{4 x}}} \left (4 e^x x-20 x+e^{4 x+e^{4 x}} \left (16 x-4 e^x x\right )\right )+5 x \log (x)-1}{e^{e^{e^{4 x}}} \left (5 e^x x-20 x\right )+5 x \log (x)} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-e^{e^{e^{4 x}}} \left (4 e^x x-20 x+e^{4 x+e^{4 x}} \left (16 x-4 e^x x\right )\right )-5 x \log (x)+1}{5 x \left (4 e^{e^{e^{4 x}}}-e^{x+e^{e^{4 x}}}-\log (x)\right )}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \frac {4 e^{e^{e^{4 x}}} \left (-e^x x+5 x-e^{4 x+e^{4 x}} \left (4 x-e^x x\right )\right )-5 x \log (x)+1}{x \left (-\log (x)+4 e^{e^{e^{4 x}}}-e^{x+e^{e^{4 x}}}\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (4 e^{x-3 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )^2+4 e^{2 x-2 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )-4 e^{4 x+e^{4 x}}+4 e^{3 x-e^{e^{4 x}}+e^{4 x}} \log (x)+4 e^{-4 e^{e^{4 x}}} \left (-e^{e^{4 x}} \log ^4(x)+12 e^{e^{e^{4 x}}+e^{4 x}} \log ^3(x)-48 e^{2 e^{e^{4 x}}+e^{4 x}} \log ^2(x)+64 e^{3 e^{e^{4 x}}+e^{4 x}} \log (x)+e^{4 e^{e^{4 x}}}\right )-\frac {e^{-4 e^{e^{4 x}}} \left (-4 e^{e^{4 x}} x \log ^5(x)+64 e^{e^{e^{4 x}}+e^{4 x}} x \log ^4(x)-384 e^{2 e^{e^{4 x}}+e^{4 x}} x \log ^3(x)+1024 e^{3 e^{e^{4 x}}+e^{4 x}} x \log ^2(x)-e^{4 e^{e^{4 x}}} x \log (x)-1024 e^{4 e^{e^{4 x}}+e^{4 x}} x \log (x)+e^{4 e^{e^{4 x}}}+4 e^{5 e^{e^{4 x}}} x\right )}{x \left (\log (x)-4 e^{e^{e^{4 x}}}+e^{x+e^{e^{4 x}}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-20 e^{e^{e^{4 x}}} x+4 e^{x+e^{e^{4 x}}} x+16 e^{4 x+e^{e^{4 x}}+e^{4 x}} x-4 e^{5 x+e^{e^{4 x}}+e^{4 x}} x+5 \log (x) x-1}{x \left (e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (4 e^{x-3 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )^2+4 e^{2 x-2 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )-4 e^{4 x+e^{4 x}}+4 e^{3 x-e^{e^{4 x}}+e^{4 x}} \log (x)+4 e^{-4 e^{e^{4 x}}} \left (-e^{e^{4 x}} \log ^4(x)+12 e^{e^{e^{4 x}}+e^{4 x}} \log ^3(x)-48 e^{2 e^{e^{4 x}}+e^{4 x}} \log ^2(x)+64 e^{3 e^{e^{4 x}}+e^{4 x}} \log (x)+e^{4 e^{e^{4 x}}}\right )-\frac {e^{-4 e^{e^{4 x}}} \left (-4 e^{e^{4 x}} x \log ^5(x)+64 e^{e^{e^{4 x}}+e^{4 x}} x \log ^4(x)-384 e^{2 e^{e^{4 x}}+e^{4 x}} x \log ^3(x)+1024 e^{3 e^{e^{4 x}}+e^{4 x}} x \log ^2(x)-e^{4 e^{e^{4 x}}} x \log (x)-1024 e^{4 e^{e^{4 x}}+e^{4 x}} x \log (x)+e^{4 e^{e^{4 x}}}+4 e^{5 e^{e^{4 x}}} x\right )}{x \left (\log (x)-4 e^{e^{e^{4 x}}}+e^{x+e^{e^{4 x}}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-20 e^{e^{e^{4 x}}} x+4 e^{x+e^{e^{4 x}}} x+16 e^{4 x+e^{e^{4 x}}+e^{4 x}} x-4 e^{5 x+e^{e^{4 x}}+e^{4 x}} x+5 \log (x) x-1}{x \left (e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (4 e^{x-3 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )^2+4 e^{2 x-2 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )-4 e^{4 x+e^{4 x}}+4 e^{3 x-e^{e^{4 x}}+e^{4 x}} \log (x)+4 e^{-4 e^{e^{4 x}}} \left (-e^{e^{4 x}} \log ^4(x)+12 e^{e^{e^{4 x}}+e^{4 x}} \log ^3(x)-48 e^{2 e^{e^{4 x}}+e^{4 x}} \log ^2(x)+64 e^{3 e^{e^{4 x}}+e^{4 x}} \log (x)+e^{4 e^{e^{4 x}}}\right )-\frac {e^{-4 e^{e^{4 x}}} \left (-4 e^{e^{4 x}} x \log ^5(x)+64 e^{e^{e^{4 x}}+e^{4 x}} x \log ^4(x)-384 e^{2 e^{e^{4 x}}+e^{4 x}} x \log ^3(x)+1024 e^{3 e^{e^{4 x}}+e^{4 x}} x \log ^2(x)-e^{4 e^{e^{4 x}}} x \log (x)-1024 e^{4 e^{e^{4 x}}+e^{4 x}} x \log (x)+e^{4 e^{e^{4 x}}}+4 e^{5 e^{e^{4 x}}} x\right )}{x \left (\log (x)-4 e^{e^{e^{4 x}}}+e^{x+e^{e^{4 x}}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-20 e^{e^{e^{4 x}}} x+4 e^{x+e^{e^{4 x}}} x+16 e^{4 x+e^{e^{4 x}}+e^{4 x}} x-4 e^{5 x+e^{e^{4 x}}+e^{4 x}} x+5 \log (x) x-1}{x \left (e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (4 e^{x-3 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )^2+4 e^{2 x-2 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )-4 e^{4 x+e^{4 x}}+4 e^{3 x-e^{e^{4 x}}+e^{4 x}} \log (x)+4 e^{-4 e^{e^{4 x}}} \left (-e^{e^{4 x}} \log ^4(x)+12 e^{e^{e^{4 x}}+e^{4 x}} \log ^3(x)-48 e^{2 e^{e^{4 x}}+e^{4 x}} \log ^2(x)+64 e^{3 e^{e^{4 x}}+e^{4 x}} \log (x)+e^{4 e^{e^{4 x}}}\right )-\frac {e^{-4 e^{e^{4 x}}} \left (-4 e^{e^{4 x}} x \log ^5(x)+64 e^{e^{e^{4 x}}+e^{4 x}} x \log ^4(x)-384 e^{2 e^{e^{4 x}}+e^{4 x}} x \log ^3(x)+1024 e^{3 e^{e^{4 x}}+e^{4 x}} x \log ^2(x)-e^{4 e^{e^{4 x}}} x \log (x)-1024 e^{4 e^{e^{4 x}}+e^{4 x}} x \log (x)+e^{4 e^{e^{4 x}}}+4 e^{5 e^{e^{4 x}}} x\right )}{x \left (\log (x)-4 e^{e^{e^{4 x}}}+e^{x+e^{e^{4 x}}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-20 e^{e^{e^{4 x}}} x+4 e^{x+e^{e^{4 x}}} x+16 e^{4 x+e^{e^{4 x}}+e^{4 x}} x-4 e^{5 x+e^{e^{4 x}}+e^{4 x}} x+5 \log (x) x-1}{x \left (e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (4 e^{x-3 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )^2+4 e^{2 x-2 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )-4 e^{4 x+e^{4 x}}+4 e^{3 x-e^{e^{4 x}}+e^{4 x}} \log (x)+4 e^{-4 e^{e^{4 x}}} \left (-e^{e^{4 x}} \log ^4(x)+12 e^{e^{e^{4 x}}+e^{4 x}} \log ^3(x)-48 e^{2 e^{e^{4 x}}+e^{4 x}} \log ^2(x)+64 e^{3 e^{e^{4 x}}+e^{4 x}} \log (x)+e^{4 e^{e^{4 x}}}\right )-\frac {e^{-4 e^{e^{4 x}}} \left (-4 e^{e^{4 x}} x \log ^5(x)+64 e^{e^{e^{4 x}}+e^{4 x}} x \log ^4(x)-384 e^{2 e^{e^{4 x}}+e^{4 x}} x \log ^3(x)+1024 e^{3 e^{e^{4 x}}+e^{4 x}} x \log ^2(x)-e^{4 e^{e^{4 x}}} x \log (x)-1024 e^{4 e^{e^{4 x}}+e^{4 x}} x \log (x)+e^{4 e^{e^{4 x}}}+4 e^{5 e^{e^{4 x}}} x\right )}{x \left (\log (x)-4 e^{e^{e^{4 x}}}+e^{x+e^{e^{4 x}}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-20 e^{e^{e^{4 x}}} x+4 e^{x+e^{e^{4 x}}} x+16 e^{4 x+e^{e^{4 x}}+e^{4 x}} x-4 e^{5 x+e^{e^{4 x}}+e^{4 x}} x+5 \log (x) x-1}{x \left (e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (4 e^{x-3 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )^2+4 e^{2 x-2 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )-4 e^{4 x+e^{4 x}}+4 e^{3 x-e^{e^{4 x}}+e^{4 x}} \log (x)+4 e^{-4 e^{e^{4 x}}} \left (-e^{e^{4 x}} \log ^4(x)+12 e^{e^{e^{4 x}}+e^{4 x}} \log ^3(x)-48 e^{2 e^{e^{4 x}}+e^{4 x}} \log ^2(x)+64 e^{3 e^{e^{4 x}}+e^{4 x}} \log (x)+e^{4 e^{e^{4 x}}}\right )-\frac {e^{-4 e^{e^{4 x}}} \left (-4 e^{e^{4 x}} x \log ^5(x)+64 e^{e^{e^{4 x}}+e^{4 x}} x \log ^4(x)-384 e^{2 e^{e^{4 x}}+e^{4 x}} x \log ^3(x)+1024 e^{3 e^{e^{4 x}}+e^{4 x}} x \log ^2(x)-e^{4 e^{e^{4 x}}} x \log (x)-1024 e^{4 e^{e^{4 x}}+e^{4 x}} x \log (x)+e^{4 e^{e^{4 x}}}+4 e^{5 e^{e^{4 x}}} x\right )}{x \left (\log (x)-4 e^{e^{e^{4 x}}}+e^{x+e^{e^{4 x}}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-20 e^{e^{e^{4 x}}} x+4 e^{x+e^{e^{4 x}}} x+16 e^{4 x+e^{e^{4 x}}+e^{4 x}} x-4 e^{5 x+e^{e^{4 x}}+e^{4 x}} x+5 \log (x) x-1}{x \left (e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (4 e^{x-3 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )^2+4 e^{2 x-2 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )-4 e^{4 x+e^{4 x}}+4 e^{3 x-e^{e^{4 x}}+e^{4 x}} \log (x)+4 e^{-4 e^{e^{4 x}}} \left (-e^{e^{4 x}} \log ^4(x)+12 e^{e^{e^{4 x}}+e^{4 x}} \log ^3(x)-48 e^{2 e^{e^{4 x}}+e^{4 x}} \log ^2(x)+64 e^{3 e^{e^{4 x}}+e^{4 x}} \log (x)+e^{4 e^{e^{4 x}}}\right )-\frac {e^{-4 e^{e^{4 x}}} \left (-4 e^{e^{4 x}} x \log ^5(x)+64 e^{e^{e^{4 x}}+e^{4 x}} x \log ^4(x)-384 e^{2 e^{e^{4 x}}+e^{4 x}} x \log ^3(x)+1024 e^{3 e^{e^{4 x}}+e^{4 x}} x \log ^2(x)-e^{4 e^{e^{4 x}}} x \log (x)-1024 e^{4 e^{e^{4 x}}+e^{4 x}} x \log (x)+e^{4 e^{e^{4 x}}}+4 e^{5 e^{e^{4 x}}} x\right )}{x \left (\log (x)-4 e^{e^{e^{4 x}}}+e^{x+e^{e^{4 x}}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-20 e^{e^{e^{4 x}}} x+4 e^{x+e^{e^{4 x}}} x+16 e^{4 x+e^{e^{4 x}}+e^{4 x}} x-4 e^{5 x+e^{e^{4 x}}+e^{4 x}} x+5 \log (x) x-1}{x \left (e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (4 e^{x-3 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )^2+4 e^{2 x-2 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )-4 e^{4 x+e^{4 x}}+4 e^{3 x-e^{e^{4 x}}+e^{4 x}} \log (x)+4 e^{-4 e^{e^{4 x}}} \left (-e^{e^{4 x}} \log ^4(x)+12 e^{e^{e^{4 x}}+e^{4 x}} \log ^3(x)-48 e^{2 e^{e^{4 x}}+e^{4 x}} \log ^2(x)+64 e^{3 e^{e^{4 x}}+e^{4 x}} \log (x)+e^{4 e^{e^{4 x}}}\right )-\frac {e^{-4 e^{e^{4 x}}} \left (-4 e^{e^{4 x}} x \log ^5(x)+64 e^{e^{e^{4 x}}+e^{4 x}} x \log ^4(x)-384 e^{2 e^{e^{4 x}}+e^{4 x}} x \log ^3(x)+1024 e^{3 e^{e^{4 x}}+e^{4 x}} x \log ^2(x)-e^{4 e^{e^{4 x}}} x \log (x)-1024 e^{4 e^{e^{4 x}}+e^{4 x}} x \log (x)+e^{4 e^{e^{4 x}}}+4 e^{5 e^{e^{4 x}}} x\right )}{x \left (\log (x)-4 e^{e^{e^{4 x}}}+e^{x+e^{e^{4 x}}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-20 e^{e^{e^{4 x}}} x+4 e^{x+e^{e^{4 x}}} x+16 e^{4 x+e^{e^{4 x}}+e^{4 x}} x-4 e^{5 x+e^{e^{4 x}}+e^{4 x}} x+5 \log (x) x-1}{x \left (e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (4 e^{x-3 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )^2+4 e^{2 x-2 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )-4 e^{4 x+e^{4 x}}+4 e^{3 x-e^{e^{4 x}}+e^{4 x}} \log (x)+4 e^{-4 e^{e^{4 x}}} \left (-e^{e^{4 x}} \log ^4(x)+12 e^{e^{e^{4 x}}+e^{4 x}} \log ^3(x)-48 e^{2 e^{e^{4 x}}+e^{4 x}} \log ^2(x)+64 e^{3 e^{e^{4 x}}+e^{4 x}} \log (x)+e^{4 e^{e^{4 x}}}\right )-\frac {e^{-4 e^{e^{4 x}}} \left (-4 e^{e^{4 x}} x \log ^5(x)+64 e^{e^{e^{4 x}}+e^{4 x}} x \log ^4(x)-384 e^{2 e^{e^{4 x}}+e^{4 x}} x \log ^3(x)+1024 e^{3 e^{e^{4 x}}+e^{4 x}} x \log ^2(x)-e^{4 e^{e^{4 x}}} x \log (x)-1024 e^{4 e^{e^{4 x}}+e^{4 x}} x \log (x)+e^{4 e^{e^{4 x}}}+4 e^{5 e^{e^{4 x}}} x\right )}{x \left (\log (x)-4 e^{e^{e^{4 x}}}+e^{x+e^{e^{4 x}}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-20 e^{e^{e^{4 x}}} x+4 e^{x+e^{e^{4 x}}} x+16 e^{4 x+e^{e^{4 x}}+e^{4 x}} x-4 e^{5 x+e^{e^{4 x}}+e^{4 x}} x+5 \log (x) x-1}{x \left (e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (4 e^{x-3 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )^2+4 e^{2 x-2 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )-4 e^{4 x+e^{4 x}}+4 e^{3 x-e^{e^{4 x}}+e^{4 x}} \log (x)+4 e^{-4 e^{e^{4 x}}} \left (-e^{e^{4 x}} \log ^4(x)+12 e^{e^{e^{4 x}}+e^{4 x}} \log ^3(x)-48 e^{2 e^{e^{4 x}}+e^{4 x}} \log ^2(x)+64 e^{3 e^{e^{4 x}}+e^{4 x}} \log (x)+e^{4 e^{e^{4 x}}}\right )-\frac {e^{-4 e^{e^{4 x}}} \left (-4 e^{e^{4 x}} x \log ^5(x)+64 e^{e^{e^{4 x}}+e^{4 x}} x \log ^4(x)-384 e^{2 e^{e^{4 x}}+e^{4 x}} x \log ^3(x)+1024 e^{3 e^{e^{4 x}}+e^{4 x}} x \log ^2(x)-e^{4 e^{e^{4 x}}} x \log (x)-1024 e^{4 e^{e^{4 x}}+e^{4 x}} x \log (x)+e^{4 e^{e^{4 x}}}+4 e^{5 e^{e^{4 x}}} x\right )}{x \left (\log (x)-4 e^{e^{e^{4 x}}}+e^{x+e^{e^{4 x}}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-20 e^{e^{e^{4 x}}} x+4 e^{x+e^{e^{4 x}}} x+16 e^{4 x+e^{e^{4 x}}+e^{4 x}} x-4 e^{5 x+e^{e^{4 x}}+e^{4 x}} x+5 \log (x) x-1}{x \left (e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (4 e^{x-3 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )^2+4 e^{2 x-2 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )-4 e^{4 x+e^{4 x}}+4 e^{3 x-e^{e^{4 x}}+e^{4 x}} \log (x)+4 e^{-4 e^{e^{4 x}}} \left (-e^{e^{4 x}} \log ^4(x)+12 e^{e^{e^{4 x}}+e^{4 x}} \log ^3(x)-48 e^{2 e^{e^{4 x}}+e^{4 x}} \log ^2(x)+64 e^{3 e^{e^{4 x}}+e^{4 x}} \log (x)+e^{4 e^{e^{4 x}}}\right )-\frac {e^{-4 e^{e^{4 x}}} \left (-4 e^{e^{4 x}} x \log ^5(x)+64 e^{e^{e^{4 x}}+e^{4 x}} x \log ^4(x)-384 e^{2 e^{e^{4 x}}+e^{4 x}} x \log ^3(x)+1024 e^{3 e^{e^{4 x}}+e^{4 x}} x \log ^2(x)-e^{4 e^{e^{4 x}}} x \log (x)-1024 e^{4 e^{e^{4 x}}+e^{4 x}} x \log (x)+e^{4 e^{e^{4 x}}}+4 e^{5 e^{e^{4 x}}} x\right )}{x \left (\log (x)-4 e^{e^{e^{4 x}}}+e^{x+e^{e^{4 x}}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-20 e^{e^{e^{4 x}}} x+4 e^{x+e^{e^{4 x}}} x+16 e^{4 x+e^{e^{4 x}}+e^{4 x}} x-4 e^{5 x+e^{e^{4 x}}+e^{4 x}} x+5 \log (x) x-1}{x \left (e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (4 e^{x-3 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )^2+4 e^{2 x-2 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )-4 e^{4 x+e^{4 x}}+4 e^{3 x-e^{e^{4 x}}+e^{4 x}} \log (x)+4 e^{-4 e^{e^{4 x}}} \left (-e^{e^{4 x}} \log ^4(x)+12 e^{e^{e^{4 x}}+e^{4 x}} \log ^3(x)-48 e^{2 e^{e^{4 x}}+e^{4 x}} \log ^2(x)+64 e^{3 e^{e^{4 x}}+e^{4 x}} \log (x)+e^{4 e^{e^{4 x}}}\right )-\frac {e^{-4 e^{e^{4 x}}} \left (-4 e^{e^{4 x}} x \log ^5(x)+64 e^{e^{e^{4 x}}+e^{4 x}} x \log ^4(x)-384 e^{2 e^{e^{4 x}}+e^{4 x}} x \log ^3(x)+1024 e^{3 e^{e^{4 x}}+e^{4 x}} x \log ^2(x)-e^{4 e^{e^{4 x}}} x \log (x)-1024 e^{4 e^{e^{4 x}}+e^{4 x}} x \log (x)+e^{4 e^{e^{4 x}}}+4 e^{5 e^{e^{4 x}}} x\right )}{x \left (\log (x)-4 e^{e^{e^{4 x}}}+e^{x+e^{e^{4 x}}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-20 e^{e^{e^{4 x}}} x+4 e^{x+e^{e^{4 x}}} x+16 e^{4 x+e^{e^{4 x}}+e^{4 x}} x-4 e^{5 x+e^{e^{4 x}}+e^{4 x}} x+5 \log (x) x-1}{x \left (e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (4 e^{x-3 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )^2+4 e^{2 x-2 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )-4 e^{4 x+e^{4 x}}+4 e^{3 x-e^{e^{4 x}}+e^{4 x}} \log (x)+4 e^{-4 e^{e^{4 x}}} \left (-e^{e^{4 x}} \log ^4(x)+12 e^{e^{e^{4 x}}+e^{4 x}} \log ^3(x)-48 e^{2 e^{e^{4 x}}+e^{4 x}} \log ^2(x)+64 e^{3 e^{e^{4 x}}+e^{4 x}} \log (x)+e^{4 e^{e^{4 x}}}\right )-\frac {e^{-4 e^{e^{4 x}}} \left (-4 e^{e^{4 x}} x \log ^5(x)+64 e^{e^{e^{4 x}}+e^{4 x}} x \log ^4(x)-384 e^{2 e^{e^{4 x}}+e^{4 x}} x \log ^3(x)+1024 e^{3 e^{e^{4 x}}+e^{4 x}} x \log ^2(x)-e^{4 e^{e^{4 x}}} x \log (x)-1024 e^{4 e^{e^{4 x}}+e^{4 x}} x \log (x)+e^{4 e^{e^{4 x}}}+4 e^{5 e^{e^{4 x}}} x\right )}{x \left (\log (x)-4 e^{e^{e^{4 x}}}+e^{x+e^{e^{4 x}}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-20 e^{e^{e^{4 x}}} x+4 e^{x+e^{e^{4 x}}} x+16 e^{4 x+e^{e^{4 x}}+e^{4 x}} x-4 e^{5 x+e^{e^{4 x}}+e^{4 x}} x+5 \log (x) x-1}{x \left (e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{5} \int \left (4 e^{x-3 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )^2+4 e^{2 x-2 e^{e^{4 x}}+e^{4 x}} \log (x) \left (4 e^{e^{e^{4 x}}}-\log (x)\right )-4 e^{4 x+e^{4 x}}+4 e^{3 x-e^{e^{4 x}}+e^{4 x}} \log (x)+4 e^{-4 e^{e^{4 x}}} \left (-e^{e^{4 x}} \log ^4(x)+12 e^{e^{e^{4 x}}+e^{4 x}} \log ^3(x)-48 e^{2 e^{e^{4 x}}+e^{4 x}} \log ^2(x)+64 e^{3 e^{e^{4 x}}+e^{4 x}} \log (x)+e^{4 e^{e^{4 x}}}\right )-\frac {e^{-4 e^{e^{4 x}}} \left (-4 e^{e^{4 x}} x \log ^5(x)+64 e^{e^{e^{4 x}}+e^{4 x}} x \log ^4(x)-384 e^{2 e^{e^{4 x}}+e^{4 x}} x \log ^3(x)+1024 e^{3 e^{e^{4 x}}+e^{4 x}} x \log ^2(x)-e^{4 e^{e^{4 x}}} x \log (x)-1024 e^{4 e^{e^{4 x}}+e^{4 x}} x \log (x)+e^{4 e^{e^{4 x}}}+4 e^{5 e^{e^{4 x}}} x\right )}{x \left (\log (x)-4 e^{e^{e^{4 x}}}+e^{x+e^{e^{4 x}}}\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{5} \int \frac {-20 e^{e^{e^{4 x}}} x+4 e^{x+e^{e^{4 x}}} x+16 e^{4 x+e^{e^{4 x}}+e^{4 x}} x-4 e^{5 x+e^{e^{4 x}}+e^{4 x}} x+5 \log (x) x-1}{x \left (e^{e^{e^{4 x}}} \left (-4+e^x\right )+\log (x)\right )}dx\) |
Int[(-1 + E^E^E^(4*x)*(-20*x + 4*E^x*x + E^(E^(4*x) + 4*x)*(16*x - 4*E^x*x )) + 5*x*Log[x])/(E^E^E^(4*x)*(-20*x + 5*E^x*x) + 5*x*Log[x]),x]
3.17.30.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 5.04 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93
| method | result | size |
| parallelrisch | \(x -\frac {\ln \left ({\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{{\mathrm e}^{4 x}}}+\ln \left (x \right )-4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{4 x}}}\right )}{5}\) | \(26\) |
| risch | \(x -\frac {\ln \left ({\mathrm e}^{x}-4\right )}{5}-\frac {\ln \left ({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{4 x}}}+\frac {\ln \left (x \right )}{{\mathrm e}^{x}-4}\right )}{5}\) | \(29\) |
int((((-4*exp(x)*x+16*x)*exp(4*x)*exp(exp(4*x))+4*exp(x)*x-20*x)*exp(exp(e xp(4*x)))+5*x*ln(x)-1)/((5*exp(x)*x-20*x)*exp(exp(exp(4*x)))+5*x*ln(x)),x, method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {-1+e^{e^{e^{4 x}}} \left (-20 x+4 e^x x+e^{e^{4 x}+4 x} \left (16 x-4 e^x x\right )\right )+5 x \log (x)}{e^{e^{e^{4 x}}} \left (-20 x+5 e^x x\right )+5 x \log (x)} \, dx=x - \frac {1}{5} \, \log \left (\frac {{\left (e^{x} - 4\right )} e^{\left (e^{\left (e^{\left (4 \, x\right )}\right )}\right )} + \log \left (x\right )}{e^{x} - 4}\right ) - \frac {1}{5} \, \log \left (e^{x} - 4\right ) \]
integrate((((-4*exp(x)*x+16*x)*exp(4*x)*exp(exp(4*x))+4*exp(x)*x-20*x)*exp (exp(exp(4*x)))+5*x*log(x)-1)/((5*exp(x)*x-20*x)*exp(exp(exp(4*x)))+5*x*lo g(x)),x, algorithm=\
Time = 0.53 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.04 \[ \int \frac {-1+e^{e^{e^{4 x}}} \left (-20 x+4 e^x x+e^{e^{4 x}+4 x} \left (16 x-4 e^x x\right )\right )+5 x \log (x)}{e^{e^{e^{4 x}}} \left (-20 x+5 e^x x\right )+5 x \log (x)} \, dx=x - \frac {\log {\left (e^{x} - 4 \right )}}{5} - \frac {\log {\left (e^{e^{e^{4 x}}} + \frac {\log {\left (x \right )}}{e^{x} - 4} \right )}}{5} \]
integrate((((-4*exp(x)*x+16*x)*exp(4*x)*exp(exp(4*x))+4*exp(x)*x-20*x)*exp (exp(exp(4*x)))+5*x*ln(x)-1)/((5*exp(x)*x-20*x)*exp(exp(exp(4*x)))+5*x*ln( x)),x)
Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {-1+e^{e^{e^{4 x}}} \left (-20 x+4 e^x x+e^{e^{4 x}+4 x} \left (16 x-4 e^x x\right )\right )+5 x \log (x)}{e^{e^{e^{4 x}}} \left (-20 x+5 e^x x\right )+5 x \log (x)} \, dx=x - \frac {1}{5} \, \log \left (\frac {{\left (e^{x} - 4\right )} e^{\left (e^{\left (e^{\left (4 \, x\right )}\right )}\right )} + \log \left (x\right )}{e^{x} - 4}\right ) - \frac {1}{5} \, \log \left (e^{x} - 4\right ) \]
integrate((((-4*exp(x)*x+16*x)*exp(4*x)*exp(exp(4*x))+4*exp(x)*x-20*x)*exp (exp(exp(4*x)))+5*x*log(x)-1)/((5*exp(x)*x-20*x)*exp(exp(exp(4*x)))+5*x*lo g(x)),x, algorithm=\
\[ \int \frac {-1+e^{e^{e^{4 x}}} \left (-20 x+4 e^x x+e^{e^{4 x}+4 x} \left (16 x-4 e^x x\right )\right )+5 x \log (x)}{e^{e^{e^{4 x}}} \left (-20 x+5 e^x x\right )+5 x \log (x)} \, dx=\int { -\frac {4 \, {\left ({\left (x e^{x} - 4 \, x\right )} e^{\left (4 \, x + e^{\left (4 \, x\right )}\right )} - x e^{x} + 5 \, x\right )} e^{\left (e^{\left (e^{\left (4 \, x\right )}\right )}\right )} - 5 \, x \log \left (x\right ) + 1}{5 \, {\left ({\left (x e^{x} - 4 \, x\right )} e^{\left (e^{\left (e^{\left (4 \, x\right )}\right )}\right )} + x \log \left (x\right )\right )}} \,d x } \]
integrate((((-4*exp(x)*x+16*x)*exp(4*x)*exp(exp(4*x))+4*exp(x)*x-20*x)*exp (exp(exp(4*x)))+5*x*log(x)-1)/((5*exp(x)*x-20*x)*exp(exp(exp(4*x)))+5*x*lo g(x)),x, algorithm=\
integrate(-1/5*(4*((x*e^x - 4*x)*e^(4*x + e^(4*x)) - x*e^x + 5*x)*e^(e^(e^ (4*x))) - 5*x*log(x) + 1)/((x*e^x - 4*x)*e^(e^(e^(4*x))) + x*log(x)), x)
Time = 8.90 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.39 \[ \int \frac {-1+e^{e^{e^{4 x}}} \left (-20 x+4 e^x x+e^{e^{4 x}+4 x} \left (16 x-4 e^x x\right )\right )+5 x \log (x)}{e^{e^{e^{4 x}}} \left (-20 x+5 e^x x\right )+5 x \log (x)} \, dx=x-\frac {\ln \left (\frac {\ln \left (x\right )-4\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}}+{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{4\,x}}}\,{\mathrm {e}}^x}{{\mathrm {e}}^x-4}\right )}{5}-\frac {\ln \left ({\mathrm {e}}^x-4\right )}{5} \]
int(-(exp(exp(exp(4*x)))*(4*x*exp(x) - 20*x + exp(4*x)*exp(exp(4*x))*(16*x - 4*x*exp(x))) + 5*x*log(x) - 1)/(exp(exp(exp(4*x)))*(20*x - 5*x*exp(x)) - 5*x*log(x)),x)