Integrand size = 147, antiderivative size = 34 \[ \int \frac {e^{\frac {-5 x-5 x^2-5 x^3+\left (-6+e^x (-2-x)-3 x\right ) \log (\log (x))}{10 x^2+5 x^3}} \left (-12-12 x-3 x^2+e^x \left (-4-4 x-x^2\right )+\left (10 x+10 x^2-5 x^3\right ) \log (x)+\left (24+24 x+6 x^2+e^x \left (8+4 x-2 x^2-x^3\right )\right ) \log (x) \log (\log (x))\right )}{\left (20 x^3+20 x^4+5 x^5\right ) \log (x)} \, dx=e^{\frac {-x+\frac {-1+x}{2+x}-\frac {\left (3+e^x\right ) \log (\log (x))}{5 x}}{x}} \]
Time = 0.29 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.03 \[ \int \frac {e^{\frac {-5 x-5 x^2-5 x^3+\left (-6+e^x (-2-x)-3 x\right ) \log (\log (x))}{10 x^2+5 x^3}} \left (-12-12 x-3 x^2+e^x \left (-4-4 x-x^2\right )+\left (10 x+10 x^2-5 x^3\right ) \log (x)+\left (24+24 x+6 x^2+e^x \left (8+4 x-2 x^2-x^3\right )\right ) \log (x) \log (\log (x))\right )}{\left (20 x^3+20 x^4+5 x^5\right ) \log (x)} \, dx=e^{-\frac {1+x+x^2}{2 x+x^2}} \log ^{-\frac {3+e^x}{5 x^2}}(x) \]
Integrate[(E^((-5*x - 5*x^2 - 5*x^3 + (-6 + E^x*(-2 - x) - 3*x)*Log[Log[x] ])/(10*x^2 + 5*x^3))*(-12 - 12*x - 3*x^2 + E^x*(-4 - 4*x - x^2) + (10*x + 10*x^2 - 5*x^3)*Log[x] + (24 + 24*x + 6*x^2 + E^x*(8 + 4*x - 2*x^2 - x^3)) *Log[x]*Log[Log[x]]))/((20*x^3 + 20*x^4 + 5*x^5)*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.
| \(\displaystyle \int \frac {\left (-3 x^2+e^x \left (-x^2-4 x-4\right )+\left (-5 x^3+10 x^2+10 x\right ) \log (x)+\left (6 x^2+e^x \left (-x^3-2 x^2+4 x+8\right )+24 x+24\right ) \log (x) \log (\log (x))-12 x-12\right ) \exp \left (\frac {-5 x^3-5 x^2-5 x+\left (e^x (-x-2)-3 x-6\right ) \log (\log (x))}{5 x^3+10 x^2}\right )}{\left (5 x^5+20 x^4+20 x^3\right ) \log (x)} \, dx\) |
| \(\Big \downarrow \) 2026 |
| \(\displaystyle \int \frac {\left (-3 x^2+e^x \left (-x^2-4 x-4\right )+\left (-5 x^3+10 x^2+10 x\right ) \log (x)+\left (6 x^2+e^x \left (-x^3-2 x^2+4 x+8\right )+24 x+24\right ) \log (x) \log (\log (x))-12 x-12\right ) \exp \left (\frac {-5 x^3-5 x^2-5 x+\left (e^x (-x-2)-3 x-6\right ) \log (\log (x))}{5 x^3+10 x^2}\right )}{x^3 \left (5 x^2+20 x+20\right ) \log (x)}dx\) |
| \(\Big \downarrow \) 2007 |
| \(\displaystyle \int \frac {\left (-3 x^2+e^x \left (-x^2-4 x-4\right )+\left (-5 x^3+10 x^2+10 x\right ) \log (x)+\left (6 x^2+e^x \left (-x^3-2 x^2+4 x+8\right )+24 x+24\right ) \log (x) \log (\log (x))-12 x-12\right ) \exp \left (\frac {-5 x^3-5 x^2-5 x+\left (e^x (-x-2)-3 x-6\right ) \log (\log (x))}{5 x^3+10 x^2}\right )}{x^3 \left (\sqrt {5} x+2 \sqrt {5}\right )^2 \log (x)}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-3 x^2+e^x \left (-x^2-4 x-4\right )+\left (-5 x^3+10 x^2+10 x\right ) \log (x)+\left (6 x^2+e^x \left (-x^3-2 x^2+4 x+8\right )+24 x+24\right ) \log (x) \log (\log (x))-12 x-12\right ) \exp \left (\frac {-5 x^3-5 x^2-5 x+\left (e^x (-x-2)-3 x-6\right ) \log (\log (x))}{x^2 (5 x+10)}\right )}{x^3 \left (\sqrt {5} x+2 \sqrt {5}\right )^2 \log (x)}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\left (-x^2+2 x+2\right ) \exp \left (\frac {-5 x^3-5 x^2-5 x+\left (e^x (-x-2)-3 x-6\right ) \log (\log (x))}{x^2 (5 x+10)}\right )}{x^2 (x+2)^2}+\frac {6 \log (\log (x)) \exp \left (\frac {-5 x^3-5 x^2-5 x+\left (e^x (-x-2)-3 x-6\right ) \log (\log (x))}{x^2 (5 x+10)}\right )}{5 x (x+2)^2}+\frac {24 \log (\log (x)) \exp \left (\frac {-5 x^3-5 x^2-5 x+\left (e^x (-x-2)-3 x-6\right ) \log (\log (x))}{x^2 (5 x+10)}\right )}{5 x^2 (x+2)^2}+\frac {24 \log (\log (x)) \exp \left (\frac {-5 x^3-5 x^2-5 x+\left (e^x (-x-2)-3 x-6\right ) \log (\log (x))}{x^2 (5 x+10)}\right )}{5 x^3 (x+2)^2}-\frac {(x \log (x) \log (\log (x))-2 \log (x) \log (\log (x))+1) \exp \left (\frac {-5 x^3-5 x^2-5 x+\left (e^x (-x-2)-3 x-6\right ) \log (\log (x))}{(5 x+10) x^2}+x\right )}{5 x^3 \log (x)}-\frac {3 \exp \left (\frac {-5 x^3-5 x^2-5 x+\left (e^x (-x-2)-3 x-6\right ) \log (\log (x))}{x^2 (5 x+10)}\right )}{5 x (x+2)^2 \log (x)}-\frac {12 \exp \left (\frac {-5 x^3-5 x^2-5 x+\left (e^x (-x-2)-3 x-6\right ) \log (\log (x))}{x^2 (5 x+10)}\right )}{5 x^2 (x+2)^2 \log (x)}-\frac {12 \exp \left (\frac {-5 x^3-5 x^2-5 x+\left (e^x (-x-2)-3 x-6\right ) \log (\log (x))}{x^2 (5 x+10)}\right )}{5 x^3 (x+2)^2 \log (x)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {e^{-\frac {x^2+x+1}{x (x+2)}} \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x) \left (\log (x) \left (-5 x \left (x^2-2 x-2\right )-\left (\left (e^x (x-2)-6\right ) (x+2)^2 \log (\log (x))\right )\right )-\left (e^x+3\right ) (x+2)^2\right )}{5 x^3 (x+2)^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int -\frac {e^{-\frac {x^2+x+1}{x (x+2)}} \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x) \left (\left (3+e^x\right ) (x+2)^2-\log (x) \left (\left (e^x (2-x)+6\right ) \log (\log (x)) (x+2)^2+5 x \left (-x^2+2 x+2\right )\right )\right )}{x^3 (x+2)^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{5} \int \frac {e^{-\frac {x^2+x+1}{x (x+2)}} \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x) \left (\left (3+e^x\right ) (x+2)^2-\log (x) \left (\left (e^x (2-x)+6\right ) \log (\log (x)) (x+2)^2+5 x \left (-x^2+2 x+2\right )\right )\right )}{x^3 (x+2)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{5} \int \left (\frac {e^{x-\frac {x^2+x+1}{x (x+2)}} (x \log (x) \log (\log (x))-2 \log (x) \log (\log (x))+1) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3}+\frac {e^{-\frac {x^2+x+1}{x (x+2)}} \left (5 \log (x) x^3-10 \log (x) x^2-6 \log (x) \log (\log (x)) x^2+3 x^2-10 \log (x) x-24 \log (x) \log (\log (x)) x+12 x-24 \log (x) \log (\log (x))+12\right ) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3 (x+2)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{5} \int \frac {e^{-\frac {x^2+x+1}{x (x+2)}} \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x) \left (\left (3+e^x\right ) (x+2)^2+\log (x) \left (\left (e^x (x-2)-6\right ) \log (\log (x)) (x+2)^2+5 x \left (x^2-2 x-2\right )\right )\right )}{x^3 (x+2)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{5} \int \left (\frac {e^{x-\frac {x^2+x+1}{x (x+2)}} (x \log (x) \log (\log (x))-2 \log (x) \log (\log (x))+1) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3}+\frac {e^{-\frac {x^2+x+1}{x (x+2)}} \left (5 \log (x) x^3-10 \log (x) x^2-6 \log (x) \log (\log (x)) x^2+3 x^2-10 \log (x) x-24 \log (x) \log (\log (x)) x+12 x-24 \log (x) \log (\log (x))+12\right ) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3 (x+2)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{5} \int \frac {e^{-\frac {x^2+x+1}{x (x+2)}} \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x) \left (\left (3+e^x\right ) (x+2)^2+\log (x) \left (\left (e^x (x-2)-6\right ) \log (\log (x)) (x+2)^2+5 x \left (x^2-2 x-2\right )\right )\right )}{x^3 (x+2)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{5} \int \left (\frac {e^{x-\frac {x^2+x+1}{x (x+2)}} (x \log (x) \log (\log (x))-2 \log (x) \log (\log (x))+1) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3}+\frac {e^{-\frac {x^2+x+1}{x (x+2)}} \left (5 \log (x) x^3-10 \log (x) x^2-6 \log (x) \log (\log (x)) x^2+3 x^2-10 \log (x) x-24 \log (x) \log (\log (x)) x+12 x-24 \log (x) \log (\log (x))+12\right ) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3 (x+2)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{5} \int \frac {e^{-\frac {x^2+x+1}{x (x+2)}} \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x) \left (\left (3+e^x\right ) (x+2)^2+\log (x) \left (\left (e^x (x-2)-6\right ) \log (\log (x)) (x+2)^2+5 x \left (x^2-2 x-2\right )\right )\right )}{x^3 (x+2)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{5} \int \left (\frac {e^{x-\frac {x^2+x+1}{x (x+2)}} (x \log (x) \log (\log (x))-2 \log (x) \log (\log (x))+1) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3}+\frac {e^{-\frac {x^2+x+1}{x (x+2)}} \left (5 \log (x) x^3-10 \log (x) x^2-6 \log (x) \log (\log (x)) x^2+3 x^2-10 \log (x) x-24 \log (x) \log (\log (x)) x+12 x-24 \log (x) \log (\log (x))+12\right ) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3 (x+2)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{5} \int \frac {e^{-\frac {x^2+x+1}{x (x+2)}} \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x) \left (\left (3+e^x\right ) (x+2)^2+\log (x) \left (\left (e^x (x-2)-6\right ) \log (\log (x)) (x+2)^2+5 x \left (x^2-2 x-2\right )\right )\right )}{x^3 (x+2)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{5} \int \left (\frac {e^{x-\frac {x^2+x+1}{x (x+2)}} (x \log (x) \log (\log (x))-2 \log (x) \log (\log (x))+1) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3}+\frac {e^{-\frac {x^2+x+1}{x (x+2)}} \left (5 \log (x) x^3-10 \log (x) x^2-6 \log (x) \log (\log (x)) x^2+3 x^2-10 \log (x) x-24 \log (x) \log (\log (x)) x+12 x-24 \log (x) \log (\log (x))+12\right ) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3 (x+2)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{5} \int \frac {e^{-\frac {x^2+x+1}{x (x+2)}} \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x) \left (\left (3+e^x\right ) (x+2)^2+\log (x) \left (\left (e^x (x-2)-6\right ) \log (\log (x)) (x+2)^2+5 x \left (x^2-2 x-2\right )\right )\right )}{x^3 (x+2)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{5} \int \left (\frac {e^{x-\frac {x^2+x+1}{x (x+2)}} (x \log (x) \log (\log (x))-2 \log (x) \log (\log (x))+1) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3}+\frac {e^{-\frac {x^2+x+1}{x (x+2)}} \left (5 \log (x) x^3-10 \log (x) x^2-6 \log (x) \log (\log (x)) x^2+3 x^2-10 \log (x) x-24 \log (x) \log (\log (x)) x+12 x-24 \log (x) \log (\log (x))+12\right ) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3 (x+2)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{5} \int \frac {e^{-\frac {x^2+x+1}{x (x+2)}} \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x) \left (\left (3+e^x\right ) (x+2)^2+\log (x) \left (\left (e^x (x-2)-6\right ) \log (\log (x)) (x+2)^2+5 x \left (x^2-2 x-2\right )\right )\right )}{x^3 (x+2)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{5} \int \left (\frac {e^{x-\frac {x^2+x+1}{x (x+2)}} (x \log (x) \log (\log (x))-2 \log (x) \log (\log (x))+1) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3}+\frac {e^{-\frac {x^2+x+1}{x (x+2)}} \left (5 \log (x) x^3-10 \log (x) x^2-6 \log (x) \log (\log (x)) x^2+3 x^2-10 \log (x) x-24 \log (x) \log (\log (x)) x+12 x-24 \log (x) \log (\log (x))+12\right ) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3 (x+2)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{5} \int \frac {e^{-\frac {x^2+x+1}{x (x+2)}} \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x) \left (\left (3+e^x\right ) (x+2)^2+\log (x) \left (\left (e^x (x-2)-6\right ) \log (\log (x)) (x+2)^2+5 x \left (x^2-2 x-2\right )\right )\right )}{x^3 (x+2)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{5} \int \left (\frac {e^{x-\frac {x^2+x+1}{x (x+2)}} (x \log (x) \log (\log (x))-2 \log (x) \log (\log (x))+1) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3}+\frac {e^{-\frac {x^2+x+1}{x (x+2)}} \left (5 \log (x) x^3-10 \log (x) x^2-6 \log (x) \log (\log (x)) x^2+3 x^2-10 \log (x) x-24 \log (x) \log (\log (x)) x+12 x-24 \log (x) \log (\log (x))+12\right ) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3 (x+2)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{5} \int \frac {e^{-\frac {x^2+x+1}{x (x+2)}} \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x) \left (\left (3+e^x\right ) (x+2)^2+\log (x) \left (\left (e^x (x-2)-6\right ) \log (\log (x)) (x+2)^2+5 x \left (x^2-2 x-2\right )\right )\right )}{x^3 (x+2)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{5} \int \left (\frac {e^{x-\frac {x^2+x+1}{x (x+2)}} (x \log (x) \log (\log (x))-2 \log (x) \log (\log (x))+1) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3}+\frac {e^{-\frac {x^2+x+1}{x (x+2)}} \left (5 \log (x) x^3-10 \log (x) x^2-6 \log (x) \log (\log (x)) x^2+3 x^2-10 \log (x) x-24 \log (x) \log (\log (x)) x+12 x-24 \log (x) \log (\log (x))+12\right ) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3 (x+2)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{5} \int \frac {e^{-\frac {x^2+x+1}{x (x+2)}} \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x) \left (\left (3+e^x\right ) (x+2)^2+\log (x) \left (\left (e^x (x-2)-6\right ) \log (\log (x)) (x+2)^2+5 x \left (x^2-2 x-2\right )\right )\right )}{x^3 (x+2)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{5} \int \left (\frac {e^{x-\frac {x^2+x+1}{x (x+2)}} (x \log (x) \log (\log (x))-2 \log (x) \log (\log (x))+1) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3}+\frac {e^{-\frac {x^2+x+1}{x (x+2)}} \left (5 \log (x) x^3-10 \log (x) x^2-6 \log (x) \log (\log (x)) x^2+3 x^2-10 \log (x) x-24 \log (x) \log (\log (x)) x+12 x-24 \log (x) \log (\log (x))+12\right ) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3 (x+2)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{5} \int \frac {e^{-\frac {x^2+x+1}{x (x+2)}} \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x) \left (\left (3+e^x\right ) (x+2)^2+\log (x) \left (\left (e^x (x-2)-6\right ) \log (\log (x)) (x+2)^2+5 x \left (x^2-2 x-2\right )\right )\right )}{x^3 (x+2)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{5} \int \left (\frac {e^{x-\frac {x^2+x+1}{x (x+2)}} (x \log (x) \log (\log (x))-2 \log (x) \log (\log (x))+1) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3}+\frac {e^{-\frac {x^2+x+1}{x (x+2)}} \left (5 \log (x) x^3-10 \log (x) x^2-6 \log (x) \log (\log (x)) x^2+3 x^2-10 \log (x) x-24 \log (x) \log (\log (x)) x+12 x-24 \log (x) \log (\log (x))+12\right ) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3 (x+2)^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{5} \int \frac {e^{-\frac {x^2+x+1}{x (x+2)}} \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x) \left (\left (3+e^x\right ) (x+2)^2+\log (x) \left (\left (e^x (x-2)-6\right ) \log (\log (x)) (x+2)^2+5 x \left (x^2-2 x-2\right )\right )\right )}{x^3 (x+2)^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{5} \int \left (\frac {e^{x-\frac {x^2+x+1}{x (x+2)}} (x \log (x) \log (\log (x))-2 \log (x) \log (\log (x))+1) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3}+\frac {e^{-\frac {x^2+x+1}{x (x+2)}} \left (5 \log (x) x^3-10 \log (x) x^2-6 \log (x) \log (\log (x)) x^2+3 x^2-10 \log (x) x-24 \log (x) \log (\log (x)) x+12 x-24 \log (x) \log (\log (x))+12\right ) \log ^{-\frac {5 x^2+e^x+3}{5 x^2}}(x)}{x^3 (x+2)^2}\right )dx\) |
Int[(E^((-5*x - 5*x^2 - 5*x^3 + (-6 + E^x*(-2 - x) - 3*x)*Log[Log[x]])/(10 *x^2 + 5*x^3))*(-12 - 12*x - 3*x^2 + E^x*(-4 - 4*x - x^2) + (10*x + 10*x^2 - 5*x^3)*Log[x] + (24 + 24*x + 6*x^2 + E^x*(8 + 4*x - 2*x^2 - x^3))*Log[x ]*Log[Log[x]]))/((20*x^3 + 20*x^4 + 5*x^5)*Log[x]),x]
3.4.40.3.1 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_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.50
\[{\mathrm e}^{-\frac {\ln \left (\ln \left (x \right )\right ) {\mathrm e}^{x} x +5 x^{3}+2 \,{\mathrm e}^{x} \ln \left (\ln \left (x \right )\right )+3 x \ln \left (\ln \left (x \right )\right )+5 x^{2}+6 \ln \left (\ln \left (x \right )\right )+5 x}{5 x^{2} \left (2+x \right )}}\]
int((((-x^3-2*x^2+4*x+8)*exp(x)+6*x^2+24*x+24)*ln(x)*ln(ln(x))+(-5*x^3+10* x^2+10*x)*ln(x)+(-x^2-4*x-4)*exp(x)-3*x^2-12*x-12)*exp((((-2-x)*exp(x)-3*x -6)*ln(ln(x))-5*x^3-5*x^2-5*x)/(5*x^3+10*x^2))/(5*x^5+20*x^4+20*x^3)/ln(x) ,x)
exp(-1/5*(ln(ln(x))*exp(x)*x+5*x^3+2*exp(x)*ln(ln(x))+3*x*ln(ln(x))+5*x^2+ 6*ln(ln(x))+5*x)/x^2/(2+x))
Time = 0.23 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.26 \[ \int \frac {e^{\frac {-5 x-5 x^2-5 x^3+\left (-6+e^x (-2-x)-3 x\right ) \log (\log (x))}{10 x^2+5 x^3}} \left (-12-12 x-3 x^2+e^x \left (-4-4 x-x^2\right )+\left (10 x+10 x^2-5 x^3\right ) \log (x)+\left (24+24 x+6 x^2+e^x \left (8+4 x-2 x^2-x^3\right )\right ) \log (x) \log (\log (x))\right )}{\left (20 x^3+20 x^4+5 x^5\right ) \log (x)} \, dx=e^{\left (-\frac {5 \, x^{3} + 5 \, x^{2} + {\left ({\left (x + 2\right )} e^{x} + 3 \, x + 6\right )} \log \left (\log \left (x\right )\right ) + 5 \, x}{5 \, {\left (x^{3} + 2 \, x^{2}\right )}}\right )} \]
integrate((((-x^3-2*x^2+4*x+8)*exp(x)+6*x^2+24*x+24)*log(x)*log(log(x))+(- 5*x^3+10*x^2+10*x)*log(x)+(-x^2-4*x-4)*exp(x)-3*x^2-12*x-12)*exp((((-2-x)* exp(x)-3*x-6)*log(log(x))-5*x^3-5*x^2-5*x)/(5*x^3+10*x^2))/(5*x^5+20*x^4+2 0*x^3)/log(x),x, algorithm=\
Time = 1.64 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.24 \[ \int \frac {e^{\frac {-5 x-5 x^2-5 x^3+\left (-6+e^x (-2-x)-3 x\right ) \log (\log (x))}{10 x^2+5 x^3}} \left (-12-12 x-3 x^2+e^x \left (-4-4 x-x^2\right )+\left (10 x+10 x^2-5 x^3\right ) \log (x)+\left (24+24 x+6 x^2+e^x \left (8+4 x-2 x^2-x^3\right )\right ) \log (x) \log (\log (x))\right )}{\left (20 x^3+20 x^4+5 x^5\right ) \log (x)} \, dx=e^{\frac {- 5 x^{3} - 5 x^{2} - 5 x + \left (- 3 x + \left (- x - 2\right ) e^{x} - 6\right ) \log {\left (\log {\left (x \right )} \right )}}{5 x^{3} + 10 x^{2}}} \]
integrate((((-x**3-2*x**2+4*x+8)*exp(x)+6*x**2+24*x+24)*ln(x)*ln(ln(x))+(- 5*x**3+10*x**2+10*x)*ln(x)+(-x**2-4*x-4)*exp(x)-3*x**2-12*x-12)*exp((((-2- x)*exp(x)-3*x-6)*ln(ln(x))-5*x**3-5*x**2-5*x)/(5*x**3+10*x**2))/(5*x**5+20 *x**4+20*x**3)/ln(x),x)
Time = 0.47 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.97 \[ \int \frac {e^{\frac {-5 x-5 x^2-5 x^3+\left (-6+e^x (-2-x)-3 x\right ) \log (\log (x))}{10 x^2+5 x^3}} \left (-12-12 x-3 x^2+e^x \left (-4-4 x-x^2\right )+\left (10 x+10 x^2-5 x^3\right ) \log (x)+\left (24+24 x+6 x^2+e^x \left (8+4 x-2 x^2-x^3\right )\right ) \log (x) \log (\log (x))\right )}{\left (20 x^3+20 x^4+5 x^5\right ) \log (x)} \, dx=e^{\left (-\frac {e^{x} \log \left (\log \left (x\right )\right )}{5 \, x^{2}} + \frac {3}{2 \, {\left (x + 2\right )}} - \frac {1}{2 \, x} - \frac {3 \, \log \left (\log \left (x\right )\right )}{5 \, x^{2}} - 1\right )} \]
integrate((((-x^3-2*x^2+4*x+8)*exp(x)+6*x^2+24*x+24)*log(x)*log(log(x))+(- 5*x^3+10*x^2+10*x)*log(x)+(-x^2-4*x-4)*exp(x)-3*x^2-12*x-12)*exp((((-2-x)* exp(x)-3*x-6)*log(log(x))-5*x^3-5*x^2-5*x)/(5*x^3+10*x^2))/(5*x^5+20*x^4+2 0*x^3)/log(x),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 118 vs. \(2 (31) = 62\).
Time = 4.09 (sec) , antiderivative size = 118, normalized size of antiderivative = 3.47 \[ \int \frac {e^{\frac {-5 x-5 x^2-5 x^3+\left (-6+e^x (-2-x)-3 x\right ) \log (\log (x))}{10 x^2+5 x^3}} \left (-12-12 x-3 x^2+e^x \left (-4-4 x-x^2\right )+\left (10 x+10 x^2-5 x^3\right ) \log (x)+\left (24+24 x+6 x^2+e^x \left (8+4 x-2 x^2-x^3\right )\right ) \log (x) \log (\log (x))\right )}{\left (20 x^3+20 x^4+5 x^5\right ) \log (x)} \, dx=e^{\left (-\frac {x^{3}}{x^{3} + 2 \, x^{2}} - \frac {x e^{x} \log \left (\log \left (x\right )\right )}{5 \, {\left (x^{3} + 2 \, x^{2}\right )}} - \frac {x^{2}}{x^{3} + 2 \, x^{2}} - \frac {3 \, x \log \left (\log \left (x\right )\right )}{5 \, {\left (x^{3} + 2 \, x^{2}\right )}} - \frac {2 \, e^{x} \log \left (\log \left (x\right )\right )}{5 \, {\left (x^{3} + 2 \, x^{2}\right )}} - \frac {x}{x^{3} + 2 \, x^{2}} - \frac {6 \, \log \left (\log \left (x\right )\right )}{5 \, {\left (x^{3} + 2 \, x^{2}\right )}}\right )} \]
integrate((((-x^3-2*x^2+4*x+8)*exp(x)+6*x^2+24*x+24)*log(x)*log(log(x))+(- 5*x^3+10*x^2+10*x)*log(x)+(-x^2-4*x-4)*exp(x)-3*x^2-12*x-12)*exp((((-2-x)* exp(x)-3*x-6)*log(log(x))-5*x^3-5*x^2-5*x)/(5*x^3+10*x^2))/(5*x^5+20*x^4+2 0*x^3)/log(x),x, algorithm=\
e^(-x^3/(x^3 + 2*x^2) - 1/5*x*e^x*log(log(x))/(x^3 + 2*x^2) - x^2/(x^3 + 2 *x^2) - 3/5*x*log(log(x))/(x^3 + 2*x^2) - 2/5*e^x*log(log(x))/(x^3 + 2*x^2 ) - x/(x^3 + 2*x^2) - 6/5*log(log(x))/(x^3 + 2*x^2))
Time = 8.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.85 \[ \int \frac {e^{\frac {-5 x-5 x^2-5 x^3+\left (-6+e^x (-2-x)-3 x\right ) \log (\log (x))}{10 x^2+5 x^3}} \left (-12-12 x-3 x^2+e^x \left (-4-4 x-x^2\right )+\left (10 x+10 x^2-5 x^3\right ) \log (x)+\left (24+24 x+6 x^2+e^x \left (8+4 x-2 x^2-x^3\right )\right ) \log (x) \log (\log (x))\right )}{\left (20 x^3+20 x^4+5 x^5\right ) \log (x)} \, dx=\frac {{\mathrm {e}}^{-\frac {5\,x}{5\,x^3+10\,x^2}}\,{\mathrm {e}}^{-\frac {5\,x^2}{5\,x^3+10\,x^2}}\,{\mathrm {e}}^{-\frac {5\,x^3}{5\,x^3+10\,x^2}}}{{\ln \left (x\right )}^{\frac {{\mathrm {e}}^x+3}{5\,\left (x^2+2\,x\right )}+\frac {2\,\left ({\mathrm {e}}^x+3\right )}{5\,\left (x^3+2\,x^2\right )}}} \]
int(-(exp(-(5*x + 5*x^2 + 5*x^3 + log(log(x))*(3*x + exp(x)*(x + 2) + 6))/ (10*x^2 + 5*x^3))*(12*x + exp(x)*(4*x + x^2 + 4) + 3*x^2 - log(x)*(10*x + 10*x^2 - 5*x^3) - log(log(x))*log(x)*(24*x + 6*x^2 + exp(x)*(4*x - 2*x^2 - x^3 + 8) + 24) + 12))/(log(x)*(20*x^3 + 20*x^4 + 5*x^5)),x)