Integrand size = 100, antiderivative size = 30 \[ \int \frac {-3 e^2 \log (5)+e^{e^x+\frac {1}{3} e^{e^x} (-3-x)} \left (e^2 \log (5)+e^{2+x} (3+x) \log (5)\right )}{75+3 e^{\frac {2}{3} e^{e^x} (-3-x)}+30 x+3 x^2+e^{\frac {1}{3} e^{e^x} (-3-x)} (30+6 x)} \, dx=\frac {e^2 \log (5)}{5+e^{\frac {e^{e^x}}{-1+\frac {x}{3+x}}}+x} \]
Time = 6.22 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.37 \[ \int \frac {-3 e^2 \log (5)+e^{e^x+\frac {1}{3} e^{e^x} (-3-x)} \left (e^2 \log (5)+e^{2+x} (3+x) \log (5)\right )}{75+3 e^{\frac {2}{3} e^{e^x} (-3-x)}+30 x+3 x^2+e^{\frac {1}{3} e^{e^x} (-3-x)} (30+6 x)} \, dx=\frac {e^{2+\frac {1}{3} e^{e^x} (3+x)} \log (5)}{1+e^{\frac {1}{3} e^{e^x} (3+x)} (5+x)} \]
Integrate[(-3*E^2*Log[5] + E^(E^x + (E^E^x*(-3 - x))/3)*(E^2*Log[5] + E^(2 + x)*(3 + x)*Log[5]))/(75 + 3*E^((2*E^E^x*(-3 - x))/3) + 30*x + 3*x^2 + E ^((E^E^x*(-3 - x))/3)*(30 + 6*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.
| \(\displaystyle \int \frac {e^{\frac {1}{3} e^{e^x} (-x-3)+e^x} \left (e^{x+2} (x+3) \log (5)+e^2 \log (5)\right )-3 e^2 \log (5)}{3 x^2+30 x+3 e^{\frac {2}{3} e^{e^x} (-x-3)}+e^{\frac {1}{3} e^{e^x} (-x-3)} (6 x+30)+75} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{\frac {2}{3} e^{e^x} (x+3)} \left (e^{\frac {1}{3} e^{e^x} (-x-3)+e^x} \left (e^{x+2} (x+3) \log (5)+e^2 \log (5)\right )-3 e^2 \log (5)\right )}{3 \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{3} \int -\frac {e^{\frac {2}{3} e^{e^x} (x+3)} \left (3 e^2 \log (5)-e^{e^x-\frac {1}{3} e^{e^x} (x+3)} \left (e^{x+2} \log (5) (x+3)+e^2 \log (5)\right )\right )}{\left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{3} \int \frac {e^{\frac {2}{3} e^{e^x} (x+3)} \left (3 e^2 \log (5)-e^{e^x-\frac {1}{3} e^{e^x} (x+3)} \left (e^{x+2} \log (5) (x+3)+e^2 \log (5)\right )\right )}{\left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{3} \int \left (-e^{\frac {1}{3} e^{e^x} (x+3)+e^x+2} \log (5) \left (e^x x+3 e^x+1\right )+\frac {e^{\frac {2}{3} e^{e^x} (x+3)+e^x+2} (x+5) \log (5) \left (e^x x+3 e^x+1\right )}{e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1}+\frac {e^{\frac {2}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} x^2+e^{e^x} x+8 e^{x+e^x} x+5 e^{e^x}+15 e^{x+e^x}+3\right ) \log (5)}{\left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle -\frac {1}{3} \int \frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (-e^{x+e^x} (x+3)-e^{e^x}+3 e^{\frac {1}{3} e^{e^x} (x+3)}\right ) \log (5)}{\left (e^{\frac {1}{3} e^{e^x} (x+3)} (x+5)+1\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{3} \log (5) \int -\frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} (x+3)+e^{e^x}-3 e^{\frac {1}{3} e^{e^x} (x+3)}\right )}{\left (e^{\frac {1}{3} e^{e^x} (x+3)} (x+5)+1\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} \log (5) \int \frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} (x+3)+e^{e^x}-3 e^{\frac {1}{3} e^{e^x} (x+3)}\right )}{\left (e^{\frac {1}{3} e^{e^x} (x+3)} (x+5)+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{3} \log (5) \int \left (\frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} x^2+e^{e^x} x+8 e^{x+e^x} x+5 e^{e^x}+15 e^{x+e^x}+3\right )}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )^2}-\frac {3 e^{\frac {1}{3} e^{e^x} (x+3)+2}}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{3} \log (5) \int \frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} (x+3)+e^{e^x}-3 e^{\frac {1}{3} e^{e^x} (x+3)}\right )}{\left (e^{\frac {1}{3} e^{e^x} (x+3)} (x+5)+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{3} \log (5) \int \left (\frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} x^2+e^{e^x} x+8 e^{x+e^x} x+5 e^{e^x}+15 e^{x+e^x}+3\right )}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )^2}-\frac {3 e^{\frac {1}{3} e^{e^x} (x+3)+2}}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{3} \log (5) \int \frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} (x+3)+e^{e^x}-3 e^{\frac {1}{3} e^{e^x} (x+3)}\right )}{\left (e^{\frac {1}{3} e^{e^x} (x+3)} (x+5)+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{3} \log (5) \int \left (\frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} x^2+e^{e^x} x+8 e^{x+e^x} x+5 e^{e^x}+15 e^{x+e^x}+3\right )}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )^2}-\frac {3 e^{\frac {1}{3} e^{e^x} (x+3)+2}}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{3} \log (5) \int \frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} (x+3)+e^{e^x}-3 e^{\frac {1}{3} e^{e^x} (x+3)}\right )}{\left (e^{\frac {1}{3} e^{e^x} (x+3)} (x+5)+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{3} \log (5) \int \left (\frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} x^2+e^{e^x} x+8 e^{x+e^x} x+5 e^{e^x}+15 e^{x+e^x}+3\right )}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )^2}-\frac {3 e^{\frac {1}{3} e^{e^x} (x+3)+2}}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{3} \log (5) \int \frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} (x+3)+e^{e^x}-3 e^{\frac {1}{3} e^{e^x} (x+3)}\right )}{\left (e^{\frac {1}{3} e^{e^x} (x+3)} (x+5)+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{3} \log (5) \int \left (\frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} x^2+e^{e^x} x+8 e^{x+e^x} x+5 e^{e^x}+15 e^{x+e^x}+3\right )}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )^2}-\frac {3 e^{\frac {1}{3} e^{e^x} (x+3)+2}}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{3} \log (5) \int \frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} (x+3)+e^{e^x}-3 e^{\frac {1}{3} e^{e^x} (x+3)}\right )}{\left (e^{\frac {1}{3} e^{e^x} (x+3)} (x+5)+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{3} \log (5) \int \left (\frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} x^2+e^{e^x} x+8 e^{x+e^x} x+5 e^{e^x}+15 e^{x+e^x}+3\right )}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )^2}-\frac {3 e^{\frac {1}{3} e^{e^x} (x+3)+2}}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{3} \log (5) \int \frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} (x+3)+e^{e^x}-3 e^{\frac {1}{3} e^{e^x} (x+3)}\right )}{\left (e^{\frac {1}{3} e^{e^x} (x+3)} (x+5)+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{3} \log (5) \int \left (\frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} x^2+e^{e^x} x+8 e^{x+e^x} x+5 e^{e^x}+15 e^{x+e^x}+3\right )}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )^2}-\frac {3 e^{\frac {1}{3} e^{e^x} (x+3)+2}}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{3} \log (5) \int \frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} (x+3)+e^{e^x}-3 e^{\frac {1}{3} e^{e^x} (x+3)}\right )}{\left (e^{\frac {1}{3} e^{e^x} (x+3)} (x+5)+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{3} \log (5) \int \left (\frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} x^2+e^{e^x} x+8 e^{x+e^x} x+5 e^{e^x}+15 e^{x+e^x}+3\right )}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )^2}-\frac {3 e^{\frac {1}{3} e^{e^x} (x+3)+2}}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{3} \log (5) \int \frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} (x+3)+e^{e^x}-3 e^{\frac {1}{3} e^{e^x} (x+3)}\right )}{\left (e^{\frac {1}{3} e^{e^x} (x+3)} (x+5)+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{3} \log (5) \int \left (\frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} x^2+e^{e^x} x+8 e^{x+e^x} x+5 e^{e^x}+15 e^{x+e^x}+3\right )}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )^2}-\frac {3 e^{\frac {1}{3} e^{e^x} (x+3)+2}}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{3} \log (5) \int \frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} (x+3)+e^{e^x}-3 e^{\frac {1}{3} e^{e^x} (x+3)}\right )}{\left (e^{\frac {1}{3} e^{e^x} (x+3)} (x+5)+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{3} \log (5) \int \left (\frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} x^2+e^{e^x} x+8 e^{x+e^x} x+5 e^{e^x}+15 e^{x+e^x}+3\right )}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )^2}-\frac {3 e^{\frac {1}{3} e^{e^x} (x+3)+2}}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{3} \log (5) \int \frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} (x+3)+e^{e^x}-3 e^{\frac {1}{3} e^{e^x} (x+3)}\right )}{\left (e^{\frac {1}{3} e^{e^x} (x+3)} (x+5)+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{3} \log (5) \int \left (\frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} x^2+e^{e^x} x+8 e^{x+e^x} x+5 e^{e^x}+15 e^{x+e^x}+3\right )}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )^2}-\frac {3 e^{\frac {1}{3} e^{e^x} (x+3)+2}}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \frac {1}{3} \log (5) \int \frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} (x+3)+e^{e^x}-3 e^{\frac {1}{3} e^{e^x} (x+3)}\right )}{\left (e^{\frac {1}{3} e^{e^x} (x+3)} (x+5)+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \frac {1}{3} \log (5) \int \left (\frac {e^{\frac {1}{3} e^{e^x} (x+3)+2} \left (e^{x+e^x} x^2+e^{e^x} x+8 e^{x+e^x} x+5 e^{e^x}+15 e^{x+e^x}+3\right )}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )^2}-\frac {3 e^{\frac {1}{3} e^{e^x} (x+3)+2}}{(x+5) \left (e^{\frac {1}{3} e^{e^x} (x+3)} x+5 e^{\frac {1}{3} e^{e^x} (x+3)}+1\right )}\right )dx\) |
Int[(-3*E^2*Log[5] + E^(E^x + (E^E^x*(-3 - x))/3)*(E^2*Log[5] + E^(2 + x)* (3 + x)*Log[5]))/(75 + 3*E^((2*E^E^x*(-3 - x))/3) + 30*x + 3*x^2 + E^((E^E ^x*(-3 - x))/3)*(30 + 6*x)),x]
3.21.54.3.1 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 = 2.82 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.67
| method | result | size |
| risch | \(\frac {{\mathrm e}^{2} \ln \left (5\right )}{5+{\mathrm e}^{-\frac {\left (3+x \right ) {\mathrm e}^{{\mathrm e}^{x}}}{3}}+x}\) | \(20\) |
int((((3+x)*exp(2)*ln(5)*exp(x)+exp(2)*ln(5))*exp(exp(x))*exp(1/3*(-3-x)*e xp(exp(x)))-3*exp(2)*ln(5))/(3*exp(1/3*(-3-x)*exp(exp(x)))^2+(6*x+30)*exp( 1/3*(-3-x)*exp(exp(x)))+3*x^2+30*x+75),x,method=_RETURNVERBOSE)
Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.30 \[ \int \frac {-3 e^2 \log (5)+e^{e^x+\frac {1}{3} e^{e^x} (-3-x)} \left (e^2 \log (5)+e^{2+x} (3+x) \log (5)\right )}{75+3 e^{\frac {2}{3} e^{e^x} (-3-x)}+30 x+3 x^2+e^{\frac {1}{3} e^{e^x} (-3-x)} (30+6 x)} \, dx=\frac {e^{\left (e^{x} + 2\right )} \log \left (5\right )}{{\left (x + 5\right )} e^{\left (e^{x}\right )} + e^{\left (-\frac {1}{3} \, {\left ({\left (x + 3\right )} e^{\left (e^{x} + 2\right )} - 3 \, e^{\left (x + 2\right )}\right )} e^{\left (-2\right )}\right )}} \]
integrate((((3+x)*exp(2)*log(5)*exp(x)+exp(2)*log(5))*exp(exp(x))*exp(1/3* (-3-x)*exp(exp(x)))-3*exp(2)*log(5))/(3*exp(1/3*(-3-x)*exp(exp(x)))^2+(6*x +30)*exp(1/3*(-3-x)*exp(exp(x)))+3*x^2+30*x+75),x, algorithm=\
Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.73 \[ \int \frac {-3 e^2 \log (5)+e^{e^x+\frac {1}{3} e^{e^x} (-3-x)} \left (e^2 \log (5)+e^{2+x} (3+x) \log (5)\right )}{75+3 e^{\frac {2}{3} e^{e^x} (-3-x)}+30 x+3 x^2+e^{\frac {1}{3} e^{e^x} (-3-x)} (30+6 x)} \, dx=\frac {e^{2} \log {\left (5 \right )}}{x + e^{\left (- \frac {x}{3} - 1\right ) e^{e^{x}}} + 5} \]
integrate((((3+x)*exp(2)*ln(5)*exp(x)+exp(2)*ln(5))*exp(exp(x))*exp(1/3*(- 3-x)*exp(exp(x)))-3*exp(2)*ln(5))/(3*exp(1/3*(-3-x)*exp(exp(x)))**2+(6*x+3 0)*exp(1/3*(-3-x)*exp(exp(x)))+3*x**2+30*x+75),x)
Time = 0.38 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13 \[ \int \frac {-3 e^2 \log (5)+e^{e^x+\frac {1}{3} e^{e^x} (-3-x)} \left (e^2 \log (5)+e^{2+x} (3+x) \log (5)\right )}{75+3 e^{\frac {2}{3} e^{e^x} (-3-x)}+30 x+3 x^2+e^{\frac {1}{3} e^{e^x} (-3-x)} (30+6 x)} \, dx=\frac {e^{\left (\frac {1}{3} \, x e^{\left (e^{x}\right )} + e^{\left (e^{x}\right )} + 2\right )} \log \left (5\right )}{{\left (x + 5\right )} e^{\left (\frac {1}{3} \, x e^{\left (e^{x}\right )} + e^{\left (e^{x}\right )}\right )} + 1} \]
integrate((((3+x)*exp(2)*log(5)*exp(x)+exp(2)*log(5))*exp(exp(x))*exp(1/3* (-3-x)*exp(exp(x)))-3*exp(2)*log(5))/(3*exp(1/3*(-3-x)*exp(exp(x)))^2+(6*x +30)*exp(1/3*(-3-x)*exp(exp(x)))+3*x^2+30*x+75),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 2069 vs. \(2 (26) = 52\).
Time = 0.35 (sec) , antiderivative size = 2069, normalized size of antiderivative = 68.97 \[ \int \frac {-3 e^2 \log (5)+e^{e^x+\frac {1}{3} e^{e^x} (-3-x)} \left (e^2 \log (5)+e^{2+x} (3+x) \log (5)\right )}{75+3 e^{\frac {2}{3} e^{e^x} (-3-x)}+30 x+3 x^2+e^{\frac {1}{3} e^{e^x} (-3-x)} (30+6 x)} \, dx=\text {Too large to display} \]
integrate((((3+x)*exp(2)*log(5)*exp(x)+exp(2)*log(5))*exp(exp(x))*exp(1/3* (-3-x)*exp(exp(x)))-3*exp(2)*log(5))/(3*exp(1/3*(-3-x)*exp(exp(x)))^2+(6*x +30)*exp(1/3*(-3-x)*exp(exp(x)))+3*x^2+30*x+75),x, algorithm=\
(x^5*e^(1/3*x*e^(e^x) + 3*x + 2*e^x + e^(e^x) + 2)*log(5) + 21*x^4*e^(1/3* x*e^(e^x) + 3*x + 2*e^x + e^(e^x) + 2)*log(5) + 2*x^4*e^(1/3*x*e^(e^x) + 2 *x + 2*e^x + e^(e^x) + 2)*log(5) + x^4*e^(3*x + 2*e^x + 2)*log(5) + 174*x^ 3*e^(1/3*x*e^(e^x) + 3*x + 2*e^x + e^(e^x) + 2)*log(5) + 36*x^3*e^(1/3*x*e ^(e^x) + 2*x + 2*e^x + e^(e^x) + 2)*log(5) + 6*x^3*e^(1/3*x*e^(e^x) + 2*x + e^x + e^(e^x) + 2)*log(5) + x^3*e^(1/3*x*e^(e^x) + x + 2*e^x + e^(e^x) + 2)*log(5) + 16*x^3*e^(3*x + 2*e^x + 2)*log(5) + 2*x^3*e^(2*x + 2*e^x + 2) *log(5) + 710*x^2*e^(1/3*x*e^(e^x) + 3*x + 2*e^x + e^(e^x) + 2)*log(5) + 2 40*x^2*e^(1/3*x*e^(e^x) + 2*x + 2*e^x + e^(e^x) + 2)*log(5) + 78*x^2*e^(1/ 3*x*e^(e^x) + 2*x + e^x + e^(e^x) + 2)*log(5) + 15*x^2*e^(1/3*x*e^(e^x) + x + 2*e^x + e^(e^x) + 2)*log(5) + 6*x^2*e^(1/3*x*e^(e^x) + x + e^x + e^(e^ x) + 2)*log(5) + 94*x^2*e^(3*x + 2*e^x + 2)*log(5) + 26*x^2*e^(2*x + 2*e^x + 2)*log(5) + 6*x^2*e^(2*x + e^x + 2)*log(5) + x^2*e^(x + 2*e^x + 2)*log( 5) + 1425*x*e^(1/3*x*e^(e^x) + 3*x + 2*e^x + e^(e^x) + 2)*log(5) + 700*x*e ^(1/3*x*e^(e^x) + 2*x + 2*e^x + e^(e^x) + 2)*log(5) + 330*x*e^(1/3*x*e^(e^ x) + 2*x + e^x + e^(e^x) + 2)*log(5) + 75*x*e^(1/3*x*e^(e^x) + x + 2*e^x + e^(e^x) + 2)*log(5) + 60*x*e^(1/3*x*e^(e^x) + x + e^x + e^(e^x) + 2)*log( 5) + 9*x*e^(1/3*x*e^(e^x) + x + e^(e^x) + 2)*log(5) + 240*x*e^(3*x + 2*e^x + 2)*log(5) + 110*x*e^(2*x + 2*e^x + 2)*log(5) + 48*x*e^(2*x + e^x + 2)*l og(5) + 10*x*e^(x + 2*e^x + 2)*log(5) + 6*x*e^(x + e^x + 2)*log(5) + 11...
Time = 8.48 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.77 \[ \int \frac {-3 e^2 \log (5)+e^{e^x+\frac {1}{3} e^{e^x} (-3-x)} \left (e^2 \log (5)+e^{2+x} (3+x) \log (5)\right )}{75+3 e^{\frac {2}{3} e^{e^x} (-3-x)}+30 x+3 x^2+e^{\frac {1}{3} e^{e^x} (-3-x)} (30+6 x)} \, dx=\frac {{\mathrm {e}}^2\,\ln \left (5\right )}{x+{\mathrm {e}}^{-{\mathrm {e}}^{{\mathrm {e}}^x}-\frac {x\,{\mathrm {e}}^{{\mathrm {e}}^x}}{3}}+5} \]