Integrand size = 287, antiderivative size = 31 \[ \int \frac {\log (x)+e^{e^{3 x}} \left (-2 e^5+\left (2-2 e^5+6 e^{5+3 x} x\right ) \log (x)+\left (2-6 e^{3 x} x\right ) \log ^2(x)\right )+e^{e^{3 x}} \left (-e^5+\left (1-e^5+3 e^{5+3 x} x\right ) \log (x)+\left (1-3 e^{3 x} x\right ) \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )+\left (2 e^5+\left (-2+2 e^5\right ) \log (x)-2 \log ^2(x)+\left (e^5+\left (-1+e^5\right ) \log (x)-\log ^2(x)\right ) \log \left (-e^5+\log (x)\right )\right ) \log \left (2+\log \left (-e^5+\log (x)\right )\right )}{e^{e^{3 x}} \left (2 e^5 x \log (x)-2 x \log ^2(x)\right )+e^{e^{3 x}} \left (e^5 x \log (x)-x \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )+\left (-2 e^5 x \log (x)+2 x \log ^2(x)+\left (-e^5 x \log (x)+x \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )\right ) \log \left (2+\log \left (-e^5+\log (x)\right )\right )} \, dx=\log \left (\frac {e^{e^{3 x}}-\log \left (2+\log \left (-e^5+\log (x)\right )\right )}{x \log (x)}\right ) \]
Time = 0.43 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.06 \[ \int \frac {\log (x)+e^{e^{3 x}} \left (-2 e^5+\left (2-2 e^5+6 e^{5+3 x} x\right ) \log (x)+\left (2-6 e^{3 x} x\right ) \log ^2(x)\right )+e^{e^{3 x}} \left (-e^5+\left (1-e^5+3 e^{5+3 x} x\right ) \log (x)+\left (1-3 e^{3 x} x\right ) \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )+\left (2 e^5+\left (-2+2 e^5\right ) \log (x)-2 \log ^2(x)+\left (e^5+\left (-1+e^5\right ) \log (x)-\log ^2(x)\right ) \log \left (-e^5+\log (x)\right )\right ) \log \left (2+\log \left (-e^5+\log (x)\right )\right )}{e^{e^{3 x}} \left (2 e^5 x \log (x)-2 x \log ^2(x)\right )+e^{e^{3 x}} \left (e^5 x \log (x)-x \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )+\left (-2 e^5 x \log (x)+2 x \log ^2(x)+\left (-e^5 x \log (x)+x \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )\right ) \log \left (2+\log \left (-e^5+\log (x)\right )\right )} \, dx=-\log (x)-\log (\log (x))+\log \left (e^{e^{3 x}}-\log \left (2+\log \left (-e^5+\log (x)\right )\right )\right ) \]
Integrate[(Log[x] + E^E^(3*x)*(-2*E^5 + (2 - 2*E^5 + 6*E^(5 + 3*x)*x)*Log[ x] + (2 - 6*E^(3*x)*x)*Log[x]^2) + E^E^(3*x)*(-E^5 + (1 - E^5 + 3*E^(5 + 3 *x)*x)*Log[x] + (1 - 3*E^(3*x)*x)*Log[x]^2)*Log[-E^5 + Log[x]] + (2*E^5 + (-2 + 2*E^5)*Log[x] - 2*Log[x]^2 + (E^5 + (-1 + E^5)*Log[x] - Log[x]^2)*Lo g[-E^5 + Log[x]])*Log[2 + Log[-E^5 + Log[x]]])/(E^E^(3*x)*(2*E^5*x*Log[x] - 2*x*Log[x]^2) + E^E^(3*x)*(E^5*x*Log[x] - x*Log[x]^2)*Log[-E^5 + Log[x]] + (-2*E^5*x*Log[x] + 2*x*Log[x]^2 + (-(E^5*x*Log[x]) + x*Log[x]^2)*Log[-E ^5 + Log[x]])*Log[2 + Log[-E^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.
| \(\displaystyle \int \frac {e^{e^{3 x}} \left (\left (2-6 e^{3 x} x\right ) \log ^2(x)+\left (6 e^{3 x+5} x-2 e^5+2\right ) \log (x)-2 e^5\right )+e^{e^{3 x}} \left (\left (1-3 e^{3 x} x\right ) \log ^2(x)+\left (3 e^{3 x+5} x-e^5+1\right ) \log (x)-e^5\right ) \log \left (\log (x)-e^5\right )+\left (-2 \log ^2(x)+\left (-\log ^2(x)+\left (e^5-1\right ) \log (x)+e^5\right ) \log \left (\log (x)-e^5\right )+\left (2 e^5-2\right ) \log (x)+2 e^5\right ) \log \left (\log \left (\log (x)-e^5\right )+2\right )+\log (x)}{e^{e^{3 x}} \left (2 e^5 x \log (x)-2 x \log ^2(x)\right )+e^{e^{3 x}} \left (e^5 x \log (x)-x \log ^2(x)\right ) \log \left (\log (x)-e^5\right )+\left (2 x \log ^2(x)+\left (x \log ^2(x)-e^5 x \log (x)\right ) \log \left (\log (x)-e^5\right )-2 e^5 x \log (x)\right ) \log \left (\log \left (\log (x)-e^5\right )+2\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{e^{3 x}} \left (\left (2-6 e^{3 x} x\right ) \log ^2(x)+\left (6 e^{3 x+5} x-2 e^5+2\right ) \log (x)-2 e^5\right )+e^{e^{3 x}} \left (\left (1-3 e^{3 x} x\right ) \log ^2(x)+\left (3 e^{3 x+5} x-e^5+1\right ) \log (x)-e^5\right ) \log \left (\log (x)-e^5\right )+\left (-2 \log ^2(x)+\left (-\log ^2(x)+\left (e^5-1\right ) \log (x)+e^5\right ) \log \left (\log (x)-e^5\right )+\left (2 e^5-2\right ) \log (x)+2 e^5\right ) \log \left (\log \left (\log (x)-e^5\right )+2\right )+\log (x)}{x \left (e^5-\log (x)\right ) \log (x) \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{e^{3 x}} \log \left (\log (x)-e^5\right ) \log (x)}{x \left (e^5-\log (x)\right ) \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}+\frac {2 e^{e^{3 x}} \log (x)}{x \left (e^5-\log (x)\right ) \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}+\frac {3 e^{3 x+e^{3 x}}}{e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )}+\frac {\left (1-e^5\right ) e^{e^{3 x}} \log \left (\log (x)-e^5\right )}{x \left (e^5-\log (x)\right ) \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}+\frac {2 \left (1-e^5\right ) e^{e^{3 x}}}{x \left (e^5-\log (x)\right ) \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}+\frac {1}{x \left (e^5-\log (x)\right ) \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}+\frac {(\log (x)+1) \log \left (\log \left (\log (x)-e^5\right )+2\right )}{x \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right ) \log (x)}-\frac {e^{e^{3 x}+5} \log \left (\log (x)-e^5\right )}{x \left (e^5-\log (x)\right ) \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right ) \log (x)}-\frac {2 e^{e^{3 x}+5}}{x \left (e^5-\log (x)\right ) \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right ) \log (x)}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {-\left (\left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}} \left (3 e^{3 x} x-1\right )+\log \left (\log \left (\log (x)-e^5\right )+2\right )\right ) \log ^2(x)\right )+\left (6 e^{3 x+e^{3 x}+5} x+2 e^{e^{3 x}}-2 e^{e^{3 x}+5}+2 \left (e^5-1\right ) \log \left (\log \left (\log (x)-e^5\right )+2\right )+\log \left (\log (x)-e^5\right ) \left (e^{e^{3 x}} \left (3 e^{3 x+5} x-e^5+1\right )+\left (e^5-1\right ) \log \left (\log \left (\log (x)-e^5\right )+2\right )\right )+1\right ) \log (x)-e^5 \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}{x \left (e^5-\log (x)\right ) \log (x) \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\log \left (\log \left (\log (x)-e^5\right )+2\right ) \log (x)}{x \left (e^5-\log (x)\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}+\frac {e^{e^{3 x}} \log (x)}{x \left (e^5-\log (x)\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}+\frac {(e-1) \left (1+e+e^2+e^3+e^4\right ) \log \left (\log (x)-e^5\right ) \log \left (\log \left (\log (x)-e^5\right )+2\right )}{x \left (e^5-\log (x)\right ) \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}+\frac {2 (e-1) \left (1+e+e^2+e^3+e^4\right ) \log \left (\log \left (\log (x)-e^5\right )+2\right )}{x \left (e^5-\log (x)\right ) \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}+\frac {3 e^{3 x+e^{3 x}}}{e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )}+\frac {\left (1-e^5\right ) e^{e^{3 x}} \log \left (\log (x)-e^5\right )}{x \left (e^5-\log (x)\right ) \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}+\frac {2 \left (1-e^5\right ) e^{e^{3 x}}}{x \left (e^5-\log (x)\right ) \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}+\frac {1}{x \left (e^5-\log (x)\right ) \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}-\frac {e^5}{x \left (e^5-\log (x)\right ) \log (x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \int \frac {e^{3 x+e^{3 x}}}{e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )}dx+\int \frac {e^{e^{3 x}} \log (x)}{x \left (e^5-\log (x)\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}dx+\int \frac {1}{x \left (e^5-\log (x)\right ) \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}dx+2 \left (1-e^5\right ) \int \frac {e^{e^{3 x}}}{x \left (e^5-\log (x)\right ) \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}dx+\left (1-e^5\right ) \int \frac {e^{e^{3 x}} \log \left (\log (x)-e^5\right )}{x \left (e^5-\log (x)\right ) \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}dx-\int \frac {\log (x) \log \left (\log \left (\log (x)-e^5\right )+2\right )}{x \left (e^5-\log (x)\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}dx-2 \left (1-e^5\right ) \int \frac {\log \left (\log \left (\log (x)-e^5\right )+2\right )}{x \left (e^5-\log (x)\right ) \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}dx-\left (1-e^5\right ) \int \frac {\log \left (\log (x)-e^5\right ) \log \left (\log \left (\log (x)-e^5\right )+2\right )}{x \left (e^5-\log (x)\right ) \left (\log \left (\log (x)-e^5\right )+2\right ) \left (e^{e^{3 x}}-\log \left (\log \left (\log (x)-e^5\right )+2\right )\right )}dx+\log \left (e^5-\log (x)\right )-\log (\log (x))\) |
Int[(Log[x] + E^E^(3*x)*(-2*E^5 + (2 - 2*E^5 + 6*E^(5 + 3*x)*x)*Log[x] + ( 2 - 6*E^(3*x)*x)*Log[x]^2) + E^E^(3*x)*(-E^5 + (1 - E^5 + 3*E^(5 + 3*x)*x) *Log[x] + (1 - 3*E^(3*x)*x)*Log[x]^2)*Log[-E^5 + Log[x]] + (2*E^5 + (-2 + 2*E^5)*Log[x] - 2*Log[x]^2 + (E^5 + (-1 + E^5)*Log[x] - Log[x]^2)*Log[-E^5 + Log[x]])*Log[2 + Log[-E^5 + Log[x]]])/(E^E^(3*x)*(2*E^5*x*Log[x] - 2*x* Log[x]^2) + E^E^(3*x)*(E^5*x*Log[x] - x*Log[x]^2)*Log[-E^5 + Log[x]] + (-2 *E^5*x*Log[x] + 2*x*Log[x]^2 + (-(E^5*x*Log[x]) + x*Log[x]^2)*Log[-E^5 + L og[x]])*Log[2 + Log[-E^5 + Log[x]]]),x]
3.9.73.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 0.46 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00
\[-\ln \left (x \right )-\ln \left (\ln \left (x \right )\right )+\ln \left (\ln \left (\ln \left (\ln \left (x \right )-{\mathrm e}^{5}\right )+2\right )-{\mathrm e}^{{\mathrm e}^{3 x}}\right )\]
int((((-ln(x)^2+(exp(5)-1)*ln(x)+exp(5))*ln(ln(x)-exp(5))-2*ln(x)^2+(2*exp (5)-2)*ln(x)+2*exp(5))*ln(ln(ln(x)-exp(5))+2)+((-3*x*exp(3*x)+1)*ln(x)^2+( 3*x*exp(5)*exp(3*x)+1-exp(5))*ln(x)-exp(5))*exp(exp(3*x))*ln(ln(x)-exp(5)) +((-6*x*exp(3*x)+2)*ln(x)^2+(6*x*exp(5)*exp(3*x)-2*exp(5)+2)*ln(x)-2*exp(5 ))*exp(exp(3*x))+ln(x))/(((x*ln(x)^2-x*exp(5)*ln(x))*ln(ln(x)-exp(5))+2*x* ln(x)^2-2*x*exp(5)*ln(x))*ln(ln(ln(x)-exp(5))+2)+(-x*ln(x)^2+x*exp(5)*ln(x ))*exp(exp(3*x))*ln(ln(x)-exp(5))+(-2*x*ln(x)^2+2*x*exp(5)*ln(x))*exp(exp( 3*x))),x)
Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {\log (x)+e^{e^{3 x}} \left (-2 e^5+\left (2-2 e^5+6 e^{5+3 x} x\right ) \log (x)+\left (2-6 e^{3 x} x\right ) \log ^2(x)\right )+e^{e^{3 x}} \left (-e^5+\left (1-e^5+3 e^{5+3 x} x\right ) \log (x)+\left (1-3 e^{3 x} x\right ) \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )+\left (2 e^5+\left (-2+2 e^5\right ) \log (x)-2 \log ^2(x)+\left (e^5+\left (-1+e^5\right ) \log (x)-\log ^2(x)\right ) \log \left (-e^5+\log (x)\right )\right ) \log \left (2+\log \left (-e^5+\log (x)\right )\right )}{e^{e^{3 x}} \left (2 e^5 x \log (x)-2 x \log ^2(x)\right )+e^{e^{3 x}} \left (e^5 x \log (x)-x \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )+\left (-2 e^5 x \log (x)+2 x \log ^2(x)+\left (-e^5 x \log (x)+x \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )\right ) \log \left (2+\log \left (-e^5+\log (x)\right )\right )} \, dx=-\log \left (x\right ) + \log \left (-e^{\left (e^{\left (3 \, x\right )}\right )} + \log \left (\log \left (-e^{5} + \log \left (x\right )\right ) + 2\right )\right ) - \log \left (\log \left (x\right )\right ) \]
integrate((((-log(x)^2+(exp(5)-1)*log(x)+exp(5))*log(log(x)-exp(5))-2*log( x)^2+(2*exp(5)-2)*log(x)+2*exp(5))*log(log(log(x)-exp(5))+2)+((-3*x*exp(3* x)+1)*log(x)^2+(3*x*exp(5)*exp(3*x)+1-exp(5))*log(x)-exp(5))*exp(exp(3*x)) *log(log(x)-exp(5))+((-6*x*exp(3*x)+2)*log(x)^2+(6*x*exp(5)*exp(3*x)-2*exp (5)+2)*log(x)-2*exp(5))*exp(exp(3*x))+log(x))/(((x*log(x)^2-x*exp(5)*log(x ))*log(log(x)-exp(5))+2*x*log(x)^2-2*x*exp(5)*log(x))*log(log(log(x)-exp(5 ))+2)+(-x*log(x)^2+x*exp(5)*log(x))*exp(exp(3*x))*log(log(x)-exp(5))+(-2*x *log(x)^2+2*x*exp(5)*log(x))*exp(exp(3*x))),x, algorithm=\
Time = 6.44 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {\log (x)+e^{e^{3 x}} \left (-2 e^5+\left (2-2 e^5+6 e^{5+3 x} x\right ) \log (x)+\left (2-6 e^{3 x} x\right ) \log ^2(x)\right )+e^{e^{3 x}} \left (-e^5+\left (1-e^5+3 e^{5+3 x} x\right ) \log (x)+\left (1-3 e^{3 x} x\right ) \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )+\left (2 e^5+\left (-2+2 e^5\right ) \log (x)-2 \log ^2(x)+\left (e^5+\left (-1+e^5\right ) \log (x)-\log ^2(x)\right ) \log \left (-e^5+\log (x)\right )\right ) \log \left (2+\log \left (-e^5+\log (x)\right )\right )}{e^{e^{3 x}} \left (2 e^5 x \log (x)-2 x \log ^2(x)\right )+e^{e^{3 x}} \left (e^5 x \log (x)-x \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )+\left (-2 e^5 x \log (x)+2 x \log ^2(x)+\left (-e^5 x \log (x)+x \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )\right ) \log \left (2+\log \left (-e^5+\log (x)\right )\right )} \, dx=- \log {\left (x \right )} + \log {\left (e^{e^{3 x}} - \log {\left (\log {\left (\log {\left (x \right )} - e^{5} \right )} + 2 \right )} \right )} - \log {\left (\log {\left (x \right )} \right )} \]
integrate((((-ln(x)**2+(exp(5)-1)*ln(x)+exp(5))*ln(ln(x)-exp(5))-2*ln(x)** 2+(2*exp(5)-2)*ln(x)+2*exp(5))*ln(ln(ln(x)-exp(5))+2)+((-3*x*exp(3*x)+1)*l n(x)**2+(3*x*exp(5)*exp(3*x)+1-exp(5))*ln(x)-exp(5))*exp(exp(3*x))*ln(ln(x )-exp(5))+((-6*x*exp(3*x)+2)*ln(x)**2+(6*x*exp(5)*exp(3*x)-2*exp(5)+2)*ln( x)-2*exp(5))*exp(exp(3*x))+ln(x))/(((x*ln(x)**2-x*exp(5)*ln(x))*ln(ln(x)-e xp(5))+2*x*ln(x)**2-2*x*exp(5)*ln(x))*ln(ln(ln(x)-exp(5))+2)+(-x*ln(x)**2+ x*exp(5)*ln(x))*exp(exp(3*x))*ln(ln(x)-exp(5))+(-2*x*ln(x)**2+2*x*exp(5)*l n(x))*exp(exp(3*x))),x)
Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {\log (x)+e^{e^{3 x}} \left (-2 e^5+\left (2-2 e^5+6 e^{5+3 x} x\right ) \log (x)+\left (2-6 e^{3 x} x\right ) \log ^2(x)\right )+e^{e^{3 x}} \left (-e^5+\left (1-e^5+3 e^{5+3 x} x\right ) \log (x)+\left (1-3 e^{3 x} x\right ) \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )+\left (2 e^5+\left (-2+2 e^5\right ) \log (x)-2 \log ^2(x)+\left (e^5+\left (-1+e^5\right ) \log (x)-\log ^2(x)\right ) \log \left (-e^5+\log (x)\right )\right ) \log \left (2+\log \left (-e^5+\log (x)\right )\right )}{e^{e^{3 x}} \left (2 e^5 x \log (x)-2 x \log ^2(x)\right )+e^{e^{3 x}} \left (e^5 x \log (x)-x \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )+\left (-2 e^5 x \log (x)+2 x \log ^2(x)+\left (-e^5 x \log (x)+x \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )\right ) \log \left (2+\log \left (-e^5+\log (x)\right )\right )} \, dx=-\log \left (x\right ) + \log \left (-e^{\left (e^{\left (3 \, x\right )}\right )} + \log \left (\log \left (-e^{5} + \log \left (x\right )\right ) + 2\right )\right ) - \log \left (\log \left (x\right )\right ) \]
integrate((((-log(x)^2+(exp(5)-1)*log(x)+exp(5))*log(log(x)-exp(5))-2*log( x)^2+(2*exp(5)-2)*log(x)+2*exp(5))*log(log(log(x)-exp(5))+2)+((-3*x*exp(3* x)+1)*log(x)^2+(3*x*exp(5)*exp(3*x)+1-exp(5))*log(x)-exp(5))*exp(exp(3*x)) *log(log(x)-exp(5))+((-6*x*exp(3*x)+2)*log(x)^2+(6*x*exp(5)*exp(3*x)-2*exp (5)+2)*log(x)-2*exp(5))*exp(exp(3*x))+log(x))/(((x*log(x)^2-x*exp(5)*log(x ))*log(log(x)-exp(5))+2*x*log(x)^2-2*x*exp(5)*log(x))*log(log(log(x)-exp(5 ))+2)+(-x*log(x)^2+x*exp(5)*log(x))*exp(exp(3*x))*log(log(x)-exp(5))+(-2*x *log(x)^2+2*x*exp(5)*log(x))*exp(exp(3*x))),x, algorithm=\
Time = 0.30 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.32 \[ \int \frac {\log (x)+e^{e^{3 x}} \left (-2 e^5+\left (2-2 e^5+6 e^{5+3 x} x\right ) \log (x)+\left (2-6 e^{3 x} x\right ) \log ^2(x)\right )+e^{e^{3 x}} \left (-e^5+\left (1-e^5+3 e^{5+3 x} x\right ) \log (x)+\left (1-3 e^{3 x} x\right ) \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )+\left (2 e^5+\left (-2+2 e^5\right ) \log (x)-2 \log ^2(x)+\left (e^5+\left (-1+e^5\right ) \log (x)-\log ^2(x)\right ) \log \left (-e^5+\log (x)\right )\right ) \log \left (2+\log \left (-e^5+\log (x)\right )\right )}{e^{e^{3 x}} \left (2 e^5 x \log (x)-2 x \log ^2(x)\right )+e^{e^{3 x}} \left (e^5 x \log (x)-x \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )+\left (-2 e^5 x \log (x)+2 x \log ^2(x)+\left (-e^5 x \log (x)+x \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )\right ) \log \left (2+\log \left (-e^5+\log (x)\right )\right )} \, dx=-3 \, x + \log \left (-e^{\left (3 \, x\right )} \log \left (\log \left (-e^{5} + \log \left (x\right )\right ) + 2\right ) + e^{\left (3 \, x + e^{\left (3 \, x\right )}\right )}\right ) - \log \left (x\right ) - \log \left (\log \left (x\right )\right ) \]
integrate((((-log(x)^2+(exp(5)-1)*log(x)+exp(5))*log(log(x)-exp(5))-2*log( x)^2+(2*exp(5)-2)*log(x)+2*exp(5))*log(log(log(x)-exp(5))+2)+((-3*x*exp(3* x)+1)*log(x)^2+(3*x*exp(5)*exp(3*x)+1-exp(5))*log(x)-exp(5))*exp(exp(3*x)) *log(log(x)-exp(5))+((-6*x*exp(3*x)+2)*log(x)^2+(6*x*exp(5)*exp(3*x)-2*exp (5)+2)*log(x)-2*exp(5))*exp(exp(3*x))+log(x))/(((x*log(x)^2-x*exp(5)*log(x ))*log(log(x)-exp(5))+2*x*log(x)^2-2*x*exp(5)*log(x))*log(log(log(x)-exp(5 ))+2)+(-x*log(x)^2+x*exp(5)*log(x))*exp(exp(3*x))*log(log(x)-exp(5))+(-2*x *log(x)^2+2*x*exp(5)*log(x))*exp(exp(3*x))),x, algorithm=\
Time = 10.39 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {\log (x)+e^{e^{3 x}} \left (-2 e^5+\left (2-2 e^5+6 e^{5+3 x} x\right ) \log (x)+\left (2-6 e^{3 x} x\right ) \log ^2(x)\right )+e^{e^{3 x}} \left (-e^5+\left (1-e^5+3 e^{5+3 x} x\right ) \log (x)+\left (1-3 e^{3 x} x\right ) \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )+\left (2 e^5+\left (-2+2 e^5\right ) \log (x)-2 \log ^2(x)+\left (e^5+\left (-1+e^5\right ) \log (x)-\log ^2(x)\right ) \log \left (-e^5+\log (x)\right )\right ) \log \left (2+\log \left (-e^5+\log (x)\right )\right )}{e^{e^{3 x}} \left (2 e^5 x \log (x)-2 x \log ^2(x)\right )+e^{e^{3 x}} \left (e^5 x \log (x)-x \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )+\left (-2 e^5 x \log (x)+2 x \log ^2(x)+\left (-e^5 x \log (x)+x \log ^2(x)\right ) \log \left (-e^5+\log (x)\right )\right ) \log \left (2+\log \left (-e^5+\log (x)\right )\right )} \, dx=\ln \left (\ln \left (\ln \left (\ln \left (x\right )-{\mathrm {e}}^5\right )+2\right )-{\mathrm {e}}^{{\mathrm {e}}^{3\,x}}\right )-\ln \left (\ln \left (x\right )\right )-\ln \left (x\right ) \]
int(-(log(x) + log(log(log(x) - exp(5)) + 2)*(2*exp(5) - 2*log(x)^2 + log( log(x) - exp(5))*(exp(5) + log(x)*(exp(5) - 1) - log(x)^2) + log(x)*(2*exp (5) - 2)) - exp(exp(3*x))*(2*exp(5) + log(x)^2*(6*x*exp(3*x) - 2) - log(x) *(6*x*exp(3*x)*exp(5) - 2*exp(5) + 2)) - exp(exp(3*x))*log(log(x) - exp(5) )*(exp(5) + log(x)^2*(3*x*exp(3*x) - 1) - log(x)*(3*x*exp(3*x)*exp(5) - ex p(5) + 1)))/(exp(exp(3*x))*(2*x*log(x)^2 - 2*x*exp(5)*log(x)) - log(log(lo g(x) - exp(5)) + 2)*(2*x*log(x)^2 + log(log(x) - exp(5))*(x*log(x)^2 - x*e xp(5)*log(x)) - 2*x*exp(5)*log(x)) + exp(exp(3*x))*log(log(x) - exp(5))*(x *log(x)^2 - x*exp(5)*log(x))),x)