Integrand size = 311, antiderivative size = 30 \[ \int \frac {2 e^x x+e^{e^{2 x}+2 e^x x+x^2} \left (-6 e^{2 x} x-6 x^2+e^x \left (-6 x-6 x^2\right )\right )-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right )}{-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )} \, dx=x \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{\left (e^x+x\right )^2}}\right )\right )\right ) \]
Time = 0.65 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.27 \[ \int \frac {2 e^x x+e^{e^{2 x}+2 e^x x+x^2} \left (-6 e^{2 x} x-6 x^2+e^x \left (-6 x-6 x^2\right )\right )-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right )}{-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )} \, dx=x \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right ) \]
Integrate[(2*E^x*x + E^(E^(2*x) + 2*E^x*x + x^2)*(-6*E^(2*x)*x - 6*x^2 + E ^x*(-6*x - 6*x^2)) - x*Log[5/E^E^(E^(2*x) + 2*E^x*x + x^2)] + (-(E^x*x) - x*Log[5/E^E^(E^(2*x) + 2*E^x*x + x^2)] + (3*E^x + 3*Log[5/E^E^(E^(2*x) + 2 *E^x*x + x^2)])*Log[E^x + Log[5/E^E^(E^(2*x) + 2*E^x*x + x^2)]])*Log[-x + 3*Log[E^x + Log[5/E^E^(E^(2*x) + 2*E^x*x + x^2)]]])/(-(E^x*x) - x*Log[5/E^ E^(E^(2*x) + 2*E^x*x + x^2)] + (3*E^x + 3*Log[5/E^E^(E^(2*x) + 2*E^x*x + x ^2)])*Log[E^x + Log[5/E^E^(E^(2*x) + 2*E^x*x + x^2)]]),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.
| \(\displaystyle \int \frac {e^{x^2+2 e^x x+e^{2 x}} \left (-6 x^2+e^x \left (-6 x^2-6 x\right )-6 e^{2 x} x\right )-x \log \left (5 e^{-e^{x^2+2 e^x x+e^{2 x}}}\right )+\left (-x \log \left (5 e^{-e^{x^2+2 e^x x+e^{2 x}}}\right )+\left (3 \log \left (5 e^{-e^{x^2+2 e^x x+e^{2 x}}}\right )+3 e^x\right ) \log \left (\log \left (5 e^{-e^{x^2+2 e^x x+e^{2 x}}}\right )+e^x\right )-e^x x\right ) \log \left (3 \log \left (\log \left (5 e^{-e^{x^2+2 e^x x+e^{2 x}}}\right )+e^x\right )-x\right )+2 e^x x}{-x \log \left (5 e^{-e^{x^2+2 e^x x+e^{2 x}}}\right )+\left (3 \log \left (5 e^{-e^{x^2+2 e^x x+e^{2 x}}}\right )+3 e^x\right ) \log \left (\log \left (5 e^{-e^{x^2+2 e^x x+e^{2 x}}}\right )+e^x\right )-e^x x} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {-e^{x^2+2 e^x x+e^{2 x}} \left (-6 x^2+e^x \left (-6 x^2-6 x\right )-6 e^{2 x} x\right )+x \log \left (5 e^{-e^{x^2+2 e^x x+e^{2 x}}}\right )-\left (-x \log \left (5 e^{-e^{x^2+2 e^x x+e^{2 x}}}\right )+\left (3 \log \left (5 e^{-e^{x^2+2 e^x x+e^{2 x}}}\right )+3 e^x\right ) \log \left (\log \left (5 e^{-e^{x^2+2 e^x x+e^{2 x}}}\right )+e^x\right )-e^x x\right ) \log \left (3 \log \left (\log \left (5 e^{-e^{x^2+2 e^x x+e^{2 x}}}\right )+e^x\right )-x\right )-2 e^x x}{\left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right )\right )}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {6 e^{\left (x+e^x\right )^2} \left (e^x+1\right ) x \left (x+e^x\right )}{\left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right )\right )}+\frac {-2 e^x x+x \log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )+e^x x \log \left (3 \log \left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right )-x\right )+x \log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right ) \log \left (3 \log \left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right )-x\right )-3 e^x \log \left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right ) \log \left (3 \log \left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right )-x\right )-3 \log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right ) \log \left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right ) \log \left (3 \log \left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right )-x\right )}{\left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right )\right )}\right )dx\) |
| \(\Big \downarrow \) 7299 |
| \(\displaystyle \int \left (\frac {6 e^{\left (x+e^x\right )^2} \left (e^x+1\right ) x \left (x+e^x\right )}{\left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right )\right )}+\frac {-2 e^x x+x \log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )+e^x x \log \left (3 \log \left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right )-x\right )+x \log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right ) \log \left (3 \log \left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right )-x\right )-3 e^x \log \left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right ) \log \left (3 \log \left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right )-x\right )-3 \log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right ) \log \left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right ) \log \left (3 \log \left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right )-x\right )}{\left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right ) \left (x-3 \log \left (e^x+\log \left (5 e^{-e^{\left (x+e^x\right )^2}}\right )\right )\right )}\right )dx\) |
Int[(2*E^x*x + E^(E^(2*x) + 2*E^x*x + x^2)*(-6*E^(2*x)*x - 6*x^2 + E^x*(-6 *x - 6*x^2)) - x*Log[5/E^E^(E^(2*x) + 2*E^x*x + x^2)] + (-(E^x*x) - x*Log[ 5/E^E^(E^(2*x) + 2*E^x*x + x^2)] + (3*E^x + 3*Log[5/E^E^(E^(2*x) + 2*E^x*x + x^2)])*Log[E^x + Log[5/E^E^(E^(2*x) + 2*E^x*x + x^2)]])*Log[-x + 3*Log[ E^x + Log[5/E^E^(E^(2*x) + 2*E^x*x + x^2)]]])/(-(E^x*x) - x*Log[5/E^E^(E^( 2*x) + 2*E^x*x + x^2)] + (3*E^x + 3*Log[5/E^E^(E^(2*x) + 2*E^x*x + x^2)])* Log[E^x + Log[5/E^E^(E^(2*x) + 2*E^x*x + x^2)]]),x]
3.8.4.3.1 Defintions of rubi rules used
Time = 218.35 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.13
| method | result | size |
| risch | \(\ln \left (3 \ln \left (\ln \left (5\right )-\ln \left ({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x +x^{2}}}\right )+{\mathrm e}^{x}\right )-x \right ) x\) | \(34\) |
| parallelrisch | \(x \ln \left (3 \ln \left (\ln \left (5 \,{\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} x +x^{2}}}\right )+{\mathrm e}^{x}\right )-x \right )\) | \(34\) |
int((((3*ln(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+3*exp(x))*ln(ln(5/exp(exp (exp(x)^2+2*exp(x)*x+x^2)))+exp(x))-x*ln(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2 )))-exp(x)*x)*ln(3*ln(ln(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+exp(x))-x)-x *ln(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+(-6*x*exp(x)^2+(-6*x^2-6*x)*exp(x )-6*x^2)*exp(exp(x)^2+2*exp(x)*x+x^2)+2*exp(x)*x)/((3*ln(5/exp(exp(exp(x)^ 2+2*exp(x)*x+x^2)))+3*exp(x))*ln(ln(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+e xp(x))-x*ln(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))-exp(x)*x),x,method=_RETUR NVERBOSE)
Time = 0.36 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {2 e^x x+e^{e^{2 x}+2 e^x x+x^2} \left (-6 e^{2 x} x-6 x^2+e^x \left (-6 x-6 x^2\right )\right )-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right )}{-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )} \, dx=x \log \left (-x + 3 \, \log \left (-e^{\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )} + e^{x} + \log \left (5\right )\right )\right ) \]
integrate((((3*log(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+3*exp(x))*log(log( 5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+exp(x))-x*log(5/exp(exp(exp(x)^2+2*ex p(x)*x+x^2)))-exp(x)*x)*log(3*log(log(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2))) +exp(x))-x)-x*log(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+(-6*x*exp(x)^2+(-6* x^2-6*x)*exp(x)-6*x^2)*exp(exp(x)^2+2*exp(x)*x+x^2)+2*exp(x)*x)/((3*log(5/ exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+3*exp(x))*log(log(5/exp(exp(exp(x)^2+2* exp(x)*x+x^2)))+exp(x))-x*log(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))-exp(x)* x),x, algorithm=\
Timed out. \[ \int \frac {2 e^x x+e^{e^{2 x}+2 e^x x+x^2} \left (-6 e^{2 x} x-6 x^2+e^x \left (-6 x-6 x^2\right )\right )-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right )}{-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )} \, dx=\text {Timed out} \]
integrate((((3*ln(5/exp(exp(exp(x)**2+2*exp(x)*x+x**2)))+3*exp(x))*ln(ln(5 /exp(exp(exp(x)**2+2*exp(x)*x+x**2)))+exp(x))-x*ln(5/exp(exp(exp(x)**2+2*e xp(x)*x+x**2)))-exp(x)*x)*ln(3*ln(ln(5/exp(exp(exp(x)**2+2*exp(x)*x+x**2)) )+exp(x))-x)-x*ln(5/exp(exp(exp(x)**2+2*exp(x)*x+x**2)))+(-6*x*exp(x)**2+( -6*x**2-6*x)*exp(x)-6*x**2)*exp(exp(x)**2+2*exp(x)*x+x**2)+2*exp(x)*x)/((3 *ln(5/exp(exp(exp(x)**2+2*exp(x)*x+x**2)))+3*exp(x))*ln(ln(5/exp(exp(exp(x )**2+2*exp(x)*x+x**2)))+exp(x))-x*ln(5/exp(exp(exp(x)**2+2*exp(x)*x+x**2)) )-exp(x)*x),x)
Time = 0.52 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.03 \[ \int \frac {2 e^x x+e^{e^{2 x}+2 e^x x+x^2} \left (-6 e^{2 x} x-6 x^2+e^x \left (-6 x-6 x^2\right )\right )-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right )}{-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )} \, dx=x \log \left (-x + 3 \, \log \left (-e^{\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )} + e^{x} + \log \left (5\right )\right )\right ) \]
integrate((((3*log(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+3*exp(x))*log(log( 5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+exp(x))-x*log(5/exp(exp(exp(x)^2+2*ex p(x)*x+x^2)))-exp(x)*x)*log(3*log(log(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2))) +exp(x))-x)-x*log(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+(-6*x*exp(x)^2+(-6* x^2-6*x)*exp(x)-6*x^2)*exp(exp(x)^2+2*exp(x)*x+x^2)+2*exp(x)*x)/((3*log(5/ exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+3*exp(x))*log(log(5/exp(exp(exp(x)^2+2* exp(x)*x+x^2)))+exp(x))-x*log(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))-exp(x)* x),x, algorithm=\
\[ \int \frac {2 e^x x+e^{e^{2 x}+2 e^x x+x^2} \left (-6 e^{2 x} x-6 x^2+e^x \left (-6 x-6 x^2\right )\right )-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right )}{-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )} \, dx=\int { \frac {6 \, {\left (x^{2} + x e^{\left (2 \, x\right )} + {\left (x^{2} + x\right )} e^{x}\right )} e^{\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )} - 2 \, x e^{x} + {\left (x e^{x} + x \log \left (5 \, e^{\left (-e^{\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )}\right )}\right ) - 3 \, {\left (e^{x} + \log \left (5 \, e^{\left (-e^{\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )}\right )}\right )\right )} \log \left (e^{x} + \log \left (5 \, e^{\left (-e^{\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )}\right )}\right )\right )\right )} \log \left (-x + 3 \, \log \left (e^{x} + \log \left (5 \, e^{\left (-e^{\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )}\right )}\right )\right )\right ) + x \log \left (5 \, e^{\left (-e^{\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )}\right )}\right )}{x e^{x} + x \log \left (5 \, e^{\left (-e^{\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )}\right )}\right ) - 3 \, {\left (e^{x} + \log \left (5 \, e^{\left (-e^{\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )}\right )}\right )\right )} \log \left (e^{x} + \log \left (5 \, e^{\left (-e^{\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )}\right )}\right )\right )} \,d x } \]
integrate((((3*log(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+3*exp(x))*log(log( 5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+exp(x))-x*log(5/exp(exp(exp(x)^2+2*ex p(x)*x+x^2)))-exp(x)*x)*log(3*log(log(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2))) +exp(x))-x)-x*log(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+(-6*x*exp(x)^2+(-6* x^2-6*x)*exp(x)-6*x^2)*exp(exp(x)^2+2*exp(x)*x+x^2)+2*exp(x)*x)/((3*log(5/ exp(exp(exp(x)^2+2*exp(x)*x+x^2)))+3*exp(x))*log(log(5/exp(exp(exp(x)^2+2* exp(x)*x+x^2)))+exp(x))-x*log(5/exp(exp(exp(x)^2+2*exp(x)*x+x^2)))-exp(x)* x),x, algorithm=\
integrate((6*(x^2 + x*e^(2*x) + (x^2 + x)*e^x)*e^(x^2 + 2*x*e^x + e^(2*x)) - 2*x*e^x + (x*e^x + x*log(5*e^(-e^(x^2 + 2*x*e^x + e^(2*x)))) - 3*(e^x + log(5*e^(-e^(x^2 + 2*x*e^x + e^(2*x)))))*log(e^x + log(5*e^(-e^(x^2 + 2*x *e^x + e^(2*x))))))*log(-x + 3*log(e^x + log(5*e^(-e^(x^2 + 2*x*e^x + e^(2 *x)))))) + x*log(5*e^(-e^(x^2 + 2*x*e^x + e^(2*x)))))/(x*e^x + x*log(5*e^( -e^(x^2 + 2*x*e^x + e^(2*x)))) - 3*(e^x + log(5*e^(-e^(x^2 + 2*x*e^x + e^( 2*x)))))*log(e^x + log(5*e^(-e^(x^2 + 2*x*e^x + e^(2*x)))))), x)
Time = 11.80 (sec) , antiderivative size = 189, normalized size of antiderivative = 6.30 \[ \int \frac {2 e^x x+e^{e^{2 x}+2 e^x x+x^2} \left (-6 e^{2 x} x-6 x^2+e^x \left (-6 x-6 x^2\right )\right )-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right ) \log \left (-x+3 \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )\right )}{-e^x x-x \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )+\left (3 e^x+3 \log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right ) \log \left (e^x+\log \left (5 e^{-e^{e^{2 x}+2 e^x x+x^2}}\right )\right )} \, dx=\frac {x\,\ln \left (5\right )\,\ln \left (3\,\ln \left (\ln \left (5\right )+{\mathrm {e}}^x-{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\right )-x\right )}{\ln \left (5\right )+{\mathrm {e}}^x-{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}}+\frac {x\,{\mathrm {e}}^x\,\ln \left (3\,\ln \left (\ln \left (5\right )+{\mathrm {e}}^x-{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\right )-x\right )}{\ln \left (5\right )+{\mathrm {e}}^x-{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}}-\frac {x\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\,\ln \left (3\,\ln \left (\ln \left (5\right )+{\mathrm {e}}^x-{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}\right )-x\right )}{\ln \left (5\right )+{\mathrm {e}}^x-{\mathrm {e}}^{2\,x\,{\mathrm {e}}^x}\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}}} \]
int((exp(exp(2*x) + 2*x*exp(x) + x^2)*(6*x*exp(2*x) + exp(x)*(6*x + 6*x^2) + 6*x^2) + log(3*log(log(5*exp(-exp(exp(2*x) + 2*x*exp(x) + x^2))) + exp( x)) - x)*(x*exp(x) - log(log(5*exp(-exp(exp(2*x) + 2*x*exp(x) + x^2))) + e xp(x))*(3*log(5*exp(-exp(exp(2*x) + 2*x*exp(x) + x^2))) + 3*exp(x)) + x*lo g(5*exp(-exp(exp(2*x) + 2*x*exp(x) + x^2)))) - 2*x*exp(x) + x*log(5*exp(-e xp(exp(2*x) + 2*x*exp(x) + x^2))))/(x*exp(x) - log(log(5*exp(-exp(exp(2*x) + 2*x*exp(x) + x^2))) + exp(x))*(3*log(5*exp(-exp(exp(2*x) + 2*x*exp(x) + x^2))) + 3*exp(x)) + x*log(5*exp(-exp(exp(2*x) + 2*x*exp(x) + x^2)))),x)
(x*log(5)*log(3*log(log(5) + exp(x) - exp(2*x*exp(x))*exp(x^2)*exp(exp(2*x ))) - x))/(log(5) + exp(x) - exp(2*x*exp(x))*exp(x^2)*exp(exp(2*x))) + (x* exp(x)*log(3*log(log(5) + exp(x) - exp(2*x*exp(x))*exp(x^2)*exp(exp(2*x))) - x))/(log(5) + exp(x) - exp(2*x*exp(x))*exp(x^2)*exp(exp(2*x))) - (x*exp (2*x*exp(x))*exp(x^2)*exp(exp(2*x))*log(3*log(log(5) + exp(x) - exp(2*x*ex p(x))*exp(x^2)*exp(exp(2*x))) - x))/(log(5) + exp(x) - exp(2*x*exp(x))*exp (x^2)*exp(exp(2*x)))