Integrand size = 216, antiderivative size = 32 \[ \int \frac {400+100 e^2+100 x^2+e^{\frac {e^{2-e^{e^x}}}{10+e^{2-e^{e^x}} (1+x)}} \left (e^{4-2 e^{e^x}} x^2+10 e^{2-e^{e^x}+e^x+x} x^2\right )+e^{2-e^{e^x}} \left (80+80 x+20 x^2+20 x^3+e^2 (20+20 x)\right )+e^{4-2 e^{e^x}} \left (4+8 x+5 x^2+2 x^3+x^4+e^2 \left (1+2 x+x^2\right )\right )}{100 x^2+e^{2-e^{e^x}} \left (20 x^2+20 x^3\right )+e^{4-2 e^{e^x}} \left (x^2+2 x^3+x^4\right )} \, dx=-e^{\frac {1}{1+10 e^{-2+e^{e^x}}+x}}-\frac {4+e^2}{x}+x \]
Time = 0.74 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.28 \[ \int \frac {400+100 e^2+100 x^2+e^{\frac {e^{2-e^{e^x}}}{10+e^{2-e^{e^x}} (1+x)}} \left (e^{4-2 e^{e^x}} x^2+10 e^{2-e^{e^x}+e^x+x} x^2\right )+e^{2-e^{e^x}} \left (80+80 x+20 x^2+20 x^3+e^2 (20+20 x)\right )+e^{4-2 e^{e^x}} \left (4+8 x+5 x^2+2 x^3+x^4+e^2 \left (1+2 x+x^2\right )\right )}{100 x^2+e^{2-e^{e^x}} \left (20 x^2+20 x^3\right )+e^{4-2 e^{e^x}} \left (x^2+2 x^3+x^4\right )} \, dx=-e^{\frac {e^2}{e^2+10 e^{e^{e^x}}+e^2 x}}+\frac {-4-e^2}{x}+x \]
Integrate[(400 + 100*E^2 + 100*x^2 + E^(E^(2 - E^E^x)/(10 + E^(2 - E^E^x)* (1 + x)))*(E^(4 - 2*E^E^x)*x^2 + 10*E^(2 - E^E^x + E^x + x)*x^2) + E^(2 - E^E^x)*(80 + 80*x + 20*x^2 + 20*x^3 + E^2*(20 + 20*x)) + E^(4 - 2*E^E^x)*( 4 + 8*x + 5*x^2 + 2*x^3 + x^4 + E^2*(1 + 2*x + x^2)))/(100*x^2 + E^(2 - E^ E^x)*(20*x^2 + 20*x^3) + E^(4 - 2*E^E^x)*(x^2 + 2*x^3 + x^4)),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.
| \(\displaystyle \int \frac {100 x^2+e^{\frac {e^{2-e^{e^x}}}{e^{2-e^{e^x}} (x+1)+10}} \left (e^{4-2 e^{e^x}} x^2+10 e^{x-e^{e^x}+e^x+2} x^2\right )+e^{2-e^{e^x}} \left (20 x^3+20 x^2+80 x+e^2 (20 x+20)+80\right )+e^{4-2 e^{e^x}} \left (x^4+2 x^3+5 x^2+e^2 \left (x^2+2 x+1\right )+8 x+4\right )+100 e^2+400}{100 x^2+e^{2-e^{e^x}} \left (20 x^3+20 x^2\right )+e^{4-2 e^{e^x}} \left (x^4+2 x^3+x^2\right )} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {e^{2 e^{e^x}} \left (100 x^2+e^{\frac {e^{2-e^{e^x}}}{e^{2-e^{e^x}} (x+1)+10}} \left (e^{4-2 e^{e^x}} x^2+10 e^{x-e^{e^x}+e^x+2} x^2\right )+e^{2-e^{e^x}} \left (20 x^3+20 x^2+80 x+e^2 (20 x+20)+80\right )+e^{4-2 e^{e^x}} \left (x^4+2 x^3+5 x^2+e^2 \left (x^2+2 x+1\right )+8 x+4\right )+400 \left (1+\frac {e^2}{4}\right )\right )}{x^2 \left (e^2 x+10 e^{e^{e^x}}+e^2\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{2 e^{e^x}-2 \left (e^{e^x}-2\right )} \left (x^2+e^2+4\right ) (x+1)^2}{x^2 \left (e^2 x+10 e^{e^{e^x}}+e^2\right )^2}+\frac {20 e^{e^{e^x}+2} \left (x^2+e^2+4\right ) (x+1)}{x^2 \left (e^2 x+10 e^{e^{e^x}}+e^2\right )^2}+\frac {100 \left (4+e^2\right ) e^{2 e^{e^x}}}{x^2 \left (e^2 x+10 e^{e^{e^x}}+e^2\right )^2}+\frac {100 e^{2 e^{e^x}}}{\left (e^2 x+10 e^{e^{e^x}}+e^2\right )^2}+\frac {e^{\frac {e^2}{e^2 (x+1)+10 e^{e^{e^x}}}+2} \left (10 e^{x+e^{e^x}+e^x}+e^2\right )}{\left (e^2 x+10 e^{e^{e^x}}+e^2\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle e^4 \left (4+e^2\right ) \int \frac {1}{x^2 \left (e^2 x+10 e^{e^{e^x}}+e^2\right )^2}dx+100 \left (4+e^2\right ) \int \frac {e^{2 e^{e^x}}}{x^2 \left (e^2 x+10 e^{e^{e^x}}+e^2\right )^2}dx+20 \left (4+e^2\right ) \int \frac {e^{2+e^{e^x}}}{x^2 \left (e^2 x+10 e^{e^{e^x}}+e^2\right )^2}dx+e^4 \int \frac {x^2}{\left (e^2 x+10 e^{e^{e^x}}+e^2\right )^2}dx+e^4 \left (5+e^2\right ) \int \frac {1}{\left (e^2 x+10 e^{e^{e^x}}+e^2\right )^2}dx+100 \int \frac {e^{2 e^{e^x}}}{\left (e^2 x+10 e^{e^{e^x}}+e^2\right )^2}dx+20 \int \frac {e^{2+e^{e^x}}}{\left (e^2 x+10 e^{e^{e^x}}+e^2\right )^2}dx+2 e^4 \left (4+e^2\right ) \int \frac {1}{x \left (e^2 x+10 e^{e^{e^x}}+e^2\right )^2}dx+20 \left (4+e^2\right ) \int \frac {e^{2+e^{e^x}}}{x \left (e^2 x+10 e^{e^{e^x}}+e^2\right )^2}dx+2 e^4 \int \frac {x}{\left (e^2 x+10 e^{e^{e^x}}+e^2\right )^2}dx+20 \int \frac {e^{2+e^{e^x}} x}{\left (e^2 x+10 e^{e^{e^x}}+e^2\right )^2}dx-e^{\frac {e^2}{e^2 (x+1)+10 e^{e^{e^x}}}}\) |
Int[(400 + 100*E^2 + 100*x^2 + E^(E^(2 - E^E^x)/(10 + E^(2 - E^E^x)*(1 + x )))*(E^(4 - 2*E^E^x)*x^2 + 10*E^(2 - E^E^x + E^x + x)*x^2) + E^(2 - E^E^x) *(80 + 80*x + 20*x^2 + 20*x^3 + E^2*(20 + 20*x)) + E^(4 - 2*E^E^x)*(4 + 8* x + 5*x^2 + 2*x^3 + x^4 + E^2*(1 + 2*x + x^2)))/(100*x^2 + E^(2 - E^E^x)*( 20*x^2 + 20*x^3) + E^(4 - 2*E^E^x)*(x^2 + 2*x^3 + x^4)),x]
3.24.76.3.1 Defintions of rubi rules used
Time = 0.14 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.53
\[x -\frac {{\mathrm e}^{2}}{x}-\frac {4}{x}-{\mathrm e}^{\frac {{\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x}}+2}}{{\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x}}+2} x +{\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{x}}+2}+10}}\]
int(((x^2*exp(-exp(exp(x))+2)^2+10*x^2*exp(x)*exp(exp(x))*exp(-exp(exp(x)) +2))*exp(exp(-exp(exp(x))+2)/((1+x)*exp(-exp(exp(x))+2)+10))+((x^2+2*x+1)* exp(2)+x^4+2*x^3+5*x^2+8*x+4)*exp(-exp(exp(x))+2)^2+((20*x+20)*exp(2)+20*x ^3+20*x^2+80*x+80)*exp(-exp(exp(x))+2)+100*exp(2)+100*x^2+400)/((x^4+2*x^3 +x^2)*exp(-exp(exp(x))+2)^2+(20*x^3+20*x^2)*exp(-exp(exp(x))+2)+100*x^2),x )
Time = 0.26 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.69 \[ \int \frac {400+100 e^2+100 x^2+e^{\frac {e^{2-e^{e^x}}}{10+e^{2-e^{e^x}} (1+x)}} \left (e^{4-2 e^{e^x}} x^2+10 e^{2-e^{e^x}+e^x+x} x^2\right )+e^{2-e^{e^x}} \left (80+80 x+20 x^2+20 x^3+e^2 (20+20 x)\right )+e^{4-2 e^{e^x}} \left (4+8 x+5 x^2+2 x^3+x^4+e^2 \left (1+2 x+x^2\right )\right )}{100 x^2+e^{2-e^{e^x}} \left (20 x^2+20 x^3\right )+e^{4-2 e^{e^x}} \left (x^2+2 x^3+x^4\right )} \, dx=\frac {x^{2} - x e^{\left (\frac {e^{\left (x + e^{x} - e^{\left (e^{x}\right )} + 2\right )}}{{\left (x + 1\right )} e^{\left (x + e^{x} - e^{\left (e^{x}\right )} + 2\right )} + 10 \, e^{\left (x + e^{x}\right )}}\right )} - e^{2} - 4}{x} \]
integrate(((x^2*exp(-exp(exp(x))+2)^2+10*x^2*exp(x)*exp(exp(x))*exp(-exp(e xp(x))+2))*exp(exp(-exp(exp(x))+2)/((1+x)*exp(-exp(exp(x))+2)+10))+((x^2+2 *x+1)*exp(2)+x^4+2*x^3+5*x^2+8*x+4)*exp(-exp(exp(x))+2)^2+((20*x+20)*exp(2 )+20*x^3+20*x^2+80*x+80)*exp(-exp(exp(x))+2)+100*exp(2)+100*x^2+400)/((x^4 +2*x^3+x^2)*exp(-exp(exp(x))+2)^2+(20*x^3+20*x^2)*exp(-exp(exp(x))+2)+100* x^2),x, algorithm=\
(x^2 - x*e^(e^(x + e^x - e^(e^x) + 2)/((x + 1)*e^(x + e^x - e^(e^x) + 2) + 10*e^(x + e^x))) - e^2 - 4)/x
Time = 1.60 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.00 \[ \int \frac {400+100 e^2+100 x^2+e^{\frac {e^{2-e^{e^x}}}{10+e^{2-e^{e^x}} (1+x)}} \left (e^{4-2 e^{e^x}} x^2+10 e^{2-e^{e^x}+e^x+x} x^2\right )+e^{2-e^{e^x}} \left (80+80 x+20 x^2+20 x^3+e^2 (20+20 x)\right )+e^{4-2 e^{e^x}} \left (4+8 x+5 x^2+2 x^3+x^4+e^2 \left (1+2 x+x^2\right )\right )}{100 x^2+e^{2-e^{e^x}} \left (20 x^2+20 x^3\right )+e^{4-2 e^{e^x}} \left (x^2+2 x^3+x^4\right )} \, dx=x - e^{\frac {e^{2 - e^{e^{x}}}}{\left (x + 1\right ) e^{2 - e^{e^{x}}} + 10}} + \frac {- e^{2} - 4}{x} \]
integrate(((x**2*exp(-exp(exp(x))+2)**2+10*x**2*exp(x)*exp(exp(x))*exp(-ex p(exp(x))+2))*exp(exp(-exp(exp(x))+2)/((1+x)*exp(-exp(exp(x))+2)+10))+((x* *2+2*x+1)*exp(2)+x**4+2*x**3+5*x**2+8*x+4)*exp(-exp(exp(x))+2)**2+((20*x+2 0)*exp(2)+20*x**3+20*x**2+80*x+80)*exp(-exp(exp(x))+2)+100*exp(2)+100*x**2 +400)/((x**4+2*x**3+x**2)*exp(-exp(exp(x))+2)**2+(20*x**3+20*x**2)*exp(-ex p(exp(x))+2)+100*x**2),x)
Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {400+100 e^2+100 x^2+e^{\frac {e^{2-e^{e^x}}}{10+e^{2-e^{e^x}} (1+x)}} \left (e^{4-2 e^{e^x}} x^2+10 e^{2-e^{e^x}+e^x+x} x^2\right )+e^{2-e^{e^x}} \left (80+80 x+20 x^2+20 x^3+e^2 (20+20 x)\right )+e^{4-2 e^{e^x}} \left (4+8 x+5 x^2+2 x^3+x^4+e^2 \left (1+2 x+x^2\right )\right )}{100 x^2+e^{2-e^{e^x}} \left (20 x^2+20 x^3\right )+e^{4-2 e^{e^x}} \left (x^2+2 x^3+x^4\right )} \, dx=\frac {x^{2} - x e^{\left (\frac {e^{2}}{x e^{2} + e^{2} + 10 \, e^{\left (e^{\left (e^{x}\right )}\right )}}\right )} - e^{2} - 4}{x} \]
integrate(((x^2*exp(-exp(exp(x))+2)^2+10*x^2*exp(x)*exp(exp(x))*exp(-exp(e xp(x))+2))*exp(exp(-exp(exp(x))+2)/((1+x)*exp(-exp(exp(x))+2)+10))+((x^2+2 *x+1)*exp(2)+x^4+2*x^3+5*x^2+8*x+4)*exp(-exp(exp(x))+2)^2+((20*x+20)*exp(2 )+20*x^3+20*x^2+80*x+80)*exp(-exp(exp(x))+2)+100*exp(2)+100*x^2+400)/((x^4 +2*x^3+x^2)*exp(-exp(exp(x))+2)^2+(20*x^3+20*x^2)*exp(-exp(exp(x))+2)+100* x^2),x, algorithm=\
\[ \int \frac {400+100 e^2+100 x^2+e^{\frac {e^{2-e^{e^x}}}{10+e^{2-e^{e^x}} (1+x)}} \left (e^{4-2 e^{e^x}} x^2+10 e^{2-e^{e^x}+e^x+x} x^2\right )+e^{2-e^{e^x}} \left (80+80 x+20 x^2+20 x^3+e^2 (20+20 x)\right )+e^{4-2 e^{e^x}} \left (4+8 x+5 x^2+2 x^3+x^4+e^2 \left (1+2 x+x^2\right )\right )}{100 x^2+e^{2-e^{e^x}} \left (20 x^2+20 x^3\right )+e^{4-2 e^{e^x}} \left (x^2+2 x^3+x^4\right )} \, dx=\int { \frac {100 \, x^{2} + {\left (10 \, x^{2} e^{\left (x + e^{x} - e^{\left (e^{x}\right )} + 2\right )} + x^{2} e^{\left (-2 \, e^{\left (e^{x}\right )} + 4\right )}\right )} e^{\left (\frac {e^{\left (-e^{\left (e^{x}\right )} + 2\right )}}{{\left (x + 1\right )} e^{\left (-e^{\left (e^{x}\right )} + 2\right )} + 10}\right )} + 20 \, {\left (x^{3} + x^{2} + {\left (x + 1\right )} e^{2} + 4 \, x + 4\right )} e^{\left (-e^{\left (e^{x}\right )} + 2\right )} + {\left (x^{4} + 2 \, x^{3} + 5 \, x^{2} + {\left (x^{2} + 2 \, x + 1\right )} e^{2} + 8 \, x + 4\right )} e^{\left (-2 \, e^{\left (e^{x}\right )} + 4\right )} + 100 \, e^{2} + 400}{100 \, x^{2} + 20 \, {\left (x^{3} + x^{2}\right )} e^{\left (-e^{\left (e^{x}\right )} + 2\right )} + {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} e^{\left (-2 \, e^{\left (e^{x}\right )} + 4\right )}} \,d x } \]
integrate(((x^2*exp(-exp(exp(x))+2)^2+10*x^2*exp(x)*exp(exp(x))*exp(-exp(e xp(x))+2))*exp(exp(-exp(exp(x))+2)/((1+x)*exp(-exp(exp(x))+2)+10))+((x^2+2 *x+1)*exp(2)+x^4+2*x^3+5*x^2+8*x+4)*exp(-exp(exp(x))+2)^2+((20*x+20)*exp(2 )+20*x^3+20*x^2+80*x+80)*exp(-exp(exp(x))+2)+100*exp(2)+100*x^2+400)/((x^4 +2*x^3+x^2)*exp(-exp(exp(x))+2)^2+(20*x^3+20*x^2)*exp(-exp(exp(x))+2)+100* x^2),x, algorithm=\
integrate((100*x^2 + (10*x^2*e^(x + e^x - e^(e^x) + 2) + x^2*e^(-2*e^(e^x) + 4))*e^(e^(-e^(e^x) + 2)/((x + 1)*e^(-e^(e^x) + 2) + 10)) + 20*(x^3 + x^ 2 + (x + 1)*e^2 + 4*x + 4)*e^(-e^(e^x) + 2) + (x^4 + 2*x^3 + 5*x^2 + (x^2 + 2*x + 1)*e^2 + 8*x + 4)*e^(-2*e^(e^x) + 4) + 100*e^2 + 400)/(100*x^2 + 2 0*(x^3 + x^2)*e^(-e^(e^x) + 2) + (x^4 + 2*x^3 + x^2)*e^(-2*e^(e^x) + 4)), x)
Time = 12.22 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.44 \[ \int \frac {400+100 e^2+100 x^2+e^{\frac {e^{2-e^{e^x}}}{10+e^{2-e^{e^x}} (1+x)}} \left (e^{4-2 e^{e^x}} x^2+10 e^{2-e^{e^x}+e^x+x} x^2\right )+e^{2-e^{e^x}} \left (80+80 x+20 x^2+20 x^3+e^2 (20+20 x)\right )+e^{4-2 e^{e^x}} \left (4+8 x+5 x^2+2 x^3+x^4+e^2 \left (1+2 x+x^2\right )\right )}{100 x^2+e^{2-e^{e^x}} \left (20 x^2+20 x^3\right )+e^{4-2 e^{e^x}} \left (x^2+2 x^3+x^4\right )} \, dx=x-{\mathrm {e}}^{\frac {{\mathrm {e}}^{-{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^2}{{\mathrm {e}}^{-{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^2+x\,{\mathrm {e}}^{-{\mathrm {e}}^{{\mathrm {e}}^x}}\,{\mathrm {e}}^2+10}}-\frac {{\mathrm {e}}^2+4}{x} \]
int((100*exp(2) + exp(4 - 2*exp(exp(x)))*(8*x + exp(2)*(2*x + x^2 + 1) + 5 *x^2 + 2*x^3 + x^4 + 4) + exp(2 - exp(exp(x)))*(80*x + 20*x^2 + 20*x^3 + e xp(2)*(20*x + 20) + 80) + exp(exp(2 - exp(exp(x)))/(exp(2 - exp(exp(x)))*( x + 1) + 10))*(x^2*exp(4 - 2*exp(exp(x))) + 10*x^2*exp(exp(x))*exp(2 - exp (exp(x)))*exp(x)) + 100*x^2 + 400)/(exp(4 - 2*exp(exp(x)))*(x^2 + 2*x^3 + x^4) + exp(2 - exp(exp(x)))*(20*x^2 + 20*x^3) + 100*x^2),x)