Integrand size = 342, antiderivative size = 33 \[ \int \frac {x^2+2 x^3+x^4+e^8 \left (1+2 x+x^2\right )+e^{\frac {4}{e^4+x}} \left (e^8+2 e^4 x+x^2\right )+e^4 \left (2 x+4 x^2+2 x^3\right )+e^{\frac {2}{e^4+x}} \left (e^8 (-2-2 x)-2 x^2-2 x^3+e^4 \left (-4 x-4 x^2\right )\right )+e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}} \left (2 x^4+x^5+e^8 \left (2 x^2+x^3\right )+e^{\frac {2}{e^4+x}} \left (-2 e^8 x^2-2 x^3-4 e^4 x^3-2 x^4\right )+e^4 \left (4 x^3+2 x^4\right )\right )}{x^3+2 x^4+x^5+e^8 \left (x+2 x^2+x^3\right )+e^{\frac {4}{e^4+x}} \left (e^8 x+2 e^4 x^2+x^3\right )+e^4 \left (2 x^2+4 x^3+2 x^4\right )+e^{\frac {2}{e^4+x}} \left (-2 x^3-2 x^4+e^8 \left (-2 x-2 x^2\right )+e^4 \left (-4 x^2-4 x^3\right )\right )} \, dx=-e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}}+\log \left (-\frac {9 x}{5}\right ) \]
Time = 0.85 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {x^2+2 x^3+x^4+e^8 \left (1+2 x+x^2\right )+e^{\frac {4}{e^4+x}} \left (e^8+2 e^4 x+x^2\right )+e^4 \left (2 x+4 x^2+2 x^3\right )+e^{\frac {2}{e^4+x}} \left (e^8 (-2-2 x)-2 x^2-2 x^3+e^4 \left (-4 x-4 x^2\right )\right )+e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}} \left (2 x^4+x^5+e^8 \left (2 x^2+x^3\right )+e^{\frac {2}{e^4+x}} \left (-2 e^8 x^2-2 x^3-4 e^4 x^3-2 x^4\right )+e^4 \left (4 x^3+2 x^4\right )\right )}{x^3+2 x^4+x^5+e^8 \left (x+2 x^2+x^3\right )+e^{\frac {4}{e^4+x}} \left (e^8 x+2 e^4 x^2+x^3\right )+e^4 \left (2 x^2+4 x^3+2 x^4\right )+e^{\frac {2}{e^4+x}} \left (-2 x^3-2 x^4+e^8 \left (-2 x-2 x^2\right )+e^4 \left (-4 x^2-4 x^3\right )\right )} \, dx=-e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}}+\log (x) \]
Integrate[(x^2 + 2*x^3 + x^4 + E^8*(1 + 2*x + x^2) + E^(4/(E^4 + x))*(E^8 + 2*E^4*x + x^2) + E^4*(2*x + 4*x^2 + 2*x^3) + E^(2/(E^4 + x))*(E^8*(-2 - 2*x) - 2*x^2 - 2*x^3 + E^4*(-4*x - 4*x^2)) + E^(x^2/(-1 + E^(2/(E^4 + x)) - x))*(2*x^4 + x^5 + E^8*(2*x^2 + x^3) + E^(2/(E^4 + x))*(-2*E^8*x^2 - 2*x ^3 - 4*E^4*x^3 - 2*x^4) + E^4*(4*x^3 + 2*x^4)))/(x^3 + 2*x^4 + x^5 + E^8*( x + 2*x^2 + x^3) + E^(4/(E^4 + x))*(E^8*x + 2*E^4*x^2 + x^3) + E^4*(2*x^2 + 4*x^3 + 2*x^4) + E^(2/(E^4 + x))*(-2*x^3 - 2*x^4 + E^8*(-2*x - 2*x^2) + E^4*(-4*x^2 - 4*x^3))),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.
| \(\displaystyle \int \frac {x^4+2 x^3+x^2+e^8 \left (x^2+2 x+1\right )+e^{\frac {4}{x+e^4}} \left (x^2+2 e^4 x+e^8\right )+e^4 \left (2 x^3+4 x^2+2 x\right )+e^{\frac {2}{x+e^4}} \left (-2 x^3-2 x^2+e^4 \left (-4 x^2-4 x\right )+e^8 (-2 x-2)\right )+e^{\frac {x^2}{-x+e^{\frac {2}{x+e^4}}-1}} \left (x^5+2 x^4+e^4 \left (2 x^4+4 x^3\right )+e^8 \left (x^3+2 x^2\right )+e^{\frac {2}{x+e^4}} \left (-2 x^4-4 e^4 x^3-2 x^3-2 e^8 x^2\right )\right )}{x^5+2 x^4+x^3+e^8 \left (x^3+2 x^2+x\right )+e^{\frac {4}{x+e^4}} \left (x^3+2 e^4 x^2+e^8 x\right )+e^4 \left (2 x^4+4 x^3+2 x^2\right )+e^{\frac {2}{x+e^4}} \left (-2 x^4-2 x^3+e^8 \left (-2 x^2-2 x\right )+e^4 \left (-4 x^3-4 x^2\right )\right )} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^4+2 x^3+x^2+e^{\frac {x^2}{-x+e^{\frac {2}{x+e^4}}-1}} \left (x^3+2 x^2-2 e^{\frac {2}{x+e^4}} \left (x^2+2 e^4 x+x+e^8\right )+2 e^4 (x+2) x+e^8 (x+2)\right ) x^2+2 e^4 (x+1)^2 x+e^8 (x+1)^2+e^{\frac {4}{x+e^4}} \left (x+e^4\right )^2-2 e^{\frac {2}{x+e^4}} (x+1) \left (x+e^4\right )^2}{x \left (x+e^4\right )^2 \left (x-e^{\frac {2}{x+e^4}}+1\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^{\frac {x^2}{-x+e^{\frac {2}{x+e^4}}-1}} \left (-x^2-2 \left (1+e^4\right ) x-e^8-2\right ) x^2}{\left (x+e^4\right )^2 \left (x-e^{\frac {2}{x+e^4}}+1\right )^2}+\frac {2 e^{\frac {x^2}{-x+e^{\frac {2}{x+e^4}}-1}} \left (x^2+\left (1+2 e^4\right ) x+e^8\right ) x}{\left (x+e^4\right )^2 \left (x-e^{\frac {2}{x+e^4}}+1\right )}+\frac {1}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -2 \left (1-2 e^4\right ) \int \frac {e^{\frac {x^2}{-x+e^{\frac {2}{x+e^4}}-1}}}{\left (-x+e^{\frac {2}{x+e^4}}-1\right )^2}dx-2 \int \frac {e^{\frac {x^2}{-x+e^{\frac {2}{x+e^4}}-1}}}{-x+e^{\frac {2}{x+e^4}}-1}dx-2 \int \frac {e^{\frac {x^2}{-x+e^{\frac {2}{x+e^4}}-1}} x}{\left (-x+e^{\frac {2}{x+e^4}}-1\right )^2}dx-\int \frac {e^{\frac {x^2}{-x+e^{\frac {2}{x+e^4}}-1}} x^2}{\left (-x+e^{\frac {2}{x+e^4}}-1\right )^2}dx-2 \left (1-e^4\right ) \int \frac {e^{\frac {x^2}{-x+e^{\frac {2}{x+e^4}}-1}+8}}{\left (-x+e^{\frac {2}{x+e^4}}-1\right )^2 \left (x+e^4\right )^2}dx-2 \int \frac {e^{\frac {x^2}{-x+e^{\frac {2}{x+e^4}}-1}+8}}{\left (-x+e^{\frac {2}{x+e^4}}-1\right ) \left (x+e^4\right )^2}dx+2 \left (2-3 e^4\right ) \int \frac {e^{\frac {x^2}{-x+e^{\frac {2}{x+e^4}}-1}+4}}{\left (-x+e^{\frac {2}{x+e^4}}-1\right )^2 \left (x+e^4\right )}dx+4 \int \frac {e^{\frac {x^2}{-x+e^{\frac {2}{x+e^4}}-1}+4}}{\left (-x+e^{\frac {2}{x+e^4}}-1\right ) \left (x+e^4\right )}dx+2 \int \frac {e^{\frac {x^2}{-x+e^{\frac {2}{x+e^4}}-1}} x}{x-e^{\frac {2}{x+e^4}}+1}dx+\log (x)\) |
Int[(x^2 + 2*x^3 + x^4 + E^8*(1 + 2*x + x^2) + E^(4/(E^4 + x))*(E^8 + 2*E^ 4*x + x^2) + E^4*(2*x + 4*x^2 + 2*x^3) + E^(2/(E^4 + x))*(E^8*(-2 - 2*x) - 2*x^2 - 2*x^3 + E^4*(-4*x - 4*x^2)) + E^(x^2/(-1 + E^(2/(E^4 + x)) - x))* (2*x^4 + x^5 + E^8*(2*x^2 + x^3) + E^(2/(E^4 + x))*(-2*E^8*x^2 - 2*x^3 - 4 *E^4*x^3 - 2*x^4) + E^4*(4*x^3 + 2*x^4)))/(x^3 + 2*x^4 + x^5 + E^8*(x + 2* x^2 + x^3) + E^(4/(E^4 + x))*(E^8*x + 2*E^4*x^2 + x^3) + E^4*(2*x^2 + 4*x^ 3 + 2*x^4) + E^(2/(E^4 + x))*(-2*x^3 - 2*x^4 + E^8*(-2*x - 2*x^2) + E^4*(- 4*x^2 - 4*x^3))),x]
3.16.40.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 19.69 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82
| method | result | size |
| parallelrisch | \(\ln \left (x \right )-{\mathrm e}^{\frac {x^{2}}{{\mathrm e}^{\frac {2}{x +{\mathrm e}^{4}}}-x -1}}\) | \(27\) |
| risch | \(\ln \left (x \right )-{\mathrm e}^{-\frac {x^{2}}{-{\mathrm e}^{\frac {2}{x +{\mathrm e}^{4}}}+x +1}}\) | \(28\) |
int((((-2*x^2*exp(4)^2-4*x^3*exp(4)-2*x^4-2*x^3)*exp(2/(x+exp(4)))+(x^3+2* x^2)*exp(4)^2+(2*x^4+4*x^3)*exp(4)+x^5+2*x^4)*exp(x^2/(exp(2/(x+exp(4)))-x -1))+(exp(4)^2+2*x*exp(4)+x^2)*exp(2/(x+exp(4)))^2+((-2-2*x)*exp(4)^2+(-4* x^2-4*x)*exp(4)-2*x^3-2*x^2)*exp(2/(x+exp(4)))+(x^2+2*x+1)*exp(4)^2+(2*x^3 +4*x^2+2*x)*exp(4)+x^4+2*x^3+x^2)/((x*exp(4)^2+2*x^2*exp(4)+x^3)*exp(2/(x+ exp(4)))^2+((-2*x^2-2*x)*exp(4)^2+(-4*x^3-4*x^2)*exp(4)-2*x^4-2*x^3)*exp(2 /(x+exp(4)))+(x^3+2*x^2+x)*exp(4)^2+(2*x^4+4*x^3+2*x^2)*exp(4)+x^5+2*x^4+x ^3),x,method=_RETURNVERBOSE)
Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {x^2+2 x^3+x^4+e^8 \left (1+2 x+x^2\right )+e^{\frac {4}{e^4+x}} \left (e^8+2 e^4 x+x^2\right )+e^4 \left (2 x+4 x^2+2 x^3\right )+e^{\frac {2}{e^4+x}} \left (e^8 (-2-2 x)-2 x^2-2 x^3+e^4 \left (-4 x-4 x^2\right )\right )+e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}} \left (2 x^4+x^5+e^8 \left (2 x^2+x^3\right )+e^{\frac {2}{e^4+x}} \left (-2 e^8 x^2-2 x^3-4 e^4 x^3-2 x^4\right )+e^4 \left (4 x^3+2 x^4\right )\right )}{x^3+2 x^4+x^5+e^8 \left (x+2 x^2+x^3\right )+e^{\frac {4}{e^4+x}} \left (e^8 x+2 e^4 x^2+x^3\right )+e^4 \left (2 x^2+4 x^3+2 x^4\right )+e^{\frac {2}{e^4+x}} \left (-2 x^3-2 x^4+e^8 \left (-2 x-2 x^2\right )+e^4 \left (-4 x^2-4 x^3\right )\right )} \, dx=-e^{\left (-\frac {x^{2}}{x - e^{\left (\frac {2}{x + e^{4}}\right )} + 1}\right )} + \log \left (x\right ) \]
integrate((((-2*x^2*exp(4)^2-4*x^3*exp(4)-2*x^4-2*x^3)*exp(2/(x+exp(4)))+( x^3+2*x^2)*exp(4)^2+(2*x^4+4*x^3)*exp(4)+x^5+2*x^4)*exp(x^2/(exp(2/(x+exp( 4)))-x-1))+(exp(4)^2+2*x*exp(4)+x^2)*exp(2/(x+exp(4)))^2+((-2-2*x)*exp(4)^ 2+(-4*x^2-4*x)*exp(4)-2*x^3-2*x^2)*exp(2/(x+exp(4)))+(x^2+2*x+1)*exp(4)^2+ (2*x^3+4*x^2+2*x)*exp(4)+x^4+2*x^3+x^2)/((x*exp(4)^2+2*x^2*exp(4)+x^3)*exp (2/(x+exp(4)))^2+((-2*x^2-2*x)*exp(4)^2+(-4*x^3-4*x^2)*exp(4)-2*x^4-2*x^3) *exp(2/(x+exp(4)))+(x^3+2*x^2+x)*exp(4)^2+(2*x^4+4*x^3+2*x^2)*exp(4)+x^5+2 *x^4+x^3),x, algorithm=\
Time = 0.52 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.58 \[ \int \frac {x^2+2 x^3+x^4+e^8 \left (1+2 x+x^2\right )+e^{\frac {4}{e^4+x}} \left (e^8+2 e^4 x+x^2\right )+e^4 \left (2 x+4 x^2+2 x^3\right )+e^{\frac {2}{e^4+x}} \left (e^8 (-2-2 x)-2 x^2-2 x^3+e^4 \left (-4 x-4 x^2\right )\right )+e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}} \left (2 x^4+x^5+e^8 \left (2 x^2+x^3\right )+e^{\frac {2}{e^4+x}} \left (-2 e^8 x^2-2 x^3-4 e^4 x^3-2 x^4\right )+e^4 \left (4 x^3+2 x^4\right )\right )}{x^3+2 x^4+x^5+e^8 \left (x+2 x^2+x^3\right )+e^{\frac {4}{e^4+x}} \left (e^8 x+2 e^4 x^2+x^3\right )+e^4 \left (2 x^2+4 x^3+2 x^4\right )+e^{\frac {2}{e^4+x}} \left (-2 x^3-2 x^4+e^8 \left (-2 x-2 x^2\right )+e^4 \left (-4 x^2-4 x^3\right )\right )} \, dx=- e^{\frac {x^{2}}{- x + e^{\frac {2}{x + e^{4}}} - 1}} + \log {\left (x \right )} \]
integrate((((-2*x**2*exp(4)**2-4*x**3*exp(4)-2*x**4-2*x**3)*exp(2/(x+exp(4 )))+(x**3+2*x**2)*exp(4)**2+(2*x**4+4*x**3)*exp(4)+x**5+2*x**4)*exp(x**2/( exp(2/(x+exp(4)))-x-1))+(exp(4)**2+2*x*exp(4)+x**2)*exp(2/(x+exp(4)))**2+( (-2-2*x)*exp(4)**2+(-4*x**2-4*x)*exp(4)-2*x**3-2*x**2)*exp(2/(x+exp(4)))+( x**2+2*x+1)*exp(4)**2+(2*x**3+4*x**2+2*x)*exp(4)+x**4+2*x**3+x**2)/((x*exp (4)**2+2*x**2*exp(4)+x**3)*exp(2/(x+exp(4)))**2+((-2*x**2-2*x)*exp(4)**2+( -4*x**3-4*x**2)*exp(4)-2*x**4-2*x**3)*exp(2/(x+exp(4)))+(x**3+2*x**2+x)*ex p(4)**2+(2*x**4+4*x**3+2*x**2)*exp(4)+x**5+2*x**4+x**3),x)
\[ \int \frac {x^2+2 x^3+x^4+e^8 \left (1+2 x+x^2\right )+e^{\frac {4}{e^4+x}} \left (e^8+2 e^4 x+x^2\right )+e^4 \left (2 x+4 x^2+2 x^3\right )+e^{\frac {2}{e^4+x}} \left (e^8 (-2-2 x)-2 x^2-2 x^3+e^4 \left (-4 x-4 x^2\right )\right )+e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}} \left (2 x^4+x^5+e^8 \left (2 x^2+x^3\right )+e^{\frac {2}{e^4+x}} \left (-2 e^8 x^2-2 x^3-4 e^4 x^3-2 x^4\right )+e^4 \left (4 x^3+2 x^4\right )\right )}{x^3+2 x^4+x^5+e^8 \left (x+2 x^2+x^3\right )+e^{\frac {4}{e^4+x}} \left (e^8 x+2 e^4 x^2+x^3\right )+e^4 \left (2 x^2+4 x^3+2 x^4\right )+e^{\frac {2}{e^4+x}} \left (-2 x^3-2 x^4+e^8 \left (-2 x-2 x^2\right )+e^4 \left (-4 x^2-4 x^3\right )\right )} \, dx=\int { \frac {x^{4} + 2 \, x^{3} + x^{2} + {\left (x^{2} + 2 \, x + 1\right )} e^{8} + 2 \, {\left (x^{3} + 2 \, x^{2} + x\right )} e^{4} + {\left (x^{5} + 2 \, x^{4} + {\left (x^{3} + 2 \, x^{2}\right )} e^{8} + 2 \, {\left (x^{4} + 2 \, x^{3}\right )} e^{4} - 2 \, {\left (x^{4} + 2 \, x^{3} e^{4} + x^{3} + x^{2} e^{8}\right )} e^{\left (\frac {2}{x + e^{4}}\right )}\right )} e^{\left (-\frac {x^{2}}{x - e^{\left (\frac {2}{x + e^{4}}\right )} + 1}\right )} + {\left (x^{2} + 2 \, x e^{4} + e^{8}\right )} e^{\left (\frac {4}{x + e^{4}}\right )} - 2 \, {\left (x^{3} + x^{2} + {\left (x + 1\right )} e^{8} + 2 \, {\left (x^{2} + x\right )} e^{4}\right )} e^{\left (\frac {2}{x + e^{4}}\right )}}{x^{5} + 2 \, x^{4} + x^{3} + {\left (x^{3} + 2 \, x^{2} + x\right )} e^{8} + 2 \, {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} e^{4} + {\left (x^{3} + 2 \, x^{2} e^{4} + x e^{8}\right )} e^{\left (\frac {4}{x + e^{4}}\right )} - 2 \, {\left (x^{4} + x^{3} + {\left (x^{2} + x\right )} e^{8} + 2 \, {\left (x^{3} + x^{2}\right )} e^{4}\right )} e^{\left (\frac {2}{x + e^{4}}\right )}} \,d x } \]
integrate((((-2*x^2*exp(4)^2-4*x^3*exp(4)-2*x^4-2*x^3)*exp(2/(x+exp(4)))+( x^3+2*x^2)*exp(4)^2+(2*x^4+4*x^3)*exp(4)+x^5+2*x^4)*exp(x^2/(exp(2/(x+exp( 4)))-x-1))+(exp(4)^2+2*x*exp(4)+x^2)*exp(2/(x+exp(4)))^2+((-2-2*x)*exp(4)^ 2+(-4*x^2-4*x)*exp(4)-2*x^3-2*x^2)*exp(2/(x+exp(4)))+(x^2+2*x+1)*exp(4)^2+ (2*x^3+4*x^2+2*x)*exp(4)+x^4+2*x^3+x^2)/((x*exp(4)^2+2*x^2*exp(4)+x^3)*exp (2/(x+exp(4)))^2+((-2*x^2-2*x)*exp(4)^2+(-4*x^3-4*x^2)*exp(4)-2*x^4-2*x^3) *exp(2/(x+exp(4)))+(x^3+2*x^2+x)*exp(4)^2+(2*x^4+4*x^3+2*x^2)*exp(4)+x^5+2 *x^4+x^3),x, algorithm=\
-integrate(-(x^4*e + 2*x^3*(e^5 + e) + x^2*(e^9 + 4*e^5) + 2*x*e^9 - 2*(x^ 3*e + x^2*(2*e^5 + e) + x*e^9)*e^(2/(x + e^4)))*e^(-e^(4/(x + e^4))/(x - e ^(2/(x + e^4)) + 1) + 2*e^(2/(x + e^4))/(x - e^(2/(x + e^4)) + 1) - 1/(x - e^(2/(x + e^4)) + 1) - e^(2/(x + e^4)))/((x^2 + 2*x*e^4 + e^8)*e^(x + 4/( x + e^4)) - 2*(x^3 + x^2*(2*e^4 + 1) + x*(e^8 + 2*e^4) + e^8)*e^(x + 2/(x + e^4)) + (x^4 + 2*x^3*(e^4 + 1) + x^2*(e^8 + 4*e^4 + 1) + 2*x*(e^8 + e^4) + e^8)*e^x), x) + log(x)
Timed out. \[ \int \frac {x^2+2 x^3+x^4+e^8 \left (1+2 x+x^2\right )+e^{\frac {4}{e^4+x}} \left (e^8+2 e^4 x+x^2\right )+e^4 \left (2 x+4 x^2+2 x^3\right )+e^{\frac {2}{e^4+x}} \left (e^8 (-2-2 x)-2 x^2-2 x^3+e^4 \left (-4 x-4 x^2\right )\right )+e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}} \left (2 x^4+x^5+e^8 \left (2 x^2+x^3\right )+e^{\frac {2}{e^4+x}} \left (-2 e^8 x^2-2 x^3-4 e^4 x^3-2 x^4\right )+e^4 \left (4 x^3+2 x^4\right )\right )}{x^3+2 x^4+x^5+e^8 \left (x+2 x^2+x^3\right )+e^{\frac {4}{e^4+x}} \left (e^8 x+2 e^4 x^2+x^3\right )+e^4 \left (2 x^2+4 x^3+2 x^4\right )+e^{\frac {2}{e^4+x}} \left (-2 x^3-2 x^4+e^8 \left (-2 x-2 x^2\right )+e^4 \left (-4 x^2-4 x^3\right )\right )} \, dx=\text {Timed out} \]
integrate((((-2*x^2*exp(4)^2-4*x^3*exp(4)-2*x^4-2*x^3)*exp(2/(x+exp(4)))+( x^3+2*x^2)*exp(4)^2+(2*x^4+4*x^3)*exp(4)+x^5+2*x^4)*exp(x^2/(exp(2/(x+exp( 4)))-x-1))+(exp(4)^2+2*x*exp(4)+x^2)*exp(2/(x+exp(4)))^2+((-2-2*x)*exp(4)^ 2+(-4*x^2-4*x)*exp(4)-2*x^3-2*x^2)*exp(2/(x+exp(4)))+(x^2+2*x+1)*exp(4)^2+ (2*x^3+4*x^2+2*x)*exp(4)+x^4+2*x^3+x^2)/((x*exp(4)^2+2*x^2*exp(4)+x^3)*exp (2/(x+exp(4)))^2+((-2*x^2-2*x)*exp(4)^2+(-4*x^3-4*x^2)*exp(4)-2*x^4-2*x^3) *exp(2/(x+exp(4)))+(x^3+2*x^2+x)*exp(4)^2+(2*x^4+4*x^3+2*x^2)*exp(4)+x^5+2 *x^4+x^3),x, algorithm=\
Time = 13.76 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.82 \[ \int \frac {x^2+2 x^3+x^4+e^8 \left (1+2 x+x^2\right )+e^{\frac {4}{e^4+x}} \left (e^8+2 e^4 x+x^2\right )+e^4 \left (2 x+4 x^2+2 x^3\right )+e^{\frac {2}{e^4+x}} \left (e^8 (-2-2 x)-2 x^2-2 x^3+e^4 \left (-4 x-4 x^2\right )\right )+e^{\frac {x^2}{-1+e^{\frac {2}{e^4+x}}-x}} \left (2 x^4+x^5+e^8 \left (2 x^2+x^3\right )+e^{\frac {2}{e^4+x}} \left (-2 e^8 x^2-2 x^3-4 e^4 x^3-2 x^4\right )+e^4 \left (4 x^3+2 x^4\right )\right )}{x^3+2 x^4+x^5+e^8 \left (x+2 x^2+x^3\right )+e^{\frac {4}{e^4+x}} \left (e^8 x+2 e^4 x^2+x^3\right )+e^4 \left (2 x^2+4 x^3+2 x^4\right )+e^{\frac {2}{e^4+x}} \left (-2 x^3-2 x^4+e^8 \left (-2 x-2 x^2\right )+e^4 \left (-4 x^2-4 x^3\right )\right )} \, dx=\ln \left (x\right )-{\mathrm {e}}^{-\frac {x^2}{x-{\mathrm {e}}^{\frac {2}{x+{\mathrm {e}}^4}}+1}} \]
int((exp(4)*(2*x + 4*x^2 + 2*x^3) + exp(4/(x + exp(4)))*(exp(8) + 2*x*exp( 4) + x^2) - exp(2/(x + exp(4)))*(exp(4)*(4*x + 4*x^2) + 2*x^2 + 2*x^3 + ex p(8)*(2*x + 2)) + exp(8)*(2*x + x^2 + 1) + x^2 + 2*x^3 + x^4 + exp(-x^2/(x - exp(2/(x + exp(4))) + 1))*(exp(8)*(2*x^2 + x^3) + exp(4)*(4*x^3 + 2*x^4 ) - exp(2/(x + exp(4)))*(4*x^3*exp(4) + 2*x^2*exp(8) + 2*x^3 + 2*x^4) + 2* x^4 + x^5))/(exp(8)*(x + 2*x^2 + x^3) - exp(2/(x + exp(4)))*(exp(8)*(2*x + 2*x^2) + exp(4)*(4*x^2 + 4*x^3) + 2*x^3 + 2*x^4) + exp(4)*(2*x^2 + 4*x^3 + 2*x^4) + exp(4/(x + exp(4)))*(x*exp(8) + 2*x^2*exp(4) + x^3) + x^3 + 2*x ^4 + x^5),x)