Integrand size = 168, antiderivative size = 38 \[ \int \frac {e^{\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 e^x-2 x}} \left (-2 e^{3 x}-4 x^2-3 x^3+e^{2 x} (-4+3 x)+e^x \left (8 x+2 x^2\right )+\left (e^{2 x} (-4-2 x)-4 x^2-6 x^3+e^x \left (8 x+10 x^2-2 x^3\right )\right ) \log (x)\right )}{2 e^{2 x}-4 e^x x+2 x^2} \, dx=e^{4-e^x+x^2 \left (\frac {1}{2} \left (-1-\frac {4}{x}\right )+\frac {x}{e^x-x}\right ) \log (x)} \] Output:
exp(4-exp(x)+x^2*(x/(exp(x)-x)-2/x-1/2)*ln(x))
Time = 0.15 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.11 \[ \int \frac {e^{\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 e^x-2 x}} \left (-2 e^{3 x}-4 x^2-3 x^3+e^{2 x} (-4+3 x)+e^x \left (8 x+2 x^2\right )+\left (e^{2 x} (-4-2 x)-4 x^2-6 x^3+e^x \left (8 x+10 x^2-2 x^3\right )\right ) \log (x)\right )}{2 e^{2 x}-4 e^x x+2 x^2} \, dx=e^{4-e^x} x^{\frac {x \left (-e^x (4+x)+x (4+3 x)\right )}{2 \left (e^x-x\right )}} \] Input:
Integrate[(E^((-2*E^(2*x) - 8*x + E^x*(8 + 2*x) + (4*x^2 + 3*x^3 + E^x*(-4 *x - x^2))*Log[x])/(2*E^x - 2*x))*(-2*E^(3*x) - 4*x^2 - 3*x^3 + E^(2*x)*(- 4 + 3*x) + E^x*(8*x + 2*x^2) + (E^(2*x)*(-4 - 2*x) - 4*x^2 - 6*x^3 + E^x*( 8*x + 10*x^2 - 2*x^3))*Log[x]))/(2*E^(2*x) - 4*E^x*x + 2*x^2),x]
Output:
E^(4 - E^x)*x^((x*(-(E^x*(4 + x)) + x*(4 + 3*x)))/(2*(E^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.
| \(\displaystyle \int \frac {\left (-3 x^3-4 x^2+e^x \left (2 x^2+8 x\right )+\left (-6 x^3-4 x^2+e^x \left (-2 x^3+10 x^2+8 x\right )+e^{2 x} (-2 x-4)\right ) \log (x)-2 e^{3 x}+e^{2 x} (3 x-4)\right ) \exp \left (\frac {\left (3 x^3+4 x^2+e^x \left (-x^2-4 x\right )\right ) \log (x)-8 x-2 e^{2 x}+e^x (2 x+8)}{2 e^x-2 x}\right )}{2 x^2-4 e^x x+2 e^{2 x}} \, dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle \int \frac {\left (-3 x^3-4 x^2+e^x \left (2 x^2+8 x\right )+\left (-6 x^3-4 x^2+e^x \left (-2 x^3+10 x^2+8 x\right )+e^{2 x} (-2 x-4)\right ) \log (x)-2 e^{3 x}+e^{2 x} (3 x-4)\right ) \exp \left (\frac {\left (3 x^3+4 x^2+e^x \left (-x^2-4 x\right )\right ) \log (x)-8 x-2 e^{2 x}+e^x (2 x+8)}{2 \left (e^x-x\right )}\right )}{2 \left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \int -\frac {e^{-\frac {4 x+e^{2 x}-e^x (x+4)}{e^x-x}} x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}} \left (3 x^3+4 x^2+2 e^{3 x}+e^{2 x} (4-3 x)-2 e^x \left (x^2+4 x\right )+2 \left (3 x^3+2 x^2+e^{2 x} (x+2)-e^x \left (-x^3+5 x^2+4 x\right )\right ) \log (x)\right )}{\left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {1}{2} \int \frac {e^{-\frac {4 x+e^{2 x}-e^x (x+4)}{e^x-x}} x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}} \left (3 x^3+4 x^2+2 e^{3 x}+e^{2 x} (4-3 x)-2 e^x \left (x^2+4 x\right )+2 \left (3 x^3+2 x^2+e^{2 x} (x+2)-e^x \left (-x^3+5 x^2+4 x\right )\right ) \log (x)\right )}{\left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 7292 |
| \(\displaystyle -\frac {1}{2} \int \frac {e^{4-e^x} x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}} \left (3 x^3+4 x^2+2 e^{3 x}+e^{2 x} (4-3 x)-2 e^x \left (x^2+4 x\right )+2 \left (3 x^3+2 x^2+e^{2 x} (x+2)-e^x \left (-x^3+5 x^2+4 x\right )\right ) \log (x)\right )}{\left (e^x-x\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {1}{2} \int \left (4 e^{4-e^x} x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}}+2 e^{x-e^x+4} x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}}+4 e^{4-e^x} \log (x) x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}}+e^{4-e^x} x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}+1}+2 e^{4-e^x} \log (x) x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}+1}+\frac {2 e^{4-e^x} (x \log (x)-3 \log (x)-1) x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}+2}}{e^x-x}+\frac {2 e^{4-e^x} (x-1) \log (x) x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}+3}}{\left (e^x-x\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-4 \log (x) \int e^{4-e^x} x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}}dx-4 \int e^{4-e^x} x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}}dx-2 \int e^{x-e^x+4} x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}}dx-2 \log (x) \int e^{4-e^x} x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}+1}dx-\int e^{4-e^x} x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}+1}dx+6 \log (x) \int \frac {e^{4-e^x} x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}+2}}{e^x-x}dx+2 \int \frac {e^{4-e^x} x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}+2}}{e^x-x}dx+2 \log (x) \int \frac {e^{4-e^x} x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}+3}}{\left (e^x-x\right )^2}dx-2 \log (x) \int \frac {e^{4-e^x} x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}+3}}{e^x-x}dx-2 \log (x) \int \frac {e^{4-e^x} x^{\frac {3 x^3+4 x^2-e^x \left (x^2+4 x\right )}{2 \left (e^x-x\right )}+4}}{\left (e^x-x\right )^2}dx+4 \int \frac {\int e^{4-e^x} x^{\frac {x \left (x (3 x+4)-e^x (x+4)\right )}{2 \left (e^x-x\right )}}dx}{x}dx+2 \int \frac {\int e^{4-e^x} x^{\frac {x \left (x (3 x+4)-e^x (x+4)\right )}{2 \left (e^x-x\right )}+1}dx}{x}dx-6 \int \frac {\int \frac {e^{4-e^x} x^{\frac {x \left (x (3 x+4)-e^x (x+4)\right )}{2 \left (e^x-x\right )}+2}}{e^x-x}dx}{x}dx-2 \int \frac {\int \frac {e^{4-e^x} x^{\frac {x \left (x (3 x+4)-e^x (x+4)\right )}{2 \left (e^x-x\right )}+3}}{\left (e^x-x\right )^2}dx}{x}dx+2 \int \frac {\int \frac {e^{4-e^x} x^{\frac {x \left (x (3 x+4)-e^x (x+4)\right )}{2 \left (e^x-x\right )}+3}}{e^x-x}dx}{x}dx+2 \int \frac {\int \frac {e^{4-e^x} x^{\frac {x \left (x (3 x+4)-e^x (x+4)\right )}{2 \left (e^x-x\right )}+4}}{\left (e^x-x\right )^2}dx}{x}dx\right )\) |
Input:
Int[(E^((-2*E^(2*x) - 8*x + E^x*(8 + 2*x) + (4*x^2 + 3*x^3 + E^x*(-4*x - x ^2))*Log[x])/(2*E^x - 2*x))*(-2*E^(3*x) - 4*x^2 - 3*x^3 + E^(2*x)*(-4 + 3* x) + E^x*(8*x + 2*x^2) + (E^(2*x)*(-4 - 2*x) - 4*x^2 - 6*x^3 + E^x*(8*x + 10*x^2 - 2*x^3))*Log[x]))/(2*E^(2*x) - 4*E^x*x + 2*x^2),x]
Output:
$Aborted
Time = 59.77 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.47
| method | result | size |
| parallelrisch | \({\mathrm e}^{\frac {\left (\left (-x^{2}-4 x \right ) {\mathrm e}^{x}+3 x^{3}+4 x^{2}\right ) \ln \left (x \right )-2 \,{\mathrm e}^{2 x}+\left (2 x +8\right ) {\mathrm e}^{x}-8 x}{2 \,{\mathrm e}^{x}-2 x}}\) | \(56\) |
| risch | \({\mathrm e}^{-\frac {x^{2} {\mathrm e}^{x} \ln \left (x \right )-3 x^{3} \ln \left (x \right )+4 x \,{\mathrm e}^{x} \ln \left (x \right )-4 x^{2} \ln \left (x \right )-2 \,{\mathrm e}^{x} x +2 \,{\mathrm e}^{2 x}-8 \,{\mathrm e}^{x}+8 x}{2 \left ({\mathrm e}^{x}-x \right )}}\) | \(60\) |
Input:
int((((-2*x-4)*exp(x)^2+(-2*x^3+10*x^2+8*x)*exp(x)-6*x^3-4*x^2)*ln(x)-2*ex p(x)^3+(-4+3*x)*exp(x)^2+(2*x^2+8*x)*exp(x)-3*x^3-4*x^2)*exp((((-x^2-4*x)* exp(x)+3*x^3+4*x^2)*ln(x)-2*exp(x)^2+(2*x+8)*exp(x)-8*x)/(2*exp(x)-2*x))/( 2*exp(x)^2-4*exp(x)*x+2*x^2),x,method=_RETURNVERBOSE)
Output:
exp(1/2/(exp(x)-x)*(((-x^2-4*x)*exp(x)+3*x^3+4*x^2)*ln(x)-2*exp(x)^2+(2*x+ 8)*exp(x)-8*x))
Time = 0.09 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.39 \[ \int \frac {e^{\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 e^x-2 x}} \left (-2 e^{3 x}-4 x^2-3 x^3+e^{2 x} (-4+3 x)+e^x \left (8 x+2 x^2\right )+\left (e^{2 x} (-4-2 x)-4 x^2-6 x^3+e^x \left (8 x+10 x^2-2 x^3\right )\right ) \log (x)\right )}{2 e^{2 x}-4 e^x x+2 x^2} \, dx=e^{\left (-\frac {2 \, {\left (x + 4\right )} e^{x} + {\left (3 \, x^{3} + 4 \, x^{2} - {\left (x^{2} + 4 \, x\right )} e^{x}\right )} \log \left (x\right ) - 8 \, x - 2 \, e^{\left (2 \, x\right )}}{2 \, {\left (x - e^{x}\right )}}\right )} \] Input:
integrate((((-2*x-4)*exp(x)^2+(-2*x^3+10*x^2+8*x)*exp(x)-6*x^3-4*x^2)*log( x)-2*exp(x)^3+(-4+3*x)*exp(x)^2+(2*x^2+8*x)*exp(x)-3*x^3-4*x^2)*exp((((-x^ 2-4*x)*exp(x)+3*x^3+4*x^2)*log(x)-2*exp(x)^2+(2*x+8)*exp(x)-8*x)/(2*exp(x) -2*x))/(2*exp(x)^2-4*exp(x)*x+2*x^2),x, algorithm="fricas")
Output:
e^(-1/2*(2*(x + 4)*e^x + (3*x^3 + 4*x^2 - (x^2 + 4*x)*e^x)*log(x) - 8*x - 2*e^(2*x))/(x - e^x))
Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (26) = 52\).
Time = 0.75 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.39 \[ \int \frac {e^{\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 e^x-2 x}} \left (-2 e^{3 x}-4 x^2-3 x^3+e^{2 x} (-4+3 x)+e^x \left (8 x+2 x^2\right )+\left (e^{2 x} (-4-2 x)-4 x^2-6 x^3+e^x \left (8 x+10 x^2-2 x^3\right )\right ) \log (x)\right )}{2 e^{2 x}-4 e^x x+2 x^2} \, dx=e^{\frac {- 8 x + \left (2 x + 8\right ) e^{x} + \left (3 x^{3} + 4 x^{2} + \left (- x^{2} - 4 x\right ) e^{x}\right ) \log {\left (x \right )} - 2 e^{2 x}}{- 2 x + 2 e^{x}}} \] Input:
integrate((((-2*x-4)*exp(x)**2+(-2*x**3+10*x**2+8*x)*exp(x)-6*x**3-4*x**2) *ln(x)-2*exp(x)**3+(-4+3*x)*exp(x)**2+(2*x**2+8*x)*exp(x)-3*x**3-4*x**2)*e xp((((-x**2-4*x)*exp(x)+3*x**3+4*x**2)*ln(x)-2*exp(x)**2+(2*x+8)*exp(x)-8* x)/(2*exp(x)-2*x))/(2*exp(x)**2-4*exp(x)*x+2*x**2),x)
Output:
exp((-8*x + (2*x + 8)*exp(x) + (3*x**3 + 4*x**2 + (-x**2 - 4*x)*exp(x))*lo g(x) - 2*exp(2*x))/(-2*x + 2*exp(x)))
Time = 0.40 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.32 \[ \int \frac {e^{\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 e^x-2 x}} \left (-2 e^{3 x}-4 x^2-3 x^3+e^{2 x} (-4+3 x)+e^x \left (8 x+2 x^2\right )+\left (e^{2 x} (-4-2 x)-4 x^2-6 x^3+e^x \left (8 x+10 x^2-2 x^3\right )\right ) \log (x)\right )}{2 e^{2 x}-4 e^x x+2 x^2} \, dx=e^{\left (-\frac {3}{2} \, x^{2} \log \left (x\right ) - x e^{x} \log \left (x\right ) - 2 \, x \log \left (x\right ) - e^{\left (2 \, x\right )} \log \left (x\right ) - \frac {e^{\left (3 \, x\right )} \log \left (x\right )}{x - e^{x}} - e^{x} + 4\right )} \] Input:
integrate((((-2*x-4)*exp(x)^2+(-2*x^3+10*x^2+8*x)*exp(x)-6*x^3-4*x^2)*log( x)-2*exp(x)^3+(-4+3*x)*exp(x)^2+(2*x^2+8*x)*exp(x)-3*x^3-4*x^2)*exp((((-x^ 2-4*x)*exp(x)+3*x^3+4*x^2)*log(x)-2*exp(x)^2+(2*x+8)*exp(x)-8*x)/(2*exp(x) -2*x))/(2*exp(x)^2-4*exp(x)*x+2*x^2),x, algorithm="maxima")
Output:
e^(-3/2*x^2*log(x) - x*e^x*log(x) - 2*x*log(x) - e^(2*x)*log(x) - e^(3*x)* log(x)/(x - e^x) - e^x + 4)
Time = 0.32 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.58 \[ \int \frac {e^{\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 e^x-2 x}} \left (-2 e^{3 x}-4 x^2-3 x^3+e^{2 x} (-4+3 x)+e^x \left (8 x+2 x^2\right )+\left (e^{2 x} (-4-2 x)-4 x^2-6 x^3+e^x \left (8 x+10 x^2-2 x^3\right )\right ) \log (x)\right )}{2 e^{2 x}-4 e^x x+2 x^2} \, dx=e^{\left (-\frac {3 \, x^{3} \log \left (x\right ) - x^{2} e^{x} \log \left (x\right ) + 4 \, x^{2} \log \left (x\right ) - 4 \, x e^{x} \log \left (x\right ) + 2 \, x e^{x} - 8 \, x - 2 \, e^{\left (2 \, x\right )} + 8 \, e^{x}}{2 \, {\left (x - e^{x}\right )}}\right )} \] Input:
integrate((((-2*x-4)*exp(x)^2+(-2*x^3+10*x^2+8*x)*exp(x)-6*x^3-4*x^2)*log( x)-2*exp(x)^3+(-4+3*x)*exp(x)^2+(2*x^2+8*x)*exp(x)-3*x^3-4*x^2)*exp((((-x^ 2-4*x)*exp(x)+3*x^3+4*x^2)*log(x)-2*exp(x)^2+(2*x+8)*exp(x)-8*x)/(2*exp(x) -2*x))/(2*exp(x)^2-4*exp(x)*x+2*x^2),x, algorithm="giac")
Output:
e^(-1/2*(3*x^3*log(x) - x^2*e^x*log(x) + 4*x^2*log(x) - 4*x*e^x*log(x) + 2 *x*e^x - 8*x - 2*e^(2*x) + 8*e^x)/(x - e^x))
Time = 2.83 (sec) , antiderivative size = 98, normalized size of antiderivative = 2.58 \[ \int \frac {e^{\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 e^x-2 x}} \left (-2 e^{3 x}-4 x^2-3 x^3+e^{2 x} (-4+3 x)+e^x \left (8 x+2 x^2\right )+\left (e^{2 x} (-4-2 x)-4 x^2-6 x^3+e^x \left (8 x+10 x^2-2 x^3\right )\right ) \log (x)\right )}{2 e^{2 x}-4 e^x x+2 x^2} \, dx=x^{\frac {x^2\,{\mathrm {e}}^x+4\,x\,{\mathrm {e}}^x-4\,x^2-3\,x^3}{2\,\left (x-{\mathrm {e}}^x\right )}}\,{\mathrm {e}}^{\frac {8\,x}{2\,x-2\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {8\,{\mathrm {e}}^x}{2\,x-2\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{\frac {2\,{\mathrm {e}}^{2\,x}}{2\,x-2\,{\mathrm {e}}^x}}\,{\mathrm {e}}^{-\frac {2\,x\,{\mathrm {e}}^x}{2\,x-2\,{\mathrm {e}}^x}} \] Input:
int(-(exp((8*x + 2*exp(2*x) - log(x)*(4*x^2 - exp(x)*(4*x + x^2) + 3*x^3) - exp(x)*(2*x + 8))/(2*x - 2*exp(x)))*(2*exp(3*x) - exp(x)*(8*x + 2*x^2) + log(x)*(exp(2*x)*(2*x + 4) + 4*x^2 + 6*x^3 - exp(x)*(8*x + 10*x^2 - 2*x^3 )) - exp(2*x)*(3*x - 4) + 4*x^2 + 3*x^3))/(2*exp(2*x) - 4*x*exp(x) + 2*x^2 ),x)
Output:
x^((x^2*exp(x) + 4*x*exp(x) - 4*x^2 - 3*x^3)/(2*(x - exp(x))))*exp((8*x)/( 2*x - 2*exp(x)))*exp(-(8*exp(x))/(2*x - 2*exp(x)))*exp((2*exp(2*x))/(2*x - 2*exp(x)))*exp(-(2*x*exp(x))/(2*x - 2*exp(x)))
Time = 0.21 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.84 \[ \int \frac {e^{\frac {-2 e^{2 x}-8 x+e^x (8+2 x)+\left (4 x^2+3 x^3+e^x \left (-4 x-x^2\right )\right ) \log (x)}{2 e^x-2 x}} \left (-2 e^{3 x}-4 x^2-3 x^3+e^{2 x} (-4+3 x)+e^x \left (8 x+2 x^2\right )+\left (e^{2 x} (-4-2 x)-4 x^2-6 x^3+e^x \left (8 x+10 x^2-2 x^3\right )\right ) \log (x)\right )}{2 e^{2 x}-4 e^x x+2 x^2} \, dx=\frac {e^{\frac {3 \,\mathrm {log}\left (x \right ) x^{3}}{2 e^{x}-2 x}} e^{4}}{x^{2 x} e^{\frac {2 e^{2 x}+e^{x} \mathrm {log}\left (x \right ) x^{2}-2 e^{x} x}{2 e^{x}-2 x}}} \] Input:
int((((-2*x-4)*exp(x)^2+(-2*x^3+10*x^2+8*x)*exp(x)-6*x^3-4*x^2)*log(x)-2*e xp(x)^3+(-4+3*x)*exp(x)^2+(2*x^2+8*x)*exp(x)-3*x^3-4*x^2)*exp((((-x^2-4*x) *exp(x)+3*x^3+4*x^2)*log(x)-2*exp(x)^2+(2*x+8)*exp(x)-8*x)/(2*exp(x)-2*x)) /(2*exp(x)^2-4*exp(x)*x+2*x^2),x)
Output:
(e**((3*log(x)*x**3)/(2*e**x - 2*x))*e**4)/(x**(2*x)*e**((2*e**(2*x) + e** x*log(x)*x**2 - 2*e**x*x)/(2*e**x - 2*x)))