Integrand size = 74, antiderivative size = 34 \[ \int \frac {e^{-e^x-2 x} \left (10 e^{2 x}+5 e^{3 x} x-50 e^{e^x} x^3+e^{2 e^x+x} \left (e^{3 x} x^2+e^{2 x} \left (-x+x^2\right )\right )\right )}{x^3} \, dx=25 e^{-2 x}+\frac {e^{e^x+x}-\frac {5 e^{-e^x}}{x}}{x}+\log (3) \]
Time = 0.10 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.91 \[ \int \frac {e^{-e^x-2 x} \left (10 e^{2 x}+5 e^{3 x} x-50 e^{e^x} x^3+e^{2 e^x+x} \left (e^{3 x} x^2+e^{2 x} \left (-x+x^2\right )\right )\right )}{x^3} \, dx=25 e^{-2 x}-\frac {5 e^{-e^x}}{x^2}+\frac {e^{e^x+x}}{x} \]
Integrate[(E^(-E^x - 2*x)*(10*E^(2*x) + 5*E^(3*x)*x - 50*E^E^x*x^3 + E^(2* E^x + x)*(E^(3*x)*x^2 + E^(2*x)*(-x + x^2))))/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.
\(\displaystyle \int \frac {e^{-2 x-e^x} \left (-50 e^{e^x} x^3+e^{x+2 e^x} \left (e^{3 x} x^2+e^{2 x} \left (x^2-x\right )\right )+5 e^{3 x} x+10 e^{2 x}\right )}{x^3} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {10 e^{-e^x}}{x^3}+\frac {e^{x-e^x} \left (e^{2 e^x} x-e^{2 e^x}+5\right )}{x^2}-50 e^{-2 x}+\frac {e^{-2 x-e^x+2 \left (2 x+e^x\right )}}{x}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 10 \int \frac {e^{-e^x}}{x^3}dx+5 \int \frac {e^{x-e^x}}{x^2}dx-\int \frac {e^{x+e^x}}{x^2}dx+\int \frac {e^{x+e^x}}{x}dx+\int \frac {e^{2 x+e^x}}{x}dx+25 e^{-2 x}\) |
Int[(E^(-E^x - 2*x)*(10*E^(2*x) + 5*E^(3*x)*x - 50*E^E^x*x^3 + E^(2*E^x + x)*(E^(3*x)*x^2 + E^(2*x)*(-x + x^2))))/x^3,x]
3.30.97.3.1 Defintions of rubi rules used
Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.79
method | result | size |
risch | \(25 \,{\mathrm e}^{-2 x}+\frac {{\mathrm e}^{{\mathrm e}^{x}+x}}{x}-\frac {5 \,{\mathrm e}^{-{\mathrm e}^{x}}}{x^{2}}\) | \(27\) |
parallelrisch | \(\frac {\left ({\mathrm e}^{2 x} {\mathrm e}^{{\mathrm e}^{x}} {\mathrm e}^{{\mathrm e}^{x}+x} x +25 \,{\mathrm e}^{{\mathrm e}^{x}} x^{2}-5 \,{\mathrm e}^{2 x}\right ) {\mathrm e}^{-{\mathrm e}^{x}} {\mathrm e}^{-2 x}}{x^{2}}\) | \(43\) |
int(((x^2*exp(x)^3+(x^2-x)*exp(x)^2)*exp(exp(x))*exp(exp(x)+x)-50*x^3*exp( exp(x))+5*x*exp(x)^3+10*exp(x)^2)/x^3/exp(x)^2/exp(exp(x)),x,method=_RETUR NVERBOSE)
Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.15 \[ \int \frac {e^{-e^x-2 x} \left (10 e^{2 x}+5 e^{3 x} x-50 e^{e^x} x^3+e^{2 e^x+x} \left (e^{3 x} x^2+e^{2 x} \left (-x+x^2\right )\right )\right )}{x^3} \, dx=\frac {{\left (25 \, x^{2} e^{\left (e^{x}\right )} + x e^{\left (3 \, x + 2 \, e^{x}\right )} - 5 \, e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x - e^{x}\right )}}{x^{2}} \]
integrate(((x^2*exp(x)^3+(x^2-x)*exp(x)^2)*exp(exp(x))*exp(exp(x)+x)-50*x^ 3*exp(exp(x))+5*x*exp(x)^3+10*exp(x)^2)/x^3/exp(x)^2/exp(exp(x)),x, algori thm=\
Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int \frac {e^{-e^x-2 x} \left (10 e^{2 x}+5 e^{3 x} x-50 e^{e^x} x^3+e^{2 e^x+x} \left (e^{3 x} x^2+e^{2 x} \left (-x+x^2\right )\right )\right )}{x^3} \, dx=25 e^{- 2 x} + \frac {x^{2} e^{x} e^{e^{x}} - 5 x e^{- e^{x}}}{x^{3}} \]
integrate(((x**2*exp(x)**3+(x**2-x)*exp(x)**2)*exp(exp(x))*exp(exp(x)+x)-5 0*x**3*exp(exp(x))+5*x*exp(x)**3+10*exp(x)**2)/x**3/exp(x)**2/exp(exp(x)), x)
Time = 0.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \frac {e^{-e^x-2 x} \left (10 e^{2 x}+5 e^{3 x} x-50 e^{e^x} x^3+e^{2 e^x+x} \left (e^{3 x} x^2+e^{2 x} \left (-x+x^2\right )\right )\right )}{x^3} \, dx=\frac {x e^{\left (x + e^{x}\right )} - 5 \, e^{\left (-e^{x}\right )}}{x^{2}} + 25 \, e^{\left (-2 \, x\right )} \]
integrate(((x^2*exp(x)^3+(x^2-x)*exp(x)^2)*exp(exp(x))*exp(exp(x)+x)-50*x^ 3*exp(exp(x))+5*x*exp(x)^3+10*exp(x)^2)/x^3/exp(x)^2/exp(exp(x)),x, algori thm=\
\[ \int \frac {e^{-e^x-2 x} \left (10 e^{2 x}+5 e^{3 x} x-50 e^{e^x} x^3+e^{2 e^x+x} \left (e^{3 x} x^2+e^{2 x} \left (-x+x^2\right )\right )\right )}{x^3} \, dx=\int { -\frac {{\left (50 \, x^{3} e^{\left (e^{x}\right )} - 5 \, x e^{\left (3 \, x\right )} - {\left (x^{2} e^{\left (3 \, x\right )} + {\left (x^{2} - x\right )} e^{\left (2 \, x\right )}\right )} e^{\left (x + 2 \, e^{x}\right )} - 10 \, e^{\left (2 \, x\right )}\right )} e^{\left (-2 \, x - e^{x}\right )}}{x^{3}} \,d x } \]
integrate(((x^2*exp(x)^3+(x^2-x)*exp(x)^2)*exp(exp(x))*exp(exp(x)+x)-50*x^ 3*exp(exp(x))+5*x*exp(x)^3+10*exp(x)^2)/x^3/exp(x)^2/exp(exp(x)),x, algori thm=\
integrate(-(50*x^3*e^(e^x) - 5*x*e^(3*x) - (x^2*e^(3*x) + (x^2 - x)*e^(2*x ))*e^(x + 2*e^x) - 10*e^(2*x))*e^(-2*x - e^x)/x^3, x)
Time = 12.64 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int \frac {e^{-e^x-2 x} \left (10 e^{2 x}+5 e^{3 x} x-50 e^{e^x} x^3+e^{2 e^x+x} \left (e^{3 x} x^2+e^{2 x} \left (-x+x^2\right )\right )\right )}{x^3} \, dx=25\,{\mathrm {e}}^{-2\,x}+\frac {{\mathrm {e}}^{x+{\mathrm {e}}^x}}{x}-\frac {5\,{\mathrm {e}}^{-{\mathrm {e}}^x}}{x^2} \]