Integrand size = 102, antiderivative size = 27 \[ \int e^{-e^x+2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} \left (-6 e^5 x^4+2 e^{5+x} x^5+e^{e^x} \left (4 x+4 x^3\right )\right ) \, dx=e^{2 e^{x \left (x-e^{5-e^x} x^2\right )} x^2} \]
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int e^{-e^x+2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} \left (-6 e^5 x^4+2 e^{5+x} x^5+e^{e^x} \left (4 x+4 x^3\right )\right ) \, dx=e^{2 e^{x^2-e^{5-e^x} x^3} x^2} \]
Integrate[E^(-E^x + 2*E^((E^E^x*x^2 - E^5*x^3)/E^E^x)*x^2 + (E^E^x*x^2 - E ^5*x^3)/E^E^x)*(-6*E^5*x^4 + 2*E^(5 + x)*x^5 + E^E^x*(4*x + 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.
\(\displaystyle \int \left (2 e^{x+5} x^5-6 e^5 x^4+e^{e^x} \left (4 x^3+4 x\right )\right ) \exp \left (2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )-e^x\right ) \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (4 \left (x^2+1\right ) x \exp \left (2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )\right )+2 x^5 \exp \left (2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )+x-e^x+5\right )-6 x^4 \exp \left (2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )-e^x+5\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 4 \int \exp \left (2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )\right ) xdx+4 \int \exp \left (2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )\right ) x^3dx+2 \int \exp \left (2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+x-e^x+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )+5\right ) x^5dx-6 \int \exp \left (2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2-e^x+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )+5\right ) x^4dx\) |
Int[E^(-E^x + 2*E^((E^E^x*x^2 - E^5*x^3)/E^E^x)*x^2 + (E^E^x*x^2 - E^5*x^3 )/E^E^x)*(-6*E^5*x^4 + 2*E^(5 + x)*x^5 + E^E^x*(4*x + 4*x^3)),x]
3.16.62.3.1 Defintions of rubi rules used
Time = 0.08 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.04
\[{\mathrm e}^{2 x^{2} {\mathrm e}^{-x^{2} \left (x \,{\mathrm e}^{5}-{\mathrm e}^{{\mathrm e}^{x}}\right ) {\mathrm e}^{-{\mathrm e}^{x}}}}\]
int(((4*x^3+4*x)*exp(exp(x))+2*x^5*exp(5)*exp(x)-6*x^4*exp(5))*exp((exp(ex p(x))*x^2-x^3*exp(5))/exp(exp(x)))*exp(2*x^2*exp((exp(exp(x))*x^2-x^3*exp( 5))/exp(exp(x))))/exp(exp(x)),x)
Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (25) = 50\).
Time = 0.25 (sec) , antiderivative size = 91, normalized size of antiderivative = 3.37 \[ \int e^{-e^x+2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} \left (-6 e^5 x^4+2 e^{5+x} x^5+e^{e^x} \left (4 x+4 x^3\right )\right ) \, dx=e^{\left ({\left (x^{3} e^{5} - x^{2} e^{\left (e^{x}\right )}\right )} e^{\left (-e^{x}\right )} - {\left (x^{3} e^{10} - 2 \, x^{2} e^{\left (-{\left (x^{3} e^{5} - x^{2} e^{\left (e^{x}\right )}\right )} e^{\left (-e^{x}\right )} + e^{x} + 5\right )} - {\left (x^{2} e^{5} - e^{\left (x + 5\right )}\right )} e^{\left (e^{x}\right )}\right )} e^{\left (-e^{x} - 5\right )} + e^{x}\right )} \]
integrate(((4*x^3+4*x)*exp(exp(x))+2*x^5*exp(5)*exp(x)-6*x^4*exp(5))*exp(( exp(exp(x))*x^2-x^3*exp(5))/exp(exp(x)))*exp(2*x^2*exp((exp(exp(x))*x^2-x^ 3*exp(5))/exp(exp(x))))/exp(exp(x)),x, algorithm=\
e^((x^3*e^5 - x^2*e^(e^x))*e^(-e^x) - (x^3*e^10 - 2*x^2*e^(-(x^3*e^5 - x^2 *e^(e^x))*e^(-e^x) + e^x + 5) - (x^2*e^5 - e^(x + 5))*e^(e^x))*e^(-e^x - 5 ) + e^x)
Time = 2.76 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int e^{-e^x+2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} \left (-6 e^5 x^4+2 e^{5+x} x^5+e^{e^x} \left (4 x+4 x^3\right )\right ) \, dx=e^{2 x^{2} e^{\left (- x^{3} e^{5} + x^{2} e^{e^{x}}\right ) e^{- e^{x}}}} \]
integrate(((4*x**3+4*x)*exp(exp(x))+2*x**5*exp(5)*exp(x)-6*x**4*exp(5))*ex p((exp(exp(x))*x**2-x**3*exp(5))/exp(exp(x)))*exp(2*x**2*exp((exp(exp(x))* x**2-x**3*exp(5))/exp(exp(x))))/exp(exp(x)),x)
Time = 0.51 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int e^{-e^x+2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} \left (-6 e^5 x^4+2 e^{5+x} x^5+e^{e^x} \left (4 x+4 x^3\right )\right ) \, dx=e^{\left (2 \, x^{2} e^{\left (-x^{3} e^{\left (-e^{x} + 5\right )} + x^{2}\right )}\right )} \]
integrate(((4*x^3+4*x)*exp(exp(x))+2*x^5*exp(5)*exp(x)-6*x^4*exp(5))*exp(( exp(exp(x))*x^2-x^3*exp(5))/exp(exp(x)))*exp(2*x^2*exp((exp(exp(x))*x^2-x^ 3*exp(5))/exp(exp(x))))/exp(exp(x)),x, algorithm=\
\[ \int e^{-e^x+2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} \left (-6 e^5 x^4+2 e^{5+x} x^5+e^{e^x} \left (4 x+4 x^3\right )\right ) \, dx=\int { 2 \, {\left (x^{5} e^{\left (x + 5\right )} - 3 \, x^{4} e^{5} + 2 \, {\left (x^{3} + x\right )} e^{\left (e^{x}\right )}\right )} e^{\left (2 \, x^{2} e^{\left (-{\left (x^{3} e^{5} - x^{2} e^{\left (e^{x}\right )}\right )} e^{\left (-e^{x}\right )}\right )} - {\left (x^{3} e^{5} - x^{2} e^{\left (e^{x}\right )}\right )} e^{\left (-e^{x}\right )} - e^{x}\right )} \,d x } \]
integrate(((4*x^3+4*x)*exp(exp(x))+2*x^5*exp(5)*exp(x)-6*x^4*exp(5))*exp(( exp(exp(x))*x^2-x^3*exp(5))/exp(exp(x)))*exp(2*x^2*exp((exp(exp(x))*x^2-x^ 3*exp(5))/exp(exp(x))))/exp(exp(x)),x, algorithm=\
integrate(2*(x^5*e^(x + 5) - 3*x^4*e^5 + 2*(x^3 + x)*e^(e^x))*e^(2*x^2*e^( -(x^3*e^5 - x^2*e^(e^x))*e^(-e^x)) - (x^3*e^5 - x^2*e^(e^x))*e^(-e^x) - e^ x), x)
Time = 8.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85 \[ \int e^{-e^x+2 e^{e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} x^2+e^{-e^x} \left (e^{e^x} x^2-e^5 x^3\right )} \left (-6 e^5 x^4+2 e^{5+x} x^5+e^{e^x} \left (4 x+4 x^3\right )\right ) \, dx={\mathrm {e}}^{2\,x^2\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{-x^3\,{\mathrm {e}}^5\,{\mathrm {e}}^{-{\mathrm {e}}^x}}} \]