\(\int \frac {e^{e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} x+x^2} (18 x^2+12 e^{\frac {3-3 x}{x^2}} x^2+2 e^{\frac {2 (3-3 x)}{x^2}} x^2+e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} (9 x+e^{\frac {2 (3-3 x)}{x^2}} x-6 x^2+e^{\frac {3-3 x}{x^2}} (-12+12 x-2 x^2)))}{9 x+6 e^{\frac {3-3 x}{x^2}} x+e^{\frac {2 (3-3 x)}{x^2}} x} \, dx\) [525]

Optimal result

Integrand size = 168, antiderivative size = 36 \[ \int \frac {e^{e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} x+x^2} \left (18 x^2+12 e^{\frac {3-3 x}{x^2}} x^2+2 e^{\frac {2 (3-3 x)}{x^2}} x^2+e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} \left (9 x+e^{\frac {2 (3-3 x)}{x^2}} x-6 x^2+e^{\frac {3-3 x}{x^2}} \left (-12+12 x-2 x^2\right )\right )\right )}{9 x+6 e^{\frac {3-3 x}{x^2}} x+e^{\frac {2 (3-3 x)}{x^2}} x} \, dx=e^{x \left (e^{\frac {2 x}{-3-e^{\frac {3}{-x+\frac {x}{1-x}}}}}+x\right )} \] Output:

exp((x+exp(2*x/(-3-exp(3/(x/(1-x)-x)))))*x)
 

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {e^{e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} x+x^2} \left (18 x^2+12 e^{\frac {3-3 x}{x^2}} x^2+2 e^{\frac {2 (3-3 x)}{x^2}} x^2+e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} \left (9 x+e^{\frac {2 (3-3 x)}{x^2}} x-6 x^2+e^{\frac {3-3 x}{x^2}} \left (-12+12 x-2 x^2\right )\right )\right )}{9 x+6 e^{\frac {3-3 x}{x^2}} x+e^{\frac {2 (3-3 x)}{x^2}} x} \, dx=e^{x \left (e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}}+x\right )} \] Input:

Integrate[(E^(x/E^((2*x)/(3 + E^((3 - 3*x)/x^2))) + x^2)*(18*x^2 + 12*E^(( 
3 - 3*x)/x^2)*x^2 + 2*E^((2*(3 - 3*x))/x^2)*x^2 + (9*x + E^((2*(3 - 3*x))/ 
x^2)*x - 6*x^2 + E^((3 - 3*x)/x^2)*(-12 + 12*x - 2*x^2))/E^((2*x)/(3 + E^( 
(3 - 3*x)/x^2)))))/(9*x + 6*E^((3 - 3*x)/x^2)*x + E^((2*(3 - 3*x))/x^2)*x) 
,x]
 

Output:

E^(x*(E^((-2*x)/(3 + E^((3 - 3*x)/x^2))) + x))
 

Rubi [A] (verified)

Time = 8.92 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.006, Rules used = {7257}

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.

\(\displaystyle \int \frac {e^{x^2+e^{-\frac {2 x}{e^{\frac {3-3 x}{x^2}}+3}} x} \left (12 e^{\frac {3-3 x}{x^2}} x^2+2 e^{\frac {2 (3-3 x)}{x^2}} x^2+18 x^2+e^{-\frac {2 x}{e^{\frac {3-3 x}{x^2}}+3}} \left (-6 x^2+e^{\frac {2 (3-3 x)}{x^2}} x+e^{\frac {3-3 x}{x^2}} \left (-2 x^2+12 x-12\right )+9 x\right )\right )}{6 e^{\frac {3-3 x}{x^2}} x+e^{\frac {2 (3-3 x)}{x^2}} x+9 x} \, dx\)

\(\Big \downarrow \) 7257

\(\displaystyle e^{x^2+e^{-\frac {2 x}{e^{\frac {3 (1-x)}{x^2}}+3}} x}\)

Input:

Int[(E^(x/E^((2*x)/(3 + E^((3 - 3*x)/x^2))) + x^2)*(18*x^2 + 12*E^((3 - 3* 
x)/x^2)*x^2 + 2*E^((2*(3 - 3*x))/x^2)*x^2 + (9*x + E^((2*(3 - 3*x))/x^2)*x 
 - 6*x^2 + E^((3 - 3*x)/x^2)*(-12 + 12*x - 2*x^2))/E^((2*x)/(3 + E^((3 - 3 
*x)/x^2)))))/(9*x + 6*E^((3 - 3*x)/x^2)*x + E^((2*(3 - 3*x))/x^2)*x),x]
 

Output:

E^(x/E^((2*x)/(3 + E^((3*(1 - x))/x^2))) + x^2)
 

Defintions of rubi rules used

rule 7257

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Sim 
p[q*(F^v/Log[F]), x] /;  !FalseQ[q]] /; FreeQ[F, x]
 

Maple [A] (verified)

Time = 9.75 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.64

method	result	size
risch	\({\mathrm e}^{x \left ({\mathrm e}^{-\frac {2 x}{{\mathrm e}^{-\frac {3 \left (-1+x \right )}{x^{2}}}+3}}+x \right )}\)	\(23\)
parallelrisch	\({\mathrm e}^{x \left ({\mathrm e}^{-\frac {2 x}{{\mathrm e}^{-\frac {3 \left (-1+x \right )}{x^{2}}}+3}}+x \right )}\)	\(23\)

Input:

int(((x*exp((-3*x+3)/x^2)^2+(-2*x^2+12*x-12)*exp((-3*x+3)/x^2)-6*x^2+9*x)* 
exp(-2*x/(exp((-3*x+3)/x^2)+3))+2*x^2*exp((-3*x+3)/x^2)^2+12*x^2*exp((-3*x 
+3)/x^2)+18*x^2)*exp(x*exp(-2*x/(exp((-3*x+3)/x^2)+3))+x^2)/(x*exp((-3*x+3 
)/x^2)^2+6*x*exp((-3*x+3)/x^2)+9*x),x,method=_RETURNVERBOSE)
 

Output:

exp(x*(exp(-2*x/(exp(-3*(-1+x)/x^2)+3))+x))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.67 \[ \int \frac {e^{e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} x+x^2} \left (18 x^2+12 e^{\frac {3-3 x}{x^2}} x^2+2 e^{\frac {2 (3-3 x)}{x^2}} x^2+e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} \left (9 x+e^{\frac {2 (3-3 x)}{x^2}} x-6 x^2+e^{\frac {3-3 x}{x^2}} \left (-12+12 x-2 x^2\right )\right )\right )}{9 x+6 e^{\frac {3-3 x}{x^2}} x+e^{\frac {2 (3-3 x)}{x^2}} x} \, dx=e^{\left (x^{2} + x e^{\left (-\frac {2 \, x}{e^{\left (-\frac {3 \, {\left (x - 1\right )}}{x^{2}}\right )} + 3}\right )}\right )} \] Input:

integrate(((x*exp((-3*x+3)/x^2)^2+(-2*x^2+12*x-12)*exp((-3*x+3)/x^2)-6*x^2 
+9*x)*exp(-2*x/(exp((-3*x+3)/x^2)+3))+2*x^2*exp((-3*x+3)/x^2)^2+12*x^2*exp 
((-3*x+3)/x^2)+18*x^2)*exp(x*exp(-2*x/(exp((-3*x+3)/x^2)+3))+x^2)/(x*exp(( 
-3*x+3)/x^2)^2+6*x*exp((-3*x+3)/x^2)+9*x),x, algorithm="fricas")
 

Output:

e^(x^2 + x*e^(-2*x/(e^(-3*(x - 1)/x^2) + 3)))
 

Sympy [A] (verification not implemented)

Time = 9.16 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.61 \[ \int \frac {e^{e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} x+x^2} \left (18 x^2+12 e^{\frac {3-3 x}{x^2}} x^2+2 e^{\frac {2 (3-3 x)}{x^2}} x^2+e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} \left (9 x+e^{\frac {2 (3-3 x)}{x^2}} x-6 x^2+e^{\frac {3-3 x}{x^2}} \left (-12+12 x-2 x^2\right )\right )\right )}{9 x+6 e^{\frac {3-3 x}{x^2}} x+e^{\frac {2 (3-3 x)}{x^2}} x} \, dx=e^{x^{2} + x e^{- \frac {2 x}{e^{\frac {3 - 3 x}{x^{2}}} + 3}}} \] Input:

integrate(((x*exp((-3*x+3)/x**2)**2+(-2*x**2+12*x-12)*exp((-3*x+3)/x**2)-6 
*x**2+9*x)*exp(-2*x/(exp((-3*x+3)/x**2)+3))+2*x**2*exp((-3*x+3)/x**2)**2+1 
2*x**2*exp((-3*x+3)/x**2)+18*x**2)*exp(x*exp(-2*x/(exp((-3*x+3)/x**2)+3))+ 
x**2)/(x*exp((-3*x+3)/x**2)**2+6*x*exp((-3*x+3)/x**2)+9*x),x)
 

Output:

exp(x**2 + x*exp(-2*x/(exp((3 - 3*x)/x**2) + 3)))
                                                                                    
                                                                                    
 

Maxima [F]

\[ \int \frac {e^{e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} x+x^2} \left (18 x^2+12 e^{\frac {3-3 x}{x^2}} x^2+2 e^{\frac {2 (3-3 x)}{x^2}} x^2+e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} \left (9 x+e^{\frac {2 (3-3 x)}{x^2}} x-6 x^2+e^{\frac {3-3 x}{x^2}} \left (-12+12 x-2 x^2\right )\right )\right )}{9 x+6 e^{\frac {3-3 x}{x^2}} x+e^{\frac {2 (3-3 x)}{x^2}} x} \, dx=\int { \frac {{\left (12 \, x^{2} e^{\left (-\frac {3 \, {\left (x - 1\right )}}{x^{2}}\right )} + 2 \, x^{2} e^{\left (-\frac {6 \, {\left (x - 1\right )}}{x^{2}}\right )} + 18 \, x^{2} - {\left (6 \, x^{2} + 2 \, {\left (x^{2} - 6 \, x + 6\right )} e^{\left (-\frac {3 \, {\left (x - 1\right )}}{x^{2}}\right )} - x e^{\left (-\frac {6 \, {\left (x - 1\right )}}{x^{2}}\right )} - 9 \, x\right )} e^{\left (-\frac {2 \, x}{e^{\left (-\frac {3 \, {\left (x - 1\right )}}{x^{2}}\right )} + 3}\right )}\right )} e^{\left (x^{2} + x e^{\left (-\frac {2 \, x}{e^{\left (-\frac {3 \, {\left (x - 1\right )}}{x^{2}}\right )} + 3}\right )}\right )}}{6 \, x e^{\left (-\frac {3 \, {\left (x - 1\right )}}{x^{2}}\right )} + x e^{\left (-\frac {6 \, {\left (x - 1\right )}}{x^{2}}\right )} + 9 \, x} \,d x } \] Input:

integrate(((x*exp((-3*x+3)/x^2)^2+(-2*x^2+12*x-12)*exp((-3*x+3)/x^2)-6*x^2 
+9*x)*exp(-2*x/(exp((-3*x+3)/x^2)+3))+2*x^2*exp((-3*x+3)/x^2)^2+12*x^2*exp 
((-3*x+3)/x^2)+18*x^2)*exp(x*exp(-2*x/(exp((-3*x+3)/x^2)+3))+x^2)/(x*exp(( 
-3*x+3)/x^2)^2+6*x*exp((-3*x+3)/x^2)+9*x),x, algorithm="maxima")
 

Output:

integrate((12*x^2*e^(-3*(x - 1)/x^2) + 2*x^2*e^(-6*(x - 1)/x^2) + 18*x^2 - 
 (6*x^2 + 2*(x^2 - 6*x + 6)*e^(-3*(x - 1)/x^2) - x*e^(-6*(x - 1)/x^2) - 9* 
x)*e^(-2*x/(e^(-3*(x - 1)/x^2) + 3)))*e^(x^2 + x*e^(-2*x/(e^(-3*(x - 1)/x^ 
2) + 3)))/(6*x*e^(-3*(x - 1)/x^2) + x*e^(-6*(x - 1)/x^2) + 9*x), x)
 

Giac [A] (verification not implemented)

Time = 1.02 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \frac {e^{e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} x+x^2} \left (18 x^2+12 e^{\frac {3-3 x}{x^2}} x^2+2 e^{\frac {2 (3-3 x)}{x^2}} x^2+e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} \left (9 x+e^{\frac {2 (3-3 x)}{x^2}} x-6 x^2+e^{\frac {3-3 x}{x^2}} \left (-12+12 x-2 x^2\right )\right )\right )}{9 x+6 e^{\frac {3-3 x}{x^2}} x+e^{\frac {2 (3-3 x)}{x^2}} x} \, dx=e^{\left (x^{2} + x e^{\left (-\frac {2 \, x}{e^{\left (-\frac {3}{x} + \frac {3}{x^{2}}\right )} + 3}\right )}\right )} \] Input:

integrate(((x*exp((-3*x+3)/x^2)^2+(-2*x^2+12*x-12)*exp((-3*x+3)/x^2)-6*x^2 
+9*x)*exp(-2*x/(exp((-3*x+3)/x^2)+3))+2*x^2*exp((-3*x+3)/x^2)^2+12*x^2*exp 
((-3*x+3)/x^2)+18*x^2)*exp(x*exp(-2*x/(exp((-3*x+3)/x^2)+3))+x^2)/(x*exp(( 
-3*x+3)/x^2)^2+6*x*exp((-3*x+3)/x^2)+9*x),x, algorithm="giac")
 

Output:

e^(x^2 + x*e^(-2*x/(e^(-3/x + 3/x^2) + 3)))
 

Mupad [B] (verification not implemented)

Time = 2.63 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81 \[ \int \frac {e^{e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} x+x^2} \left (18 x^2+12 e^{\frac {3-3 x}{x^2}} x^2+2 e^{\frac {2 (3-3 x)}{x^2}} x^2+e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} \left (9 x+e^{\frac {2 (3-3 x)}{x^2}} x-6 x^2+e^{\frac {3-3 x}{x^2}} \left (-12+12 x-2 x^2\right )\right )\right )}{9 x+6 e^{\frac {3-3 x}{x^2}} x+e^{\frac {2 (3-3 x)}{x^2}} x} \, dx={\mathrm {e}}^{x^2}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-\frac {2\,x}{{\mathrm {e}}^{-\frac {3}{x}}\,{\mathrm {e}}^{\frac {3}{x^2}}+3}}} \] Input:

int((exp(x*exp(-(2*x)/(exp(-(3*x - 3)/x^2) + 3)) + x^2)*(12*x^2*exp(-(3*x 
- 3)/x^2) + 2*x^2*exp(-(2*(3*x - 3))/x^2) + 18*x^2 + exp(-(2*x)/(exp(-(3*x 
 - 3)/x^2) + 3))*(9*x - exp(-(3*x - 3)/x^2)*(2*x^2 - 12*x + 12) + x*exp(-( 
2*(3*x - 3))/x^2) - 6*x^2)))/(9*x + 6*x*exp(-(3*x - 3)/x^2) + x*exp(-(2*(3 
*x - 3))/x^2)),x)
 

Output:

exp(x^2)*exp(x*exp(-(2*x)/(exp(-3/x)*exp(3/x^2) + 3)))
 

Reduce [F]

\[ \int \frac {e^{e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} x+x^2} \left (18 x^2+12 e^{\frac {3-3 x}{x^2}} x^2+2 e^{\frac {2 (3-3 x)}{x^2}} x^2+e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} \left (9 x+e^{\frac {2 (3-3 x)}{x^2}} x-6 x^2+e^{\frac {3-3 x}{x^2}} \left (-12+12 x-2 x^2\right )\right )\right )}{9 x+6 e^{\frac {3-3 x}{x^2}} x+e^{\frac {2 (3-3 x)}{x^2}} x} \, dx=\int \frac {\left (\left (x \left ({\mathrm e}^{\frac {-3 x +3}{x^{2}}}\right )^{2}+\left (-2 x^{2}+12 x -12\right ) {\mathrm e}^{\frac {-3 x +3}{x^{2}}}-6 x^{2}+9 x \right ) {\mathrm e}^{-\frac {2 x}{{\mathrm e}^{\frac {-3 x +3}{x^{2}}}+3}}+2 x^{2} \left ({\mathrm e}^{\frac {-3 x +3}{x^{2}}}\right )^{2}+12 x^{2} {\mathrm e}^{\frac {-3 x +3}{x^{2}}}+18 x^{2}\right ) {\mathrm e}^{x \,{\mathrm e}^{-\frac {2 x}{{\mathrm e}^{\frac {-3 x +3}{x^{2}}}+3}}+x^{2}}}{x \left ({\mathrm e}^{\frac {-3 x +3}{x^{2}}}\right )^{2}+6 x \,{\mathrm e}^{\frac {-3 x +3}{x^{2}}}+9 x}d x \] Input:

int(((x*exp((-3*x+3)/x^2)^2+(-2*x^2+12*x-12)*exp((-3*x+3)/x^2)-6*x^2+9*x)* 
exp(-2*x/(exp((-3*x+3)/x^2)+3))+2*x^2*exp((-3*x+3)/x^2)^2+12*x^2*exp((-3*x 
+3)/x^2)+18*x^2)*exp(x*exp(-2*x/(exp((-3*x+3)/x^2)+3))+x^2)/(x*exp((-3*x+3 
)/x^2)^2+6*x*exp((-3*x+3)/x^2)+9*x),x)
 

Output:

int(((x*exp((-3*x+3)/x^2)^2+(-2*x^2+12*x-12)*exp((-3*x+3)/x^2)-6*x^2+9*x)* 
exp(-2*x/(exp((-3*x+3)/x^2)+3))+2*x^2*exp((-3*x+3)/x^2)^2+12*x^2*exp((-3*x 
+3)/x^2)+18*x^2)*exp(x*exp(-2*x/(exp((-3*x+3)/x^2)+3))+x^2)/(x*exp((-3*x+3 
)/x^2)^2+6*x*exp((-3*x+3)/x^2)+9*x),x)