Integrand size = 168, antiderivative size = 36 \[ \int \frac {e^{e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} x+x^2} \left (18 x^2+12 e^{\frac {3-3 x}{x^2}} x^2+2 e^{\frac {2 (3-3 x)}{x^2}} x^2+e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} \left (9 x+e^{\frac {2 (3-3 x)}{x^2}} x-6 x^2+e^{\frac {3-3 x}{x^2}} \left (-12+12 x-2 x^2\right )\right )\right )}{9 x+6 e^{\frac {3-3 x}{x^2}} x+e^{\frac {2 (3-3 x)}{x^2}} x} \, dx=e^{x \left (e^{\frac {2 x}{-3-e^{\frac {3}{-x+\frac {x}{1-x}}}}}+x\right )} \]
Time = 1.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.72 \[ \int \frac {e^{e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} x+x^2} \left (18 x^2+12 e^{\frac {3-3 x}{x^2}} x^2+2 e^{\frac {2 (3-3 x)}{x^2}} x^2+e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} \left (9 x+e^{\frac {2 (3-3 x)}{x^2}} x-6 x^2+e^{\frac {3-3 x}{x^2}} \left (-12+12 x-2 x^2\right )\right )\right )}{9 x+6 e^{\frac {3-3 x}{x^2}} x+e^{\frac {2 (3-3 x)}{x^2}} x} \, dx=e^{x \left (e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}}+x\right )} \]
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]
Time = 7.11 (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 |
\(\displaystyle e^{x^2+e^{-\frac {2 x}{e^{\frac {3 (1-x)}{x^2}}+3}} x}\) |
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]
3.6.25.3.1 Defintions of rubi rules used
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]
Time = 10.83 (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 | \({\mathrm e}^{x \left ({\mathrm e}^{-\frac {2 x}{{\mathrm e}^{-\frac {3 \left (-1+x \right )}{x^{2}}}+3}}+x \right )}\) | \(23\) |
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)
Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.67 \[ \int \frac {e^{e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} x+x^2} \left (18 x^2+12 e^{\frac {3-3 x}{x^2}} x^2+2 e^{\frac {2 (3-3 x)}{x^2}} x^2+e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} \left (9 x+e^{\frac {2 (3-3 x)}{x^2}} x-6 x^2+e^{\frac {3-3 x}{x^2}} \left (-12+12 x-2 x^2\right )\right )\right )}{9 x+6 e^{\frac {3-3 x}{x^2}} x+e^{\frac {2 (3-3 x)}{x^2}} x} \, dx=e^{\left (x^{2} + x e^{\left (-\frac {2 \, x}{e^{\left (-\frac {3 \, {\left (x - 1\right )}}{x^{2}}\right )} + 3}\right )}\right )} \]
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=\
Time = 8.54 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.61 \[ \int \frac {e^{e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} x+x^2} \left (18 x^2+12 e^{\frac {3-3 x}{x^2}} x^2+2 e^{\frac {2 (3-3 x)}{x^2}} x^2+e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} \left (9 x+e^{\frac {2 (3-3 x)}{x^2}} x-6 x^2+e^{\frac {3-3 x}{x^2}} \left (-12+12 x-2 x^2\right )\right )\right )}{9 x+6 e^{\frac {3-3 x}{x^2}} x+e^{\frac {2 (3-3 x)}{x^2}} x} \, dx=e^{x^{2} + x e^{- \frac {2 x}{e^{\frac {3 - 3 x}{x^{2}}} + 3}}} \]
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)
\[ \int \frac {e^{e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} x+x^2} \left (18 x^2+12 e^{\frac {3-3 x}{x^2}} x^2+2 e^{\frac {2 (3-3 x)}{x^2}} x^2+e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} \left (9 x+e^{\frac {2 (3-3 x)}{x^2}} x-6 x^2+e^{\frac {3-3 x}{x^2}} \left (-12+12 x-2 x^2\right )\right )\right )}{9 x+6 e^{\frac {3-3 x}{x^2}} x+e^{\frac {2 (3-3 x)}{x^2}} x} \, dx=\int { \frac {{\left (12 \, x^{2} e^{\left (-\frac {3 \, {\left (x - 1\right )}}{x^{2}}\right )} + 2 \, x^{2} e^{\left (-\frac {6 \, {\left (x - 1\right )}}{x^{2}}\right )} + 18 \, x^{2} - {\left (6 \, x^{2} + 2 \, {\left (x^{2} - 6 \, x + 6\right )} e^{\left (-\frac {3 \, {\left (x - 1\right )}}{x^{2}}\right )} - x e^{\left (-\frac {6 \, {\left (x - 1\right )}}{x^{2}}\right )} - 9 \, x\right )} e^{\left (-\frac {2 \, x}{e^{\left (-\frac {3 \, {\left (x - 1\right )}}{x^{2}}\right )} + 3}\right )}\right )} e^{\left (x^{2} + x e^{\left (-\frac {2 \, x}{e^{\left (-\frac {3 \, {\left (x - 1\right )}}{x^{2}}\right )} + 3}\right )}\right )}}{6 \, x e^{\left (-\frac {3 \, {\left (x - 1\right )}}{x^{2}}\right )} + x e^{\left (-\frac {6 \, {\left (x - 1\right )}}{x^{2}}\right )} + 9 \, x} \,d x } \]
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=\
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)
Time = 1.16 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \frac {e^{e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} x+x^2} \left (18 x^2+12 e^{\frac {3-3 x}{x^2}} x^2+2 e^{\frac {2 (3-3 x)}{x^2}} x^2+e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} \left (9 x+e^{\frac {2 (3-3 x)}{x^2}} x-6 x^2+e^{\frac {3-3 x}{x^2}} \left (-12+12 x-2 x^2\right )\right )\right )}{9 x+6 e^{\frac {3-3 x}{x^2}} x+e^{\frac {2 (3-3 x)}{x^2}} x} \, dx=e^{\left (x^{2} + x e^{\left (-\frac {2 \, x}{e^{\left (-\frac {3}{x} + \frac {3}{x^{2}}\right )} + 3}\right )}\right )} \]
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=\
Time = 9.43 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81 \[ \int \frac {e^{e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} x+x^2} \left (18 x^2+12 e^{\frac {3-3 x}{x^2}} x^2+2 e^{\frac {2 (3-3 x)}{x^2}} x^2+e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}} \left (9 x+e^{\frac {2 (3-3 x)}{x^2}} x-6 x^2+e^{\frac {3-3 x}{x^2}} \left (-12+12 x-2 x^2\right )\right )\right )}{9 x+6 e^{\frac {3-3 x}{x^2}} x+e^{\frac {2 (3-3 x)}{x^2}} x} \, dx={\mathrm {e}}^{x^2}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-\frac {2\,x}{{\mathrm {e}}^{-\frac {3}{x}}\,{\mathrm {e}}^{\frac {3}{x^2}}+3}}} \]
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)