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

Optimal. Leaf size=36 \[ e^{x \left (e^{\frac {2 x}{-3-e^{\frac {3}{-x+\frac {x}{1-x}}}}}+x\right )} \]

________________________________________________________________________________________

Rubi [A]  time = 10.72, antiderivative size = 29, normalized size of antiderivative = 0.81, number of steps used = 1, number of rules used = 1, integrand size = 168, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.006, Rules used = {6706} \begin {gather*} e^{x^2+e^{-\frac {2 x}{e^{\frac {3 (1-x)}{x^2}}+3}} x} \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

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

Rule 6706

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=e^{e^{-\frac {2 x}{3+e^{\frac {3 (1-x)}{x^2}}}} x+x^2}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 1.19, size = 26, normalized size = 0.72 \begin {gather*} e^{x \left (e^{-\frac {2 x}{3+e^{\frac {3-3 x}{x^2}}}}+x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.83, size = 24, normalized size = 0.67 \begin {gather*} e^{\left (x^{2} + x e^{\left (-\frac {2 \, x}{e^{\left (-\frac {3 \, {\left (x - 1\right )}}{x^{2}}\right )} + 3}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 1.64, size = 27, normalized size = 0.75 \begin {gather*} e^{\left (x^{2} + x e^{\left (-\frac {2 \, x}{e^{\left (-\frac {3}{x} + \frac {3}{x^{2}}\right )} + 3}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.22, size = 23, normalized size = 0.64

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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")

[Out]

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)

________________________________________________________________________________________

mupad [B]  time = 2.33, size = 29, normalized size = 0.81 \begin {gather*} {\mathrm {e}}^{x^2}\,{\mathrm {e}}^{x\,{\mathrm {e}}^{-\frac {2\,x}{{\mathrm {e}}^{-\frac {3}{x}}\,{\mathrm {e}}^{\frac {3}{x^2}}+3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 10.99, size = 22, normalized size = 0.61 \begin {gather*} e^{x^{2} + x e^{- \frac {2 x}{e^{\frac {3 - 3 x}{x^{2}}} + 3}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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)

[Out]

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

________________________________________________________________________________________