\(\int e^{-e^{2 e^{-x} x}} (3 e^x+e^{2 e^{-x} x} (-6+6 x)) \, dx\) [4573]

Optimal result

Integrand size = 37, antiderivative size = 20 \[ \int e^{-e^{2 e^{-x} x}} \left (3 e^x+e^{2 e^{-x} x} (-6+6 x)\right ) \, dx=-1+3 e^{-e^{2 e^{-x} x}+x} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.85, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {2326} \[ \int e^{-e^{2 e^{-x} x}} \left (3 e^x+e^{2 e^{-x} x} (-6+6 x)\right ) \, dx=\frac {3 e^{-e^{2 e^{-x} x}} (1-x)}{e^{-x}-e^{-x} x} \]

[In]

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

[Out]

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

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = \frac {3 e^{-e^{2 e^{-x} x}} (1-x)}{e^{-x}-e^{-x} x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int e^{-e^{2 e^{-x} x}} \left (3 e^x+e^{2 e^{-x} x} (-6+6 x)\right ) \, dx=3 e^{-e^{2 e^{-x} x}+x} \]

[In]

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

[Out]

3*E^(-E^((2*x)/E^x) + x)

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.80

method	result	size
norman	\(3 \,{\mathrm e}^{x} {\mathrm e}^{-{\mathrm e}^{2 x \,{\mathrm e}^{-x}}}\)	\(16\)
risch	\(3 \,{\mathrm e}^{x -{\mathrm e}^{2 x \,{\mathrm e}^{-x}}}\)	\(16\)
parallelrisch	\(3 \,{\mathrm e}^{x} {\mathrm e}^{-{\mathrm e}^{2 x \,{\mathrm e}^{-x}}}\)	\(16\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int e^{-e^{2 e^{-x} x}} \left (3 e^x+e^{2 e^{-x} x} (-6+6 x)\right ) \, dx=3 \, e^{\left (x - e^{\left (2 \, x e^{\left (-x\right )}\right )}\right )} \]

[In]

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

[Out]

3*e^(x - e^(2*x*e^(-x)))

Sympy [A] (verification not implemented)

Time = 0.44 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int e^{-e^{2 e^{-x} x}} \left (3 e^x+e^{2 e^{-x} x} (-6+6 x)\right ) \, dx=3 e^{x} e^{- e^{2 x e^{- x}}} \]

[In]

integrate(((6*x-6)*exp(2*x/exp(x))+3*exp(x))/exp(exp(2*x/exp(x))),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int e^{-e^{2 e^{-x} x}} \left (3 e^x+e^{2 e^{-x} x} (-6+6 x)\right ) \, dx=3 \, e^{\left (x - e^{\left (2 \, x e^{\left (-x\right )}\right )}\right )} \]

[In]

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

[Out]

3*e^(x - e^(2*x*e^(-x)))

Giac [F]

\[ \int e^{-e^{2 e^{-x} x}} \left (3 e^x+e^{2 e^{-x} x} (-6+6 x)\right ) \, dx=\int { 3 \, {\left (2 \, {\left (x - 1\right )} e^{\left (2 \, x e^{\left (-x\right )}\right )} + e^{x}\right )} e^{\left (-e^{\left (2 \, x e^{\left (-x\right )}\right )}\right )} \,d x } \]

[In]

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

[Out]

integrate(3*(2*(x - 1)*e^(2*x*e^(-x)) + e^x)*e^(-e^(2*x*e^(-x))), x)

Mupad [B] (verification not implemented)

Time = 11.21 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int e^{-e^{2 e^{-x} x}} \left (3 e^x+e^{2 e^{-x} x} (-6+6 x)\right ) \, dx=3\,{\mathrm {e}}^{-{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{-x}}}\,{\mathrm {e}}^x \]

[In]

int(exp(-exp(2*x*exp(-x)))*(3*exp(x) + exp(2*x*exp(-x))*(6*x - 6)),x)

[Out]

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