\(\int e^{2 e^{-2-x-x^2}} (1+e^{-2-x-x^2} (32+62 x-4 x^2)) \, dx\) [6610]

Optimal result

Integrand size = 42, antiderivative size = 20 \[ \int e^{2 e^{-2-x-x^2}} \left (1+e^{-2-x-x^2} \left (32+62 x-4 x^2\right )\right ) \, dx=e^{2 e^{-2-x-x^2}} (-16+x) \]

[Out]

(x-16)*exp(2*exp(-x^2-x-2))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.75, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {2326} \[ \int e^{2 e^{-2-x-x^2}} \left (1+e^{-2-x-x^2} \left (32+62 x-4 x^2\right )\right ) \, dx=-\frac {e^{2 e^{-x^2-x-2}} \left (-2 x^2+31 x+16\right )}{2 x+1} \]

[In]

Int[E^(2*E^(-2 - x - x^2))*(1 + E^(-2 - x - x^2)*(32 + 62*x - 4*x^2)),x]

[Out]

-((E^(2*E^(-2 - x - x^2))*(16 + 31*x - 2*x^2))/(1 + 2*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 {e^{2 e^{-2-x-x^2}} \left (16+31 x-2 x^2\right )}{1+2 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int e^{2 e^{-2-x-x^2}} \left (1+e^{-2-x-x^2} \left (32+62 x-4 x^2\right )\right ) \, dx=e^{2 e^{-2-x-x^2}} (-16+x) \]

[In]

Integrate[E^(2*E^(-2 - x - x^2))*(1 + E^(-2 - x - x^2)*(32 + 62*x - 4*x^2)),x]

[Out]

E^(2*E^(-2 - x - x^2))*(-16 + x)

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95

method	result	size
risch	\(\left (x -16\right ) {\mathrm e}^{2 \,{\mathrm e}^{-x^{2}-x -2}}\)	\(19\)
norman	\(x \,{\mathrm e}^{2 \,{\mathrm e}^{-x^{2}-x -2}}-16 \,{\mathrm e}^{2 \,{\mathrm e}^{-x^{2}-x -2}}\)	\(34\)
parallelrisch	\(x \,{\mathrm e}^{2 \,{\mathrm e}^{-x^{2}-x -2}}-16 \,{\mathrm e}^{2 \,{\mathrm e}^{-x^{2}-x -2}}\)	\(34\)

[In]

int(((-4*x^2+62*x+32)*exp(-x^2-x-2)+1)*exp(2*exp(-x^2-x-2)),x,method=_RETURNVERBOSE)

[Out]

(x-16)*exp(2*exp(-x^2-x-2))

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int e^{2 e^{-2-x-x^2}} \left (1+e^{-2-x-x^2} \left (32+62 x-4 x^2\right )\right ) \, dx={\left (x - 16\right )} e^{\left (2 \, e^{\left (-x^{2} - x - 2\right )}\right )} \]

[In]

integrate(((-4*x^2+62*x+32)*exp(-x^2-x-2)+1)*exp(2*exp(-x^2-x-2)),x, algorithm="fricas")

[Out]

(x - 16)*e^(2*e^(-x^2 - x - 2))

Sympy [A] (verification not implemented)

Time = 4.36 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75 \[ \int e^{2 e^{-2-x-x^2}} \left (1+e^{-2-x-x^2} \left (32+62 x-4 x^2\right )\right ) \, dx=\left (x - 16\right ) e^{2 e^{- x^{2} - x - 2}} \]

[In]

integrate(((-4*x**2+62*x+32)*exp(-x**2-x-2)+1)*exp(2*exp(-x**2-x-2)),x)

[Out]

(x - 16)*exp(2*exp(-x**2 - x - 2))

Maxima [F]

\[ \int e^{2 e^{-2-x-x^2}} \left (1+e^{-2-x-x^2} \left (32+62 x-4 x^2\right )\right ) \, dx=\int { -{\left (2 \, {\left (2 \, x^{2} - 31 \, x - 16\right )} e^{\left (-x^{2} - x - 2\right )} - 1\right )} e^{\left (2 \, e^{\left (-x^{2} - x - 2\right )}\right )} \,d x } \]

[In]

integrate(((-4*x^2+62*x+32)*exp(-x^2-x-2)+1)*exp(2*exp(-x^2-x-2)),x, algorithm="maxima")

[Out]

-integrate((2*(2*x^2 - 31*x - 16)*e^(-x^2 - x - 2) - 1)*e^(2*e^(-x^2 - x - 2)), x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.65 \[ \int e^{2 e^{-2-x-x^2}} \left (1+e^{-2-x-x^2} \left (32+62 x-4 x^2\right )\right ) \, dx=x e^{\left (2 \, e^{\left (-x^{2} - x - 2\right )}\right )} - 16 \, e^{\left (2 \, e^{\left (-x^{2} - x - 2\right )}\right )} \]

[In]

integrate(((-4*x^2+62*x+32)*exp(-x^2-x-2)+1)*exp(2*exp(-x^2-x-2)),x, algorithm="giac")

[Out]

x*e^(2*e^(-x^2 - x - 2)) - 16*e^(2*e^(-x^2 - x - 2))

Mupad [B] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.95 \[ \int e^{2 e^{-2-x-x^2}} \left (1+e^{-2-x-x^2} \left (32+62 x-4 x^2\right )\right ) \, dx={\mathrm {e}}^{2\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-2}\,{\mathrm {e}}^{-x^2}}\,\left (x-16\right ) \]

[In]

int(exp(2*exp(- x - x^2 - 2))*(exp(- x - x^2 - 2)*(62*x - 4*x^2 + 32) + 1),x)

[Out]

exp(2*exp(-x)*exp(-2)*exp(-x^2))*(x - 16)