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

Optimal. Leaf size=20 \[ e^{2 e^{-2-x-x^2}} (-16+x) \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.75, number of steps used = 1, number of rules used = 1, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.024, Rules used = {2288} \begin {gather*} -\frac {e^{2 e^{-x^2-x-2}} \left (-2 x^2+31 x+16\right )}{2 x+1} \end {gather*}

Antiderivative was successfully verified.

[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 2288

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 {gather*} \begin {aligned} \text {integral} &=-\frac {e^{2 e^{-2-x-x^2}} \left (16+31 x-2 x^2\right )}{1+2 x}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.60, size = 20, normalized size = 1.00 \begin {gather*} e^{2 e^{-2-x-x^2}} (-16+x) \end {gather*}

Antiderivative was successfully verified.

[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)

________________________________________________________________________________________

fricas [A]  time = 0.53, size = 18, normalized size = 0.90 \begin {gather*} {\left (x - 16\right )} e^{\left (2 \, e^{\left (-x^{2} - x - 2\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

giac [A]  time = 0.18, size = 33, normalized size = 1.65 \begin {gather*} x e^{\left (2 \, e^{\left (-x^{2} - x - 2\right )}\right )} - 16 \, e^{\left (2 \, e^{\left (-x^{2} - x - 2\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

maple [A]  time = 0.09, size = 19, normalized size = 0.95

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

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [B]  time = 0.10, size = 19, normalized size = 0.95 \begin {gather*} {\mathrm {e}}^{2\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^{-2}\,{\mathrm {e}}^{-x^2}}\,\left (x-16\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [A]  time = 7.42, size = 15, normalized size = 0.75 \begin {gather*} \left (x - 16\right ) e^{2 e^{- x^{2} - x - 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________