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

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

________________________________________________________________________________________

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} $$\begin{array}{c}-\frac{e^{2e^{-x^2-x-2}}(-2x^2+31x+16)}{2x+1}\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =-\frac{e^{2e^{-2-x-x^2}}(16+31x-2x^2)}{1+2x}\end{array}\end{array}$$

________________________________________________________________________________________

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

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 $$\begin{array}{c}(x-16)e^{\left(2e^{(-x^2-x-2)}\right)}\end{array}$$

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 $$\begin{array}{c}x{e}^{\left(2e^{(-x^2-x-2)}\right)}-16e^{\left(2e^{(-x^2-x-2)}\right)}\end{array}$$

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	$(x-16){\mathrm{e}}^{2{\mathrm{e}}^{-x^2-x-2}}$	$19$
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 $$\begin{array}{c}-\int (2(2x^2-31x-16)e^{(-x^2-x-2)}-1)e^{\left(2e^{(-x^2-x-2)}\right)}dx\end{array}$$

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 $$\begin{array}{c}{\mathrm{e}}^{2{\mathrm{e}}^{-x}{\mathrm{e}}^{-2}{\mathrm{e}}^{-x^2}}(x-16)\end{array}$$

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 $$\begin{array}{c}(x-16)e^{2e^{-x^2-x-2}}\end{array}$$

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))

________________________________________________________________________________________