3.50.65 \(\int e^{-4 x-x^2} (-16+e^{4 x+x^2} (2-2 x)-8 x) \, dx\)

Optimal. Leaf size=18 \[ -3+4 e^{-x (4+x)}-(-2+x) x \]

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Rubi [A]  time = 0.10, antiderivative size = 22, normalized size of antiderivative = 1.22, number of steps used = 3, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6688, 2236} \begin {gather*} -x^2+4 e^{-x^2-4 x}+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(-4*x - x^2)*(-16 + E^(4*x + x^2)*(2 - 2*x) - 8*x),x]

[Out]

4*E^(-4*x - x^2) + 2*x - x^2

Rule 2236

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(e*F^(a + b*x + c*x^2))/(
2*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && EqQ[b*e - 2*c*d, 0]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (2-2 x-8 e^{-4 x-x^2} (2+x)\right ) \, dx\\ &=2 x-x^2-8 \int e^{-4 x-x^2} (2+x) \, dx\\ &=4 e^{-4 x-x^2}+2 x-x^2\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 22, normalized size = 1.22 \begin {gather*} 4 e^{-4 x-x^2}+2 x-x^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(-4*x - x^2)*(-16 + E^(4*x + x^2)*(2 - 2*x) - 8*x),x]

[Out]

4*E^(-4*x - x^2) + 2*x - x^2

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fricas [A]  time = 0.70, size = 30, normalized size = 1.67 \begin {gather*} -{\left ({\left (x^{2} - 2 \, x\right )} e^{\left (x^{2} + 4 \, x\right )} - 4\right )} e^{\left (-x^{2} - 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+2)*exp(x^2+4*x)-8*x-16)/exp(x^2+4*x),x, algorithm="fricas")

[Out]

-((x^2 - 2*x)*e^(x^2 + 4*x) - 4)*e^(-x^2 - 4*x)

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giac [A]  time = 0.19, size = 21, normalized size = 1.17 \begin {gather*} -x^{2} + 2 \, x + 4 \, e^{\left (-x^{2} - 4 \, x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+2)*exp(x^2+4*x)-8*x-16)/exp(x^2+4*x),x, algorithm="giac")

[Out]

-x^2 + 2*x + 4*e^(-x^2 - 4*x)

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maple [A]  time = 0.04, size = 19, normalized size = 1.06

method	result	size
risch	\(-x^{2}+2 x +4 \,{\mathrm e}^{-\left (4+x \right ) x}\)	\(19\)
default	\(-x^{2}+2 x +4 \,{\mathrm e}^{-x^{2}-4 x}\)	\(22\)
norman	\(\left (4+2 x \,{\mathrm e}^{x^{2}+4 x}-x^{2} {\mathrm e}^{x^{2}+4 x}\right ) {\mathrm e}^{-x^{2}-4 x}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x+2)*exp(x^2+4*x)-8*x-16)/exp(x^2+4*x),x,method=_RETURNVERBOSE)

[Out]

-x^2+2*x+4*exp(-(4+x)*x)

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maxima [C]  time = 0.41, size = 60, normalized size = 3.33 \begin {gather*} -8 \, \sqrt {\pi } \operatorname {erf}\left (x + 2\right ) e^{4} - x^{2} - 4 i \, {\left (\frac {2 i \, \sqrt {\pi } {\left (x + 2\right )} {\left (\operatorname {erf}\left (\sqrt {{\left (x + 2\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (x + 2\right )}^{2}}} + i \, e^{\left (-{\left (x + 2\right )}^{2}\right )}\right )} e^{4} + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+2)*exp(x^2+4*x)-8*x-16)/exp(x^2+4*x),x, algorithm="maxima")

[Out]

-8*sqrt(pi)*erf(x + 2)*e^4 - x^2 - 4*I*(2*I*sqrt(pi)*(x + 2)*(erf(sqrt((x + 2)^2)) - 1)/sqrt((x + 2)^2) + I*e^
(-(x + 2)^2))*e^4 + 2*x

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mupad [B]  time = 3.34, size = 21, normalized size = 1.17 \begin {gather*} 2\,x+4\,{\mathrm {e}}^{-x^2-4\,x}-x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(- 4*x - x^2)*(8*x + exp(4*x + x^2)*(2*x - 2) + 16),x)

[Out]

2*x + 4*exp(- 4*x - x^2) - x^2

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sympy [A]  time = 0.12, size = 17, normalized size = 0.94 \begin {gather*} - x^{2} + 2 x + 4 e^{- x^{2} - 4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x+2)*exp(x**2+4*x)-8*x-16)/exp(x**2+4*x),x)

[Out]

-x**2 + 2*x + 4*exp(-x**2 - 4*x)

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