3.79.69 \(\int e^{80-2 x-x^2} (8-e^{-80+2 x+x^2}+8 x) \, dx\)

Optimal. Leaf size=18 \[ 9-4 e^{81-(1+x)^2}-x \]

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Rubi [A]  time = 0.11, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {6688, 2236} \begin {gather*} -4 e^{-x^2-2 x+80}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(80 - 2*x - x^2)*(8 - E^(-80 + 2*x + x^2) + 8*x),x]

[Out]

-4*E^(80 - 2*x - x^2) - x

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 (-1+8 e^{80-2 x-x^2} (1+x)\right ) \, dx\\ &=-x+8 \int e^{80-2 x-x^2} (1+x) \, dx\\ &=-4 e^{80-2 x-x^2}-x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 18, normalized size = 1.00 \begin {gather*} -4 e^{80-2 x-x^2}-x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(80 - 2*x - x^2)*(8 - E^(-80 + 2*x + x^2) + 8*x),x]

[Out]

-4*E^(80 - 2*x - x^2) - x

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fricas [A]  time = 0.71, size = 26, normalized size = 1.44 \begin {gather*} -{\left (x e^{\left (x^{2} + 2 \, x - 80\right )} + 4\right )} e^{\left (-x^{2} - 2 \, x + 80\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x^2+2*x-80)+8*x+8)/exp(x^2+2*x-80),x, algorithm="fricas")

[Out]

-(x*e^(x^2 + 2*x - 80) + 4)*e^(-x^2 - 2*x + 80)

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giac [A]  time = 0.15, size = 17, normalized size = 0.94 \begin {gather*} -x - 4 \, e^{\left (-x^{2} - 2 \, x + 80\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x^2+2*x-80)+8*x+8)/exp(x^2+2*x-80),x, algorithm="giac")

[Out]

-x - 4*e^(-x^2 - 2*x + 80)

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maple [A]  time = 0.03, size = 16, normalized size = 0.89

method	result	size
risch	\(-x -4 \,{\mathrm e}^{-\left (x +10\right ) \left (-8+x \right )}\)	\(16\)
default	\(-x -4 \,{\mathrm e}^{-x^{2}-2 x +80}\)	\(18\)
norman	\(\left (-4-x \,{\mathrm e}^{x^{2}+2 x -80}\right ) {\mathrm e}^{-x^{2}-2 x +80}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-exp(x^2+2*x-80)+8*x+8)/exp(x^2+2*x-80),x,method=_RETURNVERBOSE)

[Out]

-x-4*exp(-(x+10)*(-8+x))

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maxima [C]  time = 0.40, size = 55, normalized size = 3.06 \begin {gather*} 4 \, \sqrt {\pi } \operatorname {erf}\left (x + 1\right ) e^{81} + 4 i \, {\left (\frac {i \, \sqrt {\pi } {\left (x + 1\right )} {\left (\operatorname {erf}\left (\sqrt {{\left (x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (x + 1\right )}^{2}}} + i \, e^{\left (-{\left (x + 1\right )}^{2}\right )}\right )} e^{81} - x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x^2+2*x-80)+8*x+8)/exp(x^2+2*x-80),x, algorithm="maxima")

[Out]

4*sqrt(pi)*erf(x + 1)*e^81 + 4*I*(I*sqrt(pi)*(x + 1)*(erf(sqrt((x + 1)^2)) - 1)/sqrt((x + 1)^2) + I*e^(-(x + 1
)^2))*e^81 - x

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mupad [B]  time = 5.56, size = 18, normalized size = 1.00 \begin {gather*} -x-4\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{80}\,{\mathrm {e}}^{-x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(80 - x^2 - 2*x)*(8*x - exp(2*x + x^2 - 80) + 8),x)

[Out]

- x - 4*exp(-2*x)*exp(80)*exp(-x^2)

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sympy [A]  time = 0.11, size = 14, normalized size = 0.78 \begin {gather*} - x - 4 e^{- x^{2} - 2 x + 80} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-exp(x**2+2*x-80)+8*x+8)/exp(x**2+2*x-80),x)

[Out]

-x - 4*exp(-x**2 - 2*x + 80)

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