\(\int e^{80-2 x-x^2} (8-e^{-80+2 x+x^2}+8 x) \, dx\) [7869]

Optimal result

Integrand size = 30, antiderivative size = 18 \[ \int e^{80-2 x-x^2} \left (8-e^{-80+2 x+x^2}+8 x\right ) \, dx=9-4 e^{81-(1+x)^2}-x \]

[Out]

9-4/exp((1+x)^2-81)-x

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {6820, 2268} \[ \int e^{80-2 x-x^2} \left (8-e^{-80+2 x+x^2}+8 x\right ) \, dx=-4 e^{-x^2-2 x+80}-x \]

[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 2268

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 6820

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

Rubi steps \begin{align*} \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{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int e^{80-2 x-x^2} \left (8-e^{-80+2 x+x^2}+8 x\right ) \, dx=-4 e^{80-x (2+x)}-x \]

[In]

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

[Out]

-4*E^(80 - x*(2 + x)) - x

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89

method	result	size
risch	\(-x -4 \,{\mathrm e}^{-\left (x +10\right ) \left (-8+x \right )}\)	\(16\)
norman	\(\left (-4-x \,{\mathrm e}^{x^{2}+2 x -80}\right ) {\mathrm e}^{-x^{2}-2 x +80}\)	\(27\)
parallelrisch	\(-\left (x \,{\mathrm e}^{x^{2}+2 x -80}+4\right ) {\mathrm e}^{-x^{2}-2 x +80}\)	\(27\)
default	\(-x +4 \,{\mathrm e}^{80} \sqrt {\pi }\, {\mathrm e} \,\operatorname {erf}\left (1+x \right )+8 \,{\mathrm e}^{80} \left (-\frac {{\mathrm e}^{-x^{2}-2 x}}{2}-\frac {\sqrt {\pi }\, {\mathrm e} \,\operatorname {erf}\left (1+x \right )}{2}\right )\)	\(50\)
parts	\(-x +4 \,{\mathrm e}^{80} \sqrt {\pi }\, {\mathrm e} \,\operatorname {erf}\left (1+x \right )+8 \,{\mathrm e}^{80} \left (-\frac {{\mathrm e}^{-x^{2}-2 x}}{2}-\frac {\sqrt {\pi }\, {\mathrm e} \,\operatorname {erf}\left (1+x \right )}{2}\right )\)	\(50\)

[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))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.44 \[ \int e^{80-2 x-x^2} \left (8-e^{-80+2 x+x^2}+8 x\right ) \, dx=-{\left (x e^{\left (x^{2} + 2 \, x - 80\right )} + 4\right )} e^{\left (-x^{2} - 2 \, x + 80\right )} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.78 \[ \int e^{80-2 x-x^2} \left (8-e^{-80+2 x+x^2}+8 x\right ) \, dx=- x - 4 e^{- x^{2} - 2 x + 80} \]

[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)

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.23 (sec) , antiderivative size = 55, normalized size of antiderivative = 3.06 \[ \int e^{80-2 x-x^2} \left (8-e^{-80+2 x+x^2}+8 x\right ) \, dx=4 \, \sqrt {\pi } \operatorname {erf}\left (x + 1\right ) e^{81} + 4 i \, {\left (\frac {i \, \sqrt {\pi } {\left (x + 1\right )} {\left (\operatorname {erf}\left (\sqrt {{\left (x + 1\right )}^{2}}\right ) - 1\right )}}{\sqrt {{\left (x + 1\right )}^{2}}} + i \, e^{\left (-{\left (x + 1\right )}^{2}\right )}\right )} e^{81} - x \]

[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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int e^{80-2 x-x^2} \left (8-e^{-80+2 x+x^2}+8 x\right ) \, dx=-x - 4 \, e^{\left (-x^{2} - 2 \, x + 80\right )} \]

[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)

Mupad [B] (verification not implemented)

Time = 12.53 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int e^{80-2 x-x^2} \left (8-e^{-80+2 x+x^2}+8 x\right ) \, dx=-x-4\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{80}\,{\mathrm {e}}^{-x^2} \]

[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)