\(\int e^{-8 x^2-e^3 x^2} (-16 x-2 e^3 x) \, dx\) [8276]

Optimal result

Integrand size = 27, antiderivative size = 13 \[ \int e^{-8 x^2-e^3 x^2} \left (-16 x-2 e^3 x\right ) \, dx=e^{\left (-8-e^3\right ) x^2} \]

[Out]

exp(x^2*(-8-exp(3)))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6, 12, 2257, 2240} \[ \int e^{-8 x^2-e^3 x^2} \left (-16 x-2 e^3 x\right ) \, dx=e^{-\left (\left (8+e^3\right ) x^2\right )} \]

[In]

Int[E^(-8*x^2 - E^3*x^2)*(-16*x - 2*E^3*x),x]

[Out]

E^(-((8 + E^3)*x^2))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2257

Int[(F_)^(v_)*(u_)^(m_.), x_Symbol] :> Int[ExpandToSum[u, x]^m*F^ExpandToSum[v, x], x] /; FreeQ[{F, m}, x] &&
LinearQ[u, x] && BinomialQ[v, x] &&  !(LinearMatchQ[u, x] && BinomialMatchQ[v, x])

Rubi steps \begin{align*} \text {integral}& = \int e^{-8 x^2-e^3 x^2} \left (-16-2 e^3\right ) x \, dx \\ & = -\left (\left (2 \left (8+e^3\right )\right ) \int e^{-8 x^2-e^3 x^2} x \, dx\right ) \\ & = -\left (\left (2 \left (8+e^3\right )\right ) \int e^{-\left (\left (8+e^3\right ) x^2\right )} x \, dx\right ) \\ & = e^{-\left (\left (8+e^3\right ) x^2\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int e^{-8 x^2-e^3 x^2} \left (-16 x-2 e^3 x\right ) \, dx=e^{-\left (\left (8+e^3\right ) x^2\right )} \]

[In]

Integrate[E^(-8*x^2 - E^3*x^2)*(-16*x - 2*E^3*x),x]

[Out]

E^(-((8 + E^3)*x^2))

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.85

method	result	size
risch	\({\mathrm e}^{-x^{2} \left ({\mathrm e}^{3}+8\right )}\)	\(11\)
parallelrisch	\({\mathrm e}^{-x^{2} \left ({\mathrm e}^{3}+8\right )}\)	\(11\)
gosper	\({\mathrm e}^{-x^{2} {\mathrm e}^{3}-8 x^{2}}\)	\(15\)
derivativedivides	\({\mathrm e}^{-x^{2} {\mathrm e}^{3}-8 x^{2}}\)	\(15\)
norman	\({\mathrm e}^{-x^{2} {\mathrm e}^{3}-8 x^{2}}\)	\(15\)
meijerg	\(\frac {\left (-2 \,{\mathrm e}^{3}-16\right ) \left (1-{\mathrm e}^{-x^{2} \left ({\mathrm e}^{3}+8\right )}\right )}{2 \,{\mathrm e}^{3}+16}\)	\(29\)
default	\(\frac {8 \,{\mathrm e}^{-x^{2} {\mathrm e}^{3}-8 x^{2}}}{{\mathrm e}^{3}+8}+\frac {{\mathrm e}^{3} {\mathrm e}^{-x^{2} {\mathrm e}^{3}-8 x^{2}}}{{\mathrm e}^{3}+8}\)	\(47\)
parts	\(-\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {{\mathrm e}^{3}+8}\, x \right ) x \,{\mathrm e}^{3}}{\sqrt {{\mathrm e}^{3}+8}}-\frac {8 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {{\mathrm e}^{3}+8}\, x \right ) x}{\sqrt {{\mathrm e}^{3}+8}}+x \sqrt {\pi }\, \sqrt {{\mathrm e}^{3}+8}\, \operatorname {erf}\left (\sqrt {{\mathrm e}^{3}+8}\, x \right )+{\mathrm e}^{-x^{2} \left ({\mathrm e}^{3}+8\right )}\)	\(76\)

[In]

int((-2*x*exp(3)-16*x)*exp(-x^2*exp(3)-8*x^2),x,method=_RETURNVERBOSE)

[Out]

exp(-x^2*(exp(3)+8))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int e^{-8 x^2-e^3 x^2} \left (-16 x-2 e^3 x\right ) \, dx=e^{\left (-x^{2} e^{3} - 8 \, x^{2}\right )} \]

[In]

integrate((-2*x*exp(3)-16*x)*exp(-x^2*exp(3)-8*x^2),x, algorithm="fricas")

[Out]

e^(-x^2*e^3 - 8*x^2)

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int e^{-8 x^2-e^3 x^2} \left (-16 x-2 e^3 x\right ) \, dx=e^{- x^{2} e^{3} - 8 x^{2}} \]

[In]

integrate((-2*x*exp(3)-16*x)*exp(-x**2*exp(3)-8*x**2),x)

[Out]

exp(-x**2*exp(3) - 8*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int e^{-8 x^2-e^3 x^2} \left (-16 x-2 e^3 x\right ) \, dx=e^{\left (-x^{2} e^{3} - 8 \, x^{2}\right )} \]

[In]

integrate((-2*x*exp(3)-16*x)*exp(-x^2*exp(3)-8*x^2),x, algorithm="maxima")

[Out]

e^(-x^2*e^3 - 8*x^2)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int e^{-8 x^2-e^3 x^2} \left (-16 x-2 e^3 x\right ) \, dx=e^{\left (-x^{2} e^{3} - 8 \, x^{2}\right )} \]

[In]

integrate((-2*x*exp(3)-16*x)*exp(-x^2*exp(3)-8*x^2),x, algorithm="giac")

[Out]

e^(-x^2*e^3 - 8*x^2)

Mupad [B] (verification not implemented)

Time = 14.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.08 \[ \int e^{-8 x^2-e^3 x^2} \left (-16 x-2 e^3 x\right ) \, dx={\mathrm {e}}^{-x^2\,{\mathrm {e}}^3-8\,x^2} \]

[In]

int(-exp(- x^2*exp(3) - 8*x^2)*(16*x + 2*x*exp(3)),x)

[Out]

exp(- x^2*exp(3) - 8*x^2)