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

Optimal. Leaf size=13 \[ e^{\left (-8-e^3\right ) x^2} \]

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Rubi [A]  time = 0.04, antiderivative size = 12, normalized size of antiderivative = 0.92, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {6, 12, 2225, 2209} \begin {gather*} e^{-\left (\left (8+e^3\right ) x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[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 2209

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 2225

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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 12, normalized size = 0.92 \begin {gather*} e^{-\left (\left (8+e^3\right ) x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(-((8 + E^3)*x^2))

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fricas [A]  time = 0.65, size = 14, normalized size = 1.08 \begin {gather*} e^{\left (-x^{2} e^{3} - 8 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.15, size = 14, normalized size = 1.08 \begin {gather*} e^{\left (-x^{2} e^{3} - 8 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.05, size = 11, normalized size = 0.85

method	result	size
risch	\({\mathrm e}^{-x^{2} \left ({\mathrm e}^{3}+8\right )}\)	\(11\)
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	\(-\frac {8 \,{\mathrm e}^{x^{2} \left (-8-{\mathrm e}^{3}\right )}}{-8-{\mathrm e}^{3}}-\frac {{\mathrm e}^{3+x^{2} \left (-8-{\mathrm e}^{3}\right )}}{-8-{\mathrm e}^{3}}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maxima [A]  time = 0.36, size = 14, normalized size = 1.08 \begin {gather*} e^{\left (-x^{2} e^{3} - 8 \, x^{2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [B]  time = 5.31, size = 14, normalized size = 1.08 \begin {gather*} {\mathrm {e}}^{-x^2\,{\mathrm {e}}^3-8\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.12, size = 14, normalized size = 1.08 \begin {gather*} e^{- x^{2} e^{3} - 8 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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