3.46.48 \(\int -e^{-2+e^{-2-x}-x} \, dx\)

Optimal. Leaf size=9 \[ e^{e^{-2-x}} \]

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Rubi [A]  time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2282, 2194} \begin {gather*} e^{e^{-x-2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-E^(-2 + E^(-2 - x) - x),x]

[Out]

E^E^(-2 - x)

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rubi steps

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

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Mathematica [A]  time = 0.05, size = 9, normalized size = 1.00 \begin {gather*} e^{e^{-2-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-E^(-2 + E^(-2 - x) - x),x]

[Out]

E^E^(-2 - x)

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fricas [A]  time = 0.48, size = 7, normalized size = 0.78 \begin {gather*} e^{\left (e^{\left (-x - 2\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-exp(-x-2)*exp(exp(-x-2)),x, algorithm="fricas")

[Out]

e^(e^(-x - 2))

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giac [A]  time = 0.22, size = 7, normalized size = 0.78 \begin {gather*} e^{\left (e^{\left (-x - 2\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-exp(-x-2)*exp(exp(-x-2)),x, algorithm="giac")

[Out]

e^(e^(-x - 2))

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maple [A]  time = 0.02, size = 8, normalized size = 0.89

method	result	size
derivativedivides	\({\mathrm e}^{{\mathrm e}^{-x -2}}\)	\(8\)
default	\({\mathrm e}^{{\mathrm e}^{-x -2}}\)	\(8\)
norman	\({\mathrm e}^{{\mathrm e}^{-x -2}}\)	\(8\)
risch	\({\mathrm e}^{{\mathrm e}^{-x -2}}\)	\(8\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-x-2)*exp(exp(-x-2)),x,method=_RETURNVERBOSE)

[Out]

exp(exp(-x-2))

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maxima [A]  time = 0.42, size = 7, normalized size = 0.78 \begin {gather*} e^{\left (e^{\left (-x - 2\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-exp(-x-2)*exp(exp(-x-2)),x, algorithm="maxima")

[Out]

e^(e^(-x - 2))

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mupad [B]  time = 3.23, size = 7, normalized size = 0.78 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^{-x-2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(exp(- x - 2))*exp(- x - 2),x)

[Out]

exp(exp(- x - 2))

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sympy [A]  time = 0.11, size = 7, normalized size = 0.78 \begin {gather*} e^{e^{- x - 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-exp(-x-2)*exp(exp(-x-2)),x)

[Out]

exp(exp(-x - 2))

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