3.68.85 \(\int e^{7-e^{2+x}+x} \, dx\)

Optimal. Leaf size=19 \[ e^{16+e}-e^{5-e^{2+x}} \]

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Rubi [A]  time = 0.01, antiderivative size = 13, normalized size of antiderivative = 0.68, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2282, 2194} \begin {gather*} -e^{5-e^{x+2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(7 - E^(2 + x) + x),x]

[Out]

-E^(5 - 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^{7-e^2 x} \, dx,x,e^x\right )\\ &=-e^{5-e^{2+x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 13, normalized size = 0.68 \begin {gather*} -e^{5-e^{2+x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(7 - E^(2 + x) + x),x]

[Out]

-E^(5 - E^(2 + x))

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fricas [A]  time = 0.64, size = 11, normalized size = 0.58 \begin {gather*} -e^{\left (-e^{\left (x + 2\right )} + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2+x)*exp(-exp(2+x)+5),x, algorithm="fricas")

[Out]

-e^(-e^(x + 2) + 5)

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giac [A]  time = 0.13, size = 11, normalized size = 0.58 \begin {gather*} -e^{\left (-e^{\left (x + 2\right )} + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2+x)*exp(-exp(2+x)+5),x, algorithm="giac")

[Out]

-e^(-e^(x + 2) + 5)

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maple [A]  time = 0.02, size = 12, normalized size = 0.63

method	result	size
derivativedivides	\(-{\mathrm e}^{-{\mathrm e}^{2+x}+5}\)	\(12\)
default	\(-{\mathrm e}^{-{\mathrm e}^{2+x}+5}\)	\(12\)
norman	\(-{\mathrm e}^{-{\mathrm e}^{2+x}+5}\)	\(12\)
risch	\(-{\mathrm e}^{-{\mathrm e}^{2+x}+5}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(2+x)*exp(-exp(2+x)+5),x,method=_RETURNVERBOSE)

[Out]

-exp(-exp(2+x)+5)

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maxima [A]  time = 0.36, size = 11, normalized size = 0.58 \begin {gather*} -e^{\left (-e^{\left (x + 2\right )} + 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2+x)*exp(-exp(2+x)+5),x, algorithm="maxima")

[Out]

-e^(-e^(x + 2) + 5)

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mupad [B]  time = 0.05, size = 11, normalized size = 0.58 \begin {gather*} -{\mathrm {e}}^{-{\mathrm {e}}^{x+2}}\,{\mathrm {e}}^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(x + 2)*exp(5 - exp(x + 2)),x)

[Out]

-exp(-exp(x + 2))*exp(5)

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sympy [A]  time = 0.11, size = 8, normalized size = 0.42 \begin {gather*} - e^{5 - e^{x + 2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(2+x)*exp(-exp(2+x)+5),x)

[Out]

-exp(5 - exp(x + 2))

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