### 3.261 $$\int e^{e^x+x} \, dx$$

Optimal. Leaf size=5 $e^{e^x}$

[Out]

E^E^x

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Rubi [A]  time = 0.0055676, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.286, Rules used = {2282, 2194} $e^{e^x}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(E^x + x),x]

[Out]

E^E^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]]]

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]

Rubi steps

\begin{align*} \int e^{e^x+x} \, dx &=\operatorname{Subst}\left (\int e^x \, dx,x,e^x\right )\\ &=e^{e^x}\\ \end{align*}

Mathematica [A]  time = 0.0047036, size = 5, normalized size = 1. $e^{e^x}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^(E^x + x),x]

[Out]

E^E^x

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Maple [A]  time = 0.002, size = 4, normalized size = 0.8 \begin{align*}{{\rm e}^{{{\rm e}^{x}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(exp(exp(x)+x),x)

[Out]

exp(exp(x))

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Maxima [A]  time = 0.970828, size = 4, normalized size = 0.8 \begin{align*} e^{\left (e^{x}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x+exp(x)),x, algorithm="maxima")

[Out]

e^(e^x)

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Fricas [A]  time = 1.94064, size = 12, normalized size = 2.4 \begin{align*} e^{\left (e^{x}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x+exp(x)),x, algorithm="fricas")

[Out]

e^(e^x)

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Sympy [A]  time = 0.623809, size = 3, normalized size = 0.6 \begin{align*} e^{e^{x}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x+exp(x)),x)

[Out]

exp(exp(x))

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Giac [A]  time = 1.05456, size = 4, normalized size = 0.8 \begin{align*} e^{\left (e^{x}\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(x+exp(x)),x, algorithm="giac")

[Out]

e^(e^x)