3.716 \(\int e^{e^{e^x}+e^x+x} \, dx\)

Optimal. Leaf size=7 \[ e^{e^{e^x}} \]

[Out]

E^E^E^x

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Rubi [A]  time = 0.0149321, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2282, 2194} \[ e^{e^{e^x}} \]

Antiderivative was successfully verified.

[In]

Int[E^(E^E^x + E^x + x),x]

[Out]

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

Mathematica [A]  time = 0.0109893, size = 7, normalized size = 1. \[ e^{e^{e^x}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(E^E^x + E^x + x),x]

[Out]

E^E^E^x

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Maple [A]  time = 0.021, size = 5, normalized size = 0.7 \begin{align*}{{\rm e}^{{{\rm e}^{{{\rm e}^{x}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(exp(exp(x)))

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Maxima [A]  time = 0.963808, size = 5, normalized size = 0.71 \begin{align*} e^{\left (e^{\left (e^{x}\right )}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(e^(e^x))

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Fricas [A]  time = 0.839971, size = 18, normalized size = 2.57 \begin{align*} e^{\left (e^{\left (e^{x}\right )}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(e^(e^x))

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Sympy [A]  time = 0.975098, size = 5, normalized size = 0.71 \begin{align*} e^{e^{e^{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(exp(exp(x)))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int e^{\left (x + e^{x} + e^{\left (e^{x}\right )}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(e^(x + e^x + e^(e^x)), x)