3.2.91 \(\int \frac {e^{e^x+x}}{16} \, dx\)

Optimal. Leaf size=11 \[ \frac {1}{16} \left (-1+e^{e^x}\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 9, normalized size of antiderivative = 0.82, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {12, 2282, 2194} \begin {gather*} \frac {e^{e^x}}{16} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(E^x + x)/16,x]

[Out]

E^E^x/16

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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} &=\frac {1}{16} \int e^{e^x+x} \, dx\\ &=\frac {1}{16} \operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )\\ &=\frac {e^{e^x}}{16}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 0.82 \begin {gather*} \frac {e^{e^x}}{16} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(E^x + x)/16,x]

[Out]

E^E^x/16

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fricas [A]  time = 1.05, size = 5, normalized size = 0.45 \begin {gather*} \frac {1}{16} \, e^{\left (e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*exp(x)*exp(exp(x)),x, algorithm="fricas")

[Out]

1/16*e^(e^x)

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giac [A]  time = 0.36, size = 5, normalized size = 0.45 \begin {gather*} \frac {1}{16} \, e^{\left (e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*exp(x)*exp(exp(x)),x, algorithm="giac")

[Out]

1/16*e^(e^x)

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maple [A]  time = 0.02, size = 6, normalized size = 0.55

method	result	size
derivativedivides	\(\frac {{\mathrm e}^{{\mathrm e}^{x}}}{16}\)	\(6\)
default	\(\frac {{\mathrm e}^{{\mathrm e}^{x}}}{16}\)	\(6\)
norman	\(\frac {{\mathrm e}^{{\mathrm e}^{x}}}{16}\)	\(6\)
risch	\(\frac {{\mathrm e}^{{\mathrm e}^{x}}}{16}\)	\(6\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/16*exp(x)*exp(exp(x)),x,method=_RETURNVERBOSE)

[Out]

1/16*exp(exp(x))

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maxima [A]  time = 0.56, size = 5, normalized size = 0.45 \begin {gather*} \frac {1}{16} \, e^{\left (e^{x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*exp(x)*exp(exp(x)),x, algorithm="maxima")

[Out]

1/16*e^(e^x)

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mupad [B]  time = 0.26, size = 5, normalized size = 0.45 \begin {gather*} \frac {{\mathrm {e}}^{{\mathrm {e}}^x}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(exp(x))*exp(x))/16,x)

[Out]

exp(exp(x))/16

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sympy [A]  time = 0.10, size = 5, normalized size = 0.45 \begin {gather*} \frac {e^{e^{x}}}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/16*exp(x)*exp(exp(x)),x)

[Out]

exp(exp(x))/16

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