3.85.79 \(\int 9 e^{-4+e^{9 x}+9 x} \, dx\)

Optimal. Leaf size=11 \[ 2+e^{-4+e^{9 x}} \]

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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 = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {12, 2282, 2194} \begin {gather*} e^{e^{9 x}-4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[9*E^(-4 + E^(9*x) + 9*x),x]

[Out]

E^(-4 + E^(9*x))

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} &=9 \int e^{-4+e^{9 x}+9 x} \, dx\\ &=\operatorname {Subst}\left (\int e^{-4+x} \, dx,x,e^{9 x}\right )\\ &=e^{-4+e^{9 x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 9, normalized size = 0.82 \begin {gather*} e^{-4+e^{9 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[9*E^(-4 + E^(9*x) + 9*x),x]

[Out]

E^(-4 + E^(9*x))

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fricas [A]  time = 0.58, size = 7, normalized size = 0.64 \begin {gather*} e^{\left (e^{\left (9 \, x\right )} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(9*exp(9*x)*exp(exp(9*x)-4),x, algorithm="fricas")

[Out]

e^(e^(9*x) - 4)

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giac [A]  time = 0.14, size = 7, normalized size = 0.64 \begin {gather*} e^{\left (e^{\left (9 \, x\right )} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(9*exp(9*x)*exp(exp(9*x)-4),x, algorithm="giac")

[Out]

e^(e^(9*x) - 4)

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maple [A]  time = 0.02, size = 8, normalized size = 0.73

method	result	size
derivativedivides	\({\mathrm e}^{{\mathrm e}^{9 x}-4}\)	\(8\)
default	\({\mathrm e}^{{\mathrm e}^{9 x}-4}\)	\(8\)
norman	\({\mathrm e}^{{\mathrm e}^{9 x}-4}\)	\(8\)
risch	\({\mathrm e}^{{\mathrm e}^{9 x}-4}\)	\(8\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(9*exp(9*x)*exp(exp(9*x)-4),x,method=_RETURNVERBOSE)

[Out]

exp(exp(9*x)-4)

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maxima [A]  time = 0.42, size = 7, normalized size = 0.64 \begin {gather*} e^{\left (e^{\left (9 \, x\right )} - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(9*exp(9*x)*exp(exp(9*x)-4),x, algorithm="maxima")

[Out]

e^(e^(9*x) - 4)

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mupad [B]  time = 0.05, size = 8, normalized size = 0.73 \begin {gather*} {\mathrm {e}}^{-4}\,{\mathrm {e}}^{{\mathrm {e}}^{9\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(9*exp(exp(9*x) - 4)*exp(9*x),x)

[Out]

exp(-4)*exp(exp(9*x))

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sympy [A]  time = 0.10, size = 7, normalized size = 0.64 \begin {gather*} e^{e^{9 x} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(9*exp(9*x)*exp(exp(9*x)-4),x)

[Out]

exp(exp(9*x) - 4)

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