3.59.96 \(\int (1+4 e^{4 e^{e^{e^x}}+e^{e^x}+e^x+x}) \, dx\)

Optimal. Leaf size=18 \[ 260+e^{4 e^{e^{e^x}}}+x-\log (4) \]

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Rubi [A]  time = 0.05, antiderivative size = 13, normalized size of antiderivative = 0.72, number of steps used = 5, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2282, 2194} \begin {gather*} x+e^{4 e^{e^{e^x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1 + 4*E^(4*E^E^E^x + E^E^x + E^x + x),x]

[Out]

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

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Mathematica [A]  time = 0.04, size = 13, normalized size = 0.72 \begin {gather*} e^{4 e^{e^{e^x}}}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1 + 4*E^(4*E^E^E^x + E^E^x + E^x + x),x]

[Out]

E^(4*E^E^E^x) + x

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fricas [B]  time = 0.71, size = 40, normalized size = 2.22 \begin {gather*} {\left (x e^{\left (x + e^{x} + e^{\left (e^{x}\right )}\right )} + e^{\left (x + e^{x} + e^{\left (e^{x}\right )} + 4 \, e^{\left (e^{\left (e^{x}\right )}\right )}\right )}\right )} e^{\left (-x - e^{x} - e^{\left (e^{x}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(x)*exp(exp(x))*exp(exp(exp(x)))*exp(4*exp(exp(exp(x))))+1,x, algorithm="fricas")

[Out]

(x*e^(x + e^x + e^(e^x)) + e^(x + e^x + e^(e^x) + 4*e^(e^(e^x))))*e^(-x - e^x - e^(e^x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int 4 \, e^{\left (x + e^{x} + e^{\left (e^{x}\right )} + 4 \, e^{\left (e^{\left (e^{x}\right )}\right )}\right )} + 1\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(x)*exp(exp(x))*exp(exp(exp(x)))*exp(4*exp(exp(exp(x))))+1,x, algorithm="giac")

[Out]

integrate(4*e^(x + e^x + e^(e^x) + 4*e^(e^(e^x))) + 1, x)

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maple [A]  time = 0.07, size = 10, normalized size = 0.56

method	result	size
default	\(x +{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}}}\)	\(10\)
norman	\(x +{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}}}\)	\(10\)
risch	\(x +{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}}}\)	\(10\)
derivativedivides	\(\ln \left ({\mathrm e}^{x}\right )+{\mathrm e}^{4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}}}\)	\(12\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*exp(x)*exp(exp(x))*exp(exp(exp(x)))*exp(4*exp(exp(exp(x))))+1,x,method=_RETURNVERBOSE)

[Out]

x+exp(4*exp(exp(exp(x))))

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maxima [A]  time = 0.36, size = 9, normalized size = 0.50 \begin {gather*} x + e^{\left (4 \, e^{\left (e^{\left (e^{x}\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(x)*exp(exp(x))*exp(exp(exp(x)))*exp(4*exp(exp(exp(x))))+1,x, algorithm="maxima")

[Out]

x + e^(4*e^(e^(e^x)))

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mupad [B]  time = 0.07, size = 9, normalized size = 0.50 \begin {gather*} x+{\mathrm {e}}^{4\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(4*exp(exp(x))*exp(4*exp(exp(exp(x))))*exp(exp(exp(x)))*exp(x) + 1,x)

[Out]

x + exp(4*exp(exp(exp(x))))

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sympy [A]  time = 0.37, size = 10, normalized size = 0.56 \begin {gather*} x + e^{4 e^{e^{e^{x}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(4*exp(x)*exp(exp(x))*exp(exp(exp(x)))*exp(4*exp(exp(exp(x))))+1,x)

[Out]

x + exp(4*exp(exp(exp(x))))

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