3.61.11 \(\int (64 e^{4 x}-16 e^{e^{e^{e^x}}+e^{e^x}+e^x+x}) \, dx\)

Optimal. Leaf size=22 \[ 16 \left (e^3-e^{e^{e^{e^x}}}+e^{4 x}\right ) \]

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Rubi [A]  time = 0.04, antiderivative size = 19, normalized size of antiderivative = 0.86, number of steps used = 6, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2194, 2282} \begin {gather*} 16 e^{4 x}-16 e^{e^{e^{e^x}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[64*E^(4*x) - 16*E^(E^E^E^x + E^E^x + E^x + x),x]

[Out]

-16*E^E^E^E^x + 16*E^(4*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} &=-\left (16 \int e^{e^{e^{e^x}}+e^{e^x}+e^x+x} \, dx\right )+64 \int e^{4 x} \, dx\\ &=16 e^{4 x}-16 \operatorname {Subst}\left (\int e^{e^{e^x}+e^x+x} \, dx,x,e^x\right )\\ &=16 e^{4 x}-16 \operatorname {Subst}\left (\int e^{e^x+x} \, dx,x,e^{e^x}\right )\\ &=16 e^{4 x}-16 \operatorname {Subst}\left (\int e^x \, dx,x,e^{e^{e^x}}\right )\\ &=-16 e^{e^{e^{e^x}}}+16 e^{4 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 19, normalized size = 0.86 \begin {gather*} -16 e^{e^{e^{e^x}}}+16 e^{4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[64*E^(4*x) - 16*E^(E^E^E^x + E^E^x + E^x + x),x]

[Out]

-16*E^E^E^E^x + 16*E^(4*x)

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fricas [B]  time = 0.64, size = 41, normalized size = 1.86 \begin {gather*} 16 \, {\left (e^{\left (5 \, x + e^{x} + e^{\left (e^{x}\right )}\right )} - e^{\left (x + e^{x} + e^{\left (e^{x}\right )} + 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(-16*exp(x)*exp(exp(x))*exp(exp(exp(x)))*exp(exp(exp(exp(x))))+64*exp(4*x),x, algorithm="fricas")

[Out]

16*(e^(5*x + e^x + e^(e^x)) - e^(x + e^x + e^(e^x) + 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 64 \, e^{\left (4 \, x\right )} - 16 \, e^{\left (x + e^{x} + e^{\left (e^{x}\right )} + e^{\left (e^{\left (e^{x}\right )}\right )}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-16*exp(x)*exp(exp(x))*exp(exp(exp(x)))*exp(exp(exp(exp(x))))+64*exp(4*x),x, algorithm="giac")

[Out]

integrate(64*e^(4*x) - 16*e^(x + e^x + e^(e^x) + e^(e^(e^x))), x)

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maple [A]  time = 0.02, size = 15, normalized size = 0.68

method	result	size
default	\(-16 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}}}+16 \,{\mathrm e}^{4 x}\)	\(15\)
risch	\(-16 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}}}+16 \,{\mathrm e}^{4 x}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-16*exp(x)*exp(exp(x))*exp(exp(exp(x)))*exp(exp(exp(exp(x))))+64*exp(4*x),x,method=_RETURNVERBOSE)

[Out]

-16*exp(exp(exp(exp(x))))+16*exp(4*x)

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maxima [A]  time = 0.35, size = 14, normalized size = 0.64 \begin {gather*} 16 \, e^{\left (4 \, x\right )} - 16 \, e^{\left (e^{\left (e^{\left (e^{x}\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-16*exp(x)*exp(exp(x))*exp(exp(exp(x)))*exp(exp(exp(exp(x))))+64*exp(4*x),x, algorithm="maxima")

[Out]

16*e^(4*x) - 16*e^(e^(e^(e^x)))

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mupad [B]  time = 0.13, size = 14, normalized size = 0.64 \begin {gather*} 16\,{\mathrm {e}}^{4\,x}-16\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(64*exp(4*x) - 16*exp(exp(exp(exp(x))))*exp(exp(x))*exp(exp(exp(x)))*exp(x),x)

[Out]

16*exp(4*x) - 16*exp(exp(exp(exp(x))))

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sympy [A]  time = 0.54, size = 15, normalized size = 0.68 \begin {gather*} 16 e^{4 x} - 16 e^{e^{e^{e^{x}}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-16*exp(x)*exp(exp(x))*exp(exp(exp(x)))*exp(exp(exp(exp(x))))+64*exp(4*x),x)

[Out]

16*exp(4*x) - 16*exp(exp(exp(exp(x))))

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