\(\int (64 e^{4 x}-16 e^{e^{e^{e^x}}+e^{e^x}+e^x+x}) \, dx\) [6011]

Optimal result

Integrand size = 29, antiderivative size = 22 \[ \int \left (64 e^{4 x}-16 e^{e^{e^{e^x}}+e^{e^x}+e^x+x}\right ) \, dx=16 \left (e^3-e^{e^{e^{e^x}}}+e^{4 x}\right ) \]

[Out]

16*exp(3)+16*exp(4*x)-16*exp(exp(exp(exp(x))))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2225, 2320} \[ \int \left (64 e^{4 x}-16 e^{e^{e^{e^x}}+e^{e^x}+e^x+x}\right ) \, dx=16 e^{4 x}-16 e^{e^{e^{e^x}}} \]

[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 2225

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 2320

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{align*} \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 \text {Subst}\left (\int e^{e^{e^x}+e^x+x} \, dx,x,e^x\right ) \\ & = 16 e^{4 x}-16 \text {Subst}\left (\int e^{e^x+x} \, dx,x,e^{e^x}\right ) \\ & = 16 e^{4 x}-16 \text {Subst}\left (\int e^x \, dx,x,e^{e^{e^x}}\right ) \\ & = -16 e^{e^{e^{e^x}}}+16 e^{4 x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \left (64 e^{4 x}-16 e^{e^{e^{e^x}}+e^{e^x}+e^x+x}\right ) \, dx=-16 e^{e^{e^{e^x}}}+16 e^{4 x} \]

[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)

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68

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

[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)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41 vs. \(2 (18) = 36\).

Time = 0.26 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86 \[ \int \left (64 e^{4 x}-16 e^{e^{e^{e^x}}+e^{e^x}+e^x+x}\right ) \, dx=16 \, {\left (e^{\left (5 \, x + e^{x} + e^{\left (e^{x}\right )}\right )} - e^{\left (x + e^{x} + e^{\left (e^{x}\right )} + e^{\left (e^{\left (e^{x}\right )}\right )}\right )}\right )} e^{\left (-x - e^{x} - e^{\left (e^{x}\right )}\right )} \]

[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))

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.68 \[ \int \left (64 e^{4 x}-16 e^{e^{e^{e^x}}+e^{e^x}+e^x+x}\right ) \, dx=16 e^{4 x} - 16 e^{e^{e^{e^{x}}}} \]

[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))))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \left (64 e^{4 x}-16 e^{e^{e^{e^x}}+e^{e^x}+e^x+x}\right ) \, dx=16 \, e^{\left (4 \, x\right )} - 16 \, e^{\left (e^{\left (e^{\left (e^{x}\right )}\right )}\right )} \]

[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)))

Giac [F]

\[ \int \left (64 e^{4 x}-16 e^{e^{e^{e^x}}+e^{e^x}+e^x+x}\right ) \, dx=\int { 64 \, e^{\left (4 \, x\right )} - 16 \, e^{\left (x + e^{x} + e^{\left (e^{x}\right )} + e^{\left (e^{\left (e^{x}\right )}\right )}\right )} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.64 \[ \int \left (64 e^{4 x}-16 e^{e^{e^{e^x}}+e^{e^x}+e^x+x}\right ) \, dx=16\,{\mathrm {e}}^{4\,x}-16\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}} \]

[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))))