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

Optimal. Leaf size=17 \[ 4 \left (-5+e^{e^{4-x}}\right )-4 x \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 15, normalized size of antiderivative = 0.88, number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2282, 2194} \begin {gather*} 4 e^{e^{4-x}}-4 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-4 - 4*E^(4 + E^(4 - x) - x),x]

[Out]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 15, normalized size = 0.88 \begin {gather*} 4 e^{e^{4-x}}-4 x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4*E^E^(4 - x) - 4*x

________________________________________________________________________________________

fricas [B]  time = 0.89, size = 29, normalized size = 1.71 \begin {gather*} -4 \, {\left (x e^{\left (-x + 4\right )} - e^{\left (-x + e^{\left (-x + 4\right )} + 4\right )}\right )} e^{\left (x - 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.45, size = 13, normalized size = 0.76 \begin {gather*} -4 \, x + 4 \, e^{\left (e^{\left (-x + 4\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*x + 4*e^(e^(-x + 4))

________________________________________________________________________________________

maple [A]  time = 0.02, size = 14, normalized size = 0.82

method	result	size
default	\(-4 x +4 \,{\mathrm e}^{{\mathrm e}^{-x +4}}\)	\(14\)
norman	\(-4 x +4 \,{\mathrm e}^{{\mathrm e}^{-x +4}}\)	\(14\)
risch	\(-4 x +4 \,{\mathrm e}^{{\mathrm e}^{-x +4}}\)	\(14\)
derivativedivides	\(4 \ln \left ({\mathrm e}^{-x +4}\right )+4 \,{\mathrm e}^{{\mathrm e}^{-x +4}}\)	\(20\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.45, size = 13, normalized size = 0.76 \begin {gather*} -4 \, x + 4 \, e^{\left (e^{\left (-x + 4\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*x + 4*e^(e^(-x + 4))

________________________________________________________________________________________

mupad [B]  time = 0.09, size = 13, normalized size = 0.76 \begin {gather*} 4\,{\mathrm {e}}^{{\mathrm {e}}^{4-x}}-4\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*exp(exp(4 - x)) - 4*x

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 10, normalized size = 0.59 \begin {gather*} - 4 x + 4 e^{e^{4 - x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*x + 4*exp(exp(4 - x))

________________________________________________________________________________________