3.35.97 \(\int e^{4-x-e^{4-x} (4+\log (2))} (-12-3 \log (2)) \, dx\)

Optimal. Leaf size=21 \[ \frac {1}{2}-3 e^{-e^{4-x} (4+\log (2))} \]

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Rubi [A]  time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.14, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {12, 2282, 2194} \begin {gather*} -3 2^{-e^{4-x}} e^{-4 e^{4-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(4 - x - E^(4 - x)*(4 + Log[2]))*(-12 - 3*Log[2]),x]

[Out]

-3/(2^E^(4 - x)*E^(4*E^(4 - 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} &=-\left ((12+3 \log (2)) \int e^{4-x-e^{4-x} (4+\log (2))} \, dx\right )\\ &=(12+3 \log (2)) \operatorname {Subst}\left (\int e^{4-e^4 x (4+\log (2))} \, dx,x,e^{-x}\right )\\ &=-3 2^{-e^{4-x}} e^{-4 e^{4-x}}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.11, size = 17, normalized size = 0.81 \begin {gather*} -3 e^{-e^{4-x} (4+\log (2))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(4 - x - E^(4 - x)*(4 + Log[2]))*(-12 - 3*Log[2]),x]

[Out]

-3/E^(E^(4 - x)*(4 + Log[2]))

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fricas [A]  time = 0.67, size = 15, normalized size = 0.71 \begin {gather*} -3 \, e^{\left (-{\left (\log \relax (2) + 4\right )} e^{\left (-x + 4\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(2)-12)*exp(-x+4)/exp((4+log(2))*exp(-x+4)),x, algorithm="fricas")

[Out]

-3*e^(-(log(2) + 4)*e^(-x + 4))

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giac [B]  time = 0.22, size = 50, normalized size = 2.38 \begin {gather*} -\frac {3 \, {\left (\log \relax (2) + 4\right )} e^{\left (-e^{\left (-x + 4\right )} \log \relax (2) - x - 4 \, e^{\left (-x + 4\right )} + 4\right )}}{e^{\left (-x + 4\right )} \log \relax (2) + 4 \, e^{\left (-x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(2)-12)*exp(-x+4)/exp((4+log(2))*exp(-x+4)),x, algorithm="giac")

[Out]

-3*(log(2) + 4)*e^(-e^(-x + 4)*log(2) - x - 4*e^(-x + 4) + 4)/(e^(-x + 4)*log(2) + 4*e^(-x + 4))

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maple [A]  time = 0.04, size = 17, normalized size = 0.81

method	result	size
norman	\(-3 \,{\mathrm e}^{-\left (4+\ln \relax (2)\right ) {\mathrm e}^{-x +4}}\)	\(17\)
default	\(\frac {\left (-3 \ln \relax (2)-12\right ) {\mathrm e}^{-\left (4+\ln \relax (2)\right ) {\mathrm e}^{-x +4}}}{4+\ln \relax (2)}\)	\(28\)
derivativedivides	\(-\frac {\left (12+3 \ln \relax (2)\right ) {\mathrm e}^{-\left (4+\ln \relax (2)\right ) {\mathrm e}^{-x +4}}}{4+\ln \relax (2)}\)	\(29\)
risch	\(-\frac {3 \,{\mathrm e}^{-\left (4+\ln \relax (2)\right ) {\mathrm e}^{-x +4}} \ln \relax (2)}{4+\ln \relax (2)}-\frac {12 \,{\mathrm e}^{-\left (4+\ln \relax (2)\right ) {\mathrm e}^{-x +4}}}{4+\ln \relax (2)}\)	\(46\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-3*ln(2)-12)*exp(-x+4)/exp((4+ln(2))*exp(-x+4)),x,method=_RETURNVERBOSE)

[Out]

-3/exp((4+ln(2))*exp(-x+4))

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maxima [A]  time = 0.71, size = 22, normalized size = 1.05 \begin {gather*} -3 \, e^{\left (-e^{\left (-x + 4\right )} \log \relax (2) - 4 \, e^{\left (-x + 4\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*log(2)-12)*exp(-x+4)/exp((4+log(2))*exp(-x+4)),x, algorithm="maxima")

[Out]

-3*e^(-e^(-x + 4)*log(2) - 4*e^(-x + 4))

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mupad [B]  time = 0.15, size = 31, normalized size = 1.48 \begin {gather*} -\frac {{\mathrm {e}}^{-4\,{\mathrm {e}}^{4-x}}\,\left (\ln \relax (8)+12\right )}{2^{{\mathrm {e}}^{4-x}}\,\left (\ln \relax (2)+4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(-exp(4 - x)*(log(2) + 4))*exp(4 - x)*(3*log(2) + 12),x)

[Out]

-(exp(-4*exp(4 - x))*(log(8) + 12))/(2^exp(4 - x)*(log(2) + 4))

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sympy [A]  time = 0.16, size = 14, normalized size = 0.67 \begin {gather*} - 3 e^{- \left (\log {\relax (2 )} + 4\right ) e^{4 - x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-3*ln(2)-12)*exp(-x+4)/exp((4+ln(2))*exp(-x+4)),x)

[Out]

-3*exp(-(log(2) + 4)*exp(4 - x))

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