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

Optimal. Leaf size=15 \[ -e^{4+12 e^x} \log \left (\frac {4}{3}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {12, 2282, 2194} \begin {gather*} -e^{12 e^x+4} \log \left (\frac {4}{3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^(4 + 12*E^x)*Log[4/3])

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 15, normalized size = 1.00 \begin {gather*} -e^{4+12 e^x} \log \left (\frac {4}{3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(E^(4 + 12*E^x)*Log[4/3])

________________________________________________________________________________________

fricas [A]  time = 0.82, size = 15, normalized size = 1.00 \begin {gather*} x e^{\left (12 \, e^{x} - \log \relax (x) + 4\right )} \log \left (\frac {3}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*e^(12*e^x - log(x) + 4)*log(3/4)

________________________________________________________________________________________

giac [A]  time = 0.30, size = 10, normalized size = 0.67 \begin {gather*} e^{\left (12 \, e^{x} + 4\right )} \log \left (\frac {3}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(12*e^x + 4)*log(3/4)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 11, normalized size = 0.73

method	result	size
default	\(\ln \left (\frac {3}{4}\right ) {\mathrm e}^{4} {\mathrm e}^{12 \,{\mathrm e}^{x}}\)	\(11\)
norman	\(\left (\ln \relax (3)-2 \ln \relax (2)\right ) x \,{\mathrm e}^{-\ln \relax (x )+12 \,{\mathrm e}^{x}+4}\)	\(21\)
risch	\({\mathrm e}^{4+12 \,{\mathrm e}^{x}} \ln \relax (3)-2 \,{\mathrm e}^{4+12 \,{\mathrm e}^{x}} \ln \relax (2)\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(3/4)*exp(4)*exp(exp(x))^12

________________________________________________________________________________________

maxima [A]  time = 0.43, size = 10, normalized size = 0.67 \begin {gather*} e^{\left (12 \, e^{x} + 4\right )} \log \left (\frac {3}{4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(12*e^x + 4)*log(3/4)

________________________________________________________________________________________

mupad [B]  time = 0.73, size = 18, normalized size = 1.20 \begin {gather*} -{\mathrm {e}}^4\,{\mathrm {e}}^{12\,{\mathrm {e}}^x}\,\left (2\,\ln \relax (2)-\ln \relax (3)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-exp(4)*exp(12*exp(x))*(2*log(2) - log(3))

________________________________________________________________________________________

sympy [A]  time = 0.15, size = 15, normalized size = 1.00 \begin {gather*} \left (- 2 \log {\relax (2 )} + \log {\relax (3 )}\right ) e^{12 e^{x} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________