3.31.24 \(\int \frac {-16 \log (3)+e^{4+e^{\frac {x+(e+x) \log (3)}{\log (3)}}+\frac {x+(e+x) \log (3)}{\log (3)}} (16+16 \log (3))}{\log (3)} \, dx\)

Optimal. Leaf size=23 \[ 2+16 \left (e^{4+e^{e+x+\frac {x}{\log (3)}}}-x\right ) \]

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Rubi [A]  time = 0.06, antiderivative size = 29, normalized size of antiderivative = 1.26, number of steps used = 4, number of rules used = 3, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {12, 2282, 2194} \begin {gather*} 16 e^{3^{\frac {x+e}{\log (3)}} e^{\frac {x}{\log (3)}}+4}-16 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-16*Log[3] + E^(4 + E^((x + (E + x)*Log[3])/Log[3]) + (x + (E + x)*Log[3])/Log[3])*(16 + 16*Log[3]))/Log[
3],x]

[Out]

16*E^(4 + 3^((E + x)/Log[3])*E^(x/Log[3])) - 16*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} &=\frac {\int \left (-16 \log (3)+\exp \left (4+e^{\frac {x+(e+x) \log (3)}{\log (3)}}+\frac {x+(e+x) \log (3)}{\log (3)}\right ) (16+16 \log (3))\right ) \, dx}{\log (3)}\\ &=-16 x+\frac {(16 (1+\log (3))) \int \exp \left (4+e^{\frac {x+(e+x) \log (3)}{\log (3)}}+\frac {x+(e+x) \log (3)}{\log (3)}\right ) \, dx}{\log (3)}\\ &=-16 x+16 \operatorname {Subst}\left (\int e^{4+x} \, dx,x,e^{\frac {x+(e+x) \log (3)}{\log (3)}}\right )\\ &=16 e^{4+3^{\frac {e+x}{\log (3)}} e^{\frac {x}{\log (3)}}}-16 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 21, normalized size = 0.91 \begin {gather*} 16 \left (e^{4+e^{e+x+\frac {x}{\log (3)}}}-x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-16*Log[3] + E^(4 + E^((x + (E + x)*Log[3])/Log[3]) + (x + (E + x)*Log[3])/Log[3])*(16 + 16*Log[3])
)/Log[3],x]

[Out]

16*(E^(4 + E^(E + x + x/Log[3])) - x)

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fricas [B]  time = 0.84, size = 72, normalized size = 3.13 \begin {gather*} -16 \, {\left (x e^{\left (\frac {{\left (x + e\right )} \log \relax (3) + x}{\log \relax (3)}\right )} - e^{\left (\frac {{\left (x + e + 4\right )} \log \relax (3) + e^{\left (\frac {{\left (x + e\right )} \log \relax (3) + x}{\log \relax (3)}\right )} \log \relax (3) + x}{\log \relax (3)}\right )}\right )} e^{\left (-\frac {{\left (x + e\right )} \log \relax (3) + x}{\log \relax (3)}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*log(3)+16)*exp(((x+exp(1))*log(3)+x)/log(3))*exp(exp(((x+exp(1))*log(3)+x)/log(3))+4)-16*log(3)
)/log(3),x, algorithm="fricas")

[Out]

-16*(x*e^(((x + e)*log(3) + x)/log(3)) - e^(((x + e + 4)*log(3) + e^(((x + e)*log(3) + x)/log(3))*log(3) + x)/
log(3)))*e^(-((x + e)*log(3) + x)/log(3))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {16 \, {\left ({\left (\log \relax (3) + 1\right )} e^{\left (\frac {{\left (x + e\right )} \log \relax (3) + x}{\log \relax (3)} + e^{\left (\frac {{\left (x + e\right )} \log \relax (3) + x}{\log \relax (3)}\right )} + 4\right )} - \log \relax (3)\right )}}{\log \relax (3)}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*log(3)+16)*exp(((x+exp(1))*log(3)+x)/log(3))*exp(exp(((x+exp(1))*log(3)+x)/log(3))+4)-16*log(3)
)/log(3),x, algorithm="giac")

[Out]

integrate(16*((log(3) + 1)*e^(((x + e)*log(3) + x)/log(3) + e^(((x + e)*log(3) + x)/log(3)) + 4) - log(3))/log
(3), x)

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maple [A]  time = 0.16, size = 25, normalized size = 1.09

method	result	size
norman	\(-16 x +16 \,{\mathrm e}^{{\mathrm e}^{\frac {\left (x +{\mathrm e}\right ) \ln \relax (3)+x}{\ln \relax (3)}}+4}\)	\(25\)
risch	\(-16 x +16 \,{\mathrm e}^{{\mathrm e}^{\frac {\ln \relax (3) {\mathrm e}+x \ln \relax (3)+x}{\ln \relax (3)}}+4}\)	\(27\)
default	\(\frac {\frac {16 \,{\mathrm e}^{{\mathrm e}^{\frac {\left (x +{\mathrm e}\right ) \ln \relax (3)+x}{\ln \relax (3)}}+4} \ln \relax (3)^{2}}{\ln \relax (3)+1}+\frac {16 \,{\mathrm e}^{{\mathrm e}^{\frac {\left (x +{\mathrm e}\right ) \ln \relax (3)+x}{\ln \relax (3)}}+4} \ln \relax (3)}{\ln \relax (3)+1}-16 x \ln \relax (3)}{\ln \relax (3)}\)	\(70\)
derivativedivides	\(\frac {-16 \ln \relax (3) \ln \left ({\mathrm e}^{\frac {\left (x +{\mathrm e}\right ) \ln \relax (3)+x}{\ln \relax (3)}}\right )+16 \,{\mathrm e}^{{\mathrm e}^{\frac {\left (x +{\mathrm e}\right ) \ln \relax (3)+x}{\ln \relax (3)}}+4}+16 \,{\mathrm e}^{{\mathrm e}^{\frac {\left (x +{\mathrm e}\right ) \ln \relax (3)+x}{\ln \relax (3)}}+4} \ln \relax (3)}{\ln \relax (3)+1}\)	\(71\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((16*ln(3)+16)*exp(((x+exp(1))*ln(3)+x)/ln(3))*exp(exp(((x+exp(1))*ln(3)+x)/ln(3))+4)-16*ln(3))/ln(3),x,me
thod=_RETURNVERBOSE)

[Out]

-16*x+16*exp(exp(((x+exp(1))*ln(3)+x)/ln(3))+4)

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maxima [A]  time = 0.44, size = 29, normalized size = 1.26 \begin {gather*} -\frac {16 \, {\left (x \log \relax (3) - e^{\left (e^{\left (x + \frac {x}{\log \relax (3)} + e\right )} + 4\right )} \log \relax (3)\right )}}{\log \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*log(3)+16)*exp(((x+exp(1))*log(3)+x)/log(3))*exp(exp(((x+exp(1))*log(3)+x)/log(3))+4)-16*log(3)
)/log(3),x, algorithm="maxima")

[Out]

-16*(x*log(3) - e^(e^(x + x/log(3) + e) + 4)*log(3))/log(3)

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mupad [B]  time = 0.17, size = 22, normalized size = 0.96 \begin {gather*} 16\,{\mathrm {e}}^4\,{\mathrm {e}}^{{\mathrm {e}}^{\frac {x}{\ln \relax (3)}}\,{\mathrm {e}}^{\mathrm {e}}\,{\mathrm {e}}^x}-16\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(16*log(3) - exp(exp((x + log(3)*(x + exp(1)))/log(3)) + 4)*exp((x + log(3)*(x + exp(1)))/log(3))*(16*log
(3) + 16))/log(3),x)

[Out]

16*exp(4)*exp(exp(x/log(3))*exp(exp(1))*exp(x)) - 16*x

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sympy [A]  time = 0.18, size = 22, normalized size = 0.96 \begin {gather*} - 16 x + 16 e^{e^{\frac {x + \left (x + e\right ) \log {\relax (3 )}}{\log {\relax (3 )}}} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((16*ln(3)+16)*exp(((x+exp(1))*ln(3)+x)/ln(3))*exp(exp(((x+exp(1))*ln(3)+x)/ln(3))+4)-16*ln(3))/ln(3
),x)

[Out]

-16*x + 16*exp(exp((x + (x + E)*log(3))/log(3)) + 4)

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