\(\int e^{e^{e^x+2 x}+8 x \log (6)} (e^{e^x+2 x} (2+e^x)+8 \log (6)) \, dx\) [963]

Optimal result

Integrand size = 38, antiderivative size = 17 \[ \int e^{e^{e^x+2 x}+8 x \log (6)} \left (e^{e^x+2 x} \left (2+e^x\right )+8 \log (6)\right ) \, dx=e^{e^{e^x+2 x}+8 x \log (6)} \]

[Out]

exp(exp(exp(x)+2*x)+8*x*ln(6))

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {6838} \[ \int e^{e^{e^x+2 x}+8 x \log (6)} \left (e^{e^x+2 x} \left (2+e^x\right )+8 \log (6)\right ) \, dx=6^{8 x} e^{e^{2 x+e^x}} \]

[In]

Int[E^(E^(E^x + 2*x) + 8*x*Log[6])*(E^(E^x + 2*x)*(2 + E^x) + 8*Log[6]),x]

[Out]

6^(8*x)*E^E^(E^x + 2*x)

Rule 6838

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[q*(F^v/Log[F]), x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = 6^{8 x} e^{e^{e^x+2 x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int e^{e^{e^x+2 x}+8 x \log (6)} \left (e^{e^x+2 x} \left (2+e^x\right )+8 \log (6)\right ) \, dx=6^{8 x} e^{e^{e^x+2 x}} \]

[In]

Integrate[E^(E^(E^x + 2*x) + 8*x*Log[6])*(E^(E^x + 2*x)*(2 + E^x) + 8*Log[6]),x]

[Out]

6^(8*x)*E^E^(E^x + 2*x)

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}+2 x}+8 x \ln \left (6\right )}\)	\(15\)
default	\({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}+2 x}+8 x \ln \left (6\right )}\)	\(15\)
norman	\({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}+2 x}+8 x \ln \left (6\right )}\)	\(15\)
parallelrisch	\({\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}+2 x}+8 x \ln \left (6\right )}\)	\(15\)
risch	\(6561^{x} 256^{x} {\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}+2 x}}\)	\(16\)

[In]

int(((exp(x)+2)*exp(exp(x)+2*x)+8*ln(6))*exp(exp(exp(x)+2*x)+8*x*ln(6)),x,method=_RETURNVERBOSE)

[Out]

exp(exp(exp(x)+2*x)+8*x*ln(6))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int e^{e^{e^x+2 x}+8 x \log (6)} \left (e^{e^x+2 x} \left (2+e^x\right )+8 \log (6)\right ) \, dx=e^{\left (8 \, x \log \left (6\right ) + e^{\left (2 \, x + e^{x}\right )}\right )} \]

[In]

integrate(((exp(x)+2)*exp(exp(x)+2*x)+8*log(6))*exp(exp(exp(x)+2*x)+8*x*log(6)),x, algorithm="fricas")

[Out]

e^(8*x*log(6) + e^(2*x + e^x))

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int e^{e^{e^x+2 x}+8 x \log (6)} \left (e^{e^x+2 x} \left (2+e^x\right )+8 \log (6)\right ) \, dx=e^{8 x \log {\left (6 \right )} + e^{2 x + e^{x}}} \]

[In]

integrate(((exp(x)+2)*exp(exp(x)+2*x)+8*ln(6))*exp(exp(exp(x)+2*x)+8*x*ln(6)),x)

[Out]

exp(8*x*log(6) + exp(2*x + exp(x)))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int e^{e^{e^x+2 x}+8 x \log (6)} \left (e^{e^x+2 x} \left (2+e^x\right )+8 \log (6)\right ) \, dx=e^{\left (8 \, x \log \left (6\right ) + e^{\left (2 \, x + e^{x}\right )}\right )} \]

[In]

integrate(((exp(x)+2)*exp(exp(x)+2*x)+8*log(6))*exp(exp(exp(x)+2*x)+8*x*log(6)),x, algorithm="maxima")

[Out]

e^(8*x*log(6) + e^(2*x + e^x))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int e^{e^{e^x+2 x}+8 x \log (6)} \left (e^{e^x+2 x} \left (2+e^x\right )+8 \log (6)\right ) \, dx=e^{\left (8 \, x \log \left (6\right ) + e^{\left (2 \, x + e^{x}\right )}\right )} \]

[In]

integrate(((exp(x)+2)*exp(exp(x)+2*x)+8*log(6))*exp(exp(exp(x)+2*x)+8*x*log(6)),x, algorithm="giac")

[Out]

e^(8*x*log(6) + e^(2*x + e^x))

Mupad [B] (verification not implemented)

Time = 8.48 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int e^{e^{e^x+2 x}+8 x \log (6)} \left (e^{e^x+2 x} \left (2+e^x\right )+8 \log (6)\right ) \, dx=6^{8\,x}\,{\mathrm {e}}^{{\mathrm {e}}^{2\,x}\,{\mathrm {e}}^{{\mathrm {e}}^x}} \]

[In]

int(exp(exp(2*x + exp(x)) + 8*x*log(6))*(8*log(6) + exp(2*x + exp(x))*(exp(x) + 2)),x)

[Out]

6^(8*x)*exp(exp(2*x)*exp(exp(x)))