\(\int \frac {e^{5-e^{e^{\frac {e^{16} x}{3}}}-x} (-6-2 e^{16+e^{\frac {e^{16} x}{3}}+\frac {e^{16} x}{3}}-3 e^{-5+e^{e^{\frac {e^{16} x}{3}}}+x} \log (5))}{3 \log (5)} \, dx\) [5420]

Optimal result

Integrand size = 76, antiderivative size = 31 \[ \int \frac {e^{5-e^{e^{\frac {e^{16} x}{3}}}-x} \left (-6-2 e^{16+e^{\frac {e^{16} x}{3}}+\frac {e^{16} x}{3}}-3 e^{-5+e^{e^{\frac {e^{16} x}{3}}}+x} \log (5)\right )}{3 \log (5)} \, dx=-x+\frac {2 e^{5-e^{e^{\frac {e^{16} x}{3}}}-x}}{\log (5)} \]

[Out]

2/ln(5)/exp(exp(exp(1/3*x*exp(16)))+x-5)-x

Rubi [F]

\[ \int \frac {e^{5-e^{e^{\frac {e^{16} x}{3}}}-x} \left (-6-2 e^{16+e^{\frac {e^{16} x}{3}}+\frac {e^{16} x}{3}}-3 e^{-5+e^{e^{\frac {e^{16} x}{3}}}+x} \log (5)\right )}{3 \log (5)} \, dx=\int \frac {e^{5-e^{e^{\frac {e^{16} x}{3}}}-x} \left (-6-2 e^{16+e^{\frac {e^{16} x}{3}}+\frac {e^{16} x}{3}}-3 e^{-5+e^{e^{\frac {e^{16} x}{3}}}+x} \log (5)\right )}{3 \log (5)} \, dx \]

[In]

Int[(E^(5 - E^E^((E^16*x)/3) - x)*(-6 - 2*E^(16 + E^((E^16*x)/3) + (E^16*x)/3) - 3*E^(-5 + E^E^((E^16*x)/3) +
x)*Log[5]))/(3*Log[5]),x]

[Out]

-x - (6*Defer[Subst][Defer[Int][E^(5 - E^E^(E^16*x) - 3*x), x], x, x/3])/Log[5] - (2*Defer[Subst][Defer[Int][E
^(21 - E^E^(E^16*x) + E^(E^16*x) - 3*(1 - E^16/3)*x), x], x, x/3])/Log[5]

Rubi steps \begin{align*} \text {integral}& = \frac {\int e^{5-e^{e^{\frac {e^{16} x}{3}}}-x} \left (-6-2 e^{16+e^{\frac {e^{16} x}{3}}+\frac {e^{16} x}{3}}-3 e^{-5+e^{e^{\frac {e^{16} x}{3}}}+x} \log (5)\right ) \, dx}{3 \log (5)} \\ & = \frac {\text {Subst}\left (\int e^{5-e^{e^{e^{16} x}}-3 x} \left (-6-2 e^{16+e^{e^{16} x}+e^{16} x}-3 e^{-5+e^{e^{e^{16} x}}+3 x} \log (5)\right ) \, dx,x,\frac {x}{3}\right )}{\log (5)} \\ & = \frac {\text {Subst}\left (\int \left (-6 e^{5-e^{e^{e^{16} x}}-3 x}-2 e^{21-e^{e^{e^{16} x}}+e^{e^{16} x}-3 x+e^{16} x}-3 \log (5)\right ) \, dx,x,\frac {x}{3}\right )}{\log (5)} \\ & = -x-\frac {2 \text {Subst}\left (\int e^{21-e^{e^{e^{16} x}}+e^{e^{16} x}-3 x+e^{16} x} \, dx,x,\frac {x}{3}\right )}{\log (5)}-\frac {6 \text {Subst}\left (\int e^{5-e^{e^{e^{16} x}}-3 x} \, dx,x,\frac {x}{3}\right )}{\log (5)} \\ & = -x-\frac {2 \text {Subst}\left (\int e^{21-e^{e^{e^{16} x}}+e^{e^{16} x}-3 \left (1-\frac {e^{16}}{3}\right ) x} \, dx,x,\frac {x}{3}\right )}{\log (5)}-\frac {6 \text {Subst}\left (\int e^{5-e^{e^{e^{16} x}}-3 x} \, dx,x,\frac {x}{3}\right )}{\log (5)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.10 \[ \int \frac {e^{5-e^{e^{\frac {e^{16} x}{3}}}-x} \left (-6-2 e^{16+e^{\frac {e^{16} x}{3}}+\frac {e^{16} x}{3}}-3 e^{-5+e^{e^{\frac {e^{16} x}{3}}}+x} \log (5)\right )}{3 \log (5)} \, dx=-\frac {-6 e^{5-e^{e^{\frac {e^{16} x}{3}}}-x}+x \log (125)}{\log (125)} \]

[In]

Integrate[(E^(5 - E^E^((E^16*x)/3) - x)*(-6 - 2*E^(16 + E^((E^16*x)/3) + (E^16*x)/3) - 3*E^(-5 + E^E^((E^16*x)
/3) + x)*Log[5]))/(3*Log[5]),x]

[Out]

-((-6*E^(5 - E^E^((E^16*x)/3) - x) + x*Log[125])/Log[125])

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84

method	result	size
risch	\(\frac {2 \,{\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{\frac {x \,{\mathrm e}^{16}}{3}}}-x +5}}{\ln \left (5\right )}-x\)	\(26\)
norman	\(\left (\frac {2}{\ln \left (5\right )}-x \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{\frac {x \,{\mathrm e}^{16}}{3}}}+x -5}\right ) {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{\frac {x \,{\mathrm e}^{16}}{3}}}-x +5}\)	\(36\)
parallelrisch	\(-\frac {\left (3 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{\frac {x \,{\mathrm e}^{16}}{3}}}+x -5} x \ln \left (5\right )-6\right ) {\mathrm e}^{-{\mathrm e}^{{\mathrm e}^{\frac {x \,{\mathrm e}^{16}}{3}}}-x +5}}{3 \ln \left (5\right )}\)	\(38\)

[In]

int(1/3*(-3*ln(5)*exp(exp(exp(1/3*x*exp(16)))+x-5)-2*exp(16)*exp(1/3*x*exp(16))*exp(exp(1/3*x*exp(16)))-6)/ln(
5)/exp(exp(exp(1/3*x*exp(16)))+x-5),x,method=_RETURNVERBOSE)

[Out]

2/ln(5)*exp(-exp(exp(1/3*x*exp(16)))-x+5)-x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (25) = 50\).

Time = 0.26 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.81 \[ \int \frac {e^{5-e^{e^{\frac {e^{16} x}{3}}}-x} \left (-6-2 e^{16+e^{\frac {e^{16} x}{3}}+\frac {e^{16} x}{3}}-3 e^{-5+e^{e^{\frac {e^{16} x}{3}}}+x} \log (5)\right )}{3 \log (5)} \, dx=-\frac {{\left (x e^{\left ({\left ({\left (x - 5\right )} e^{\left (\frac {1}{3} \, x e^{16} + 16\right )} + e^{\left (\frac {1}{3} \, x e^{16} + e^{\left (\frac {1}{3} \, x e^{16}\right )} + 16\right )}\right )} e^{\left (-\frac {1}{3} \, x e^{16} - 16\right )}\right )} \log \left (5\right ) - 2\right )} e^{\left (-{\left ({\left (x - 5\right )} e^{\left (\frac {1}{3} \, x e^{16} + 16\right )} + e^{\left (\frac {1}{3} \, x e^{16} + e^{\left (\frac {1}{3} \, x e^{16}\right )} + 16\right )}\right )} e^{\left (-\frac {1}{3} \, x e^{16} - 16\right )}\right )}}{\log \left (5\right )} \]

[In]

integrate(1/3*(-3*log(5)*exp(exp(exp(1/3*x*exp(16)))+x-5)-2*exp(16)*exp(1/3*x*exp(16))*exp(exp(1/3*x*exp(16)))
-6)/log(5)/exp(exp(exp(1/3*x*exp(16)))+x-5),x, algorithm="fricas")

[Out]

-(x*e^(((x - 5)*e^(1/3*x*e^16 + 16) + e^(1/3*x*e^16 + e^(1/3*x*e^16) + 16))*e^(-1/3*x*e^16 - 16))*log(5) - 2)*
e^(-((x - 5)*e^(1/3*x*e^16 + 16) + e^(1/3*x*e^16 + e^(1/3*x*e^16) + 16))*e^(-1/3*x*e^16 - 16))/log(5)

Sympy [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {e^{5-e^{e^{\frac {e^{16} x}{3}}}-x} \left (-6-2 e^{16+e^{\frac {e^{16} x}{3}}+\frac {e^{16} x}{3}}-3 e^{-5+e^{e^{\frac {e^{16} x}{3}}}+x} \log (5)\right )}{3 \log (5)} \, dx=- x + \frac {2 e^{- x - e^{e^{\frac {x e^{16}}{3}}} + 5}}{\log {\left (5 \right )}} \]

[In]

integrate(1/3*(-3*ln(5)*exp(exp(exp(1/3*x*exp(16)))+x-5)-2*exp(16)*exp(1/3*x*exp(16))*exp(exp(1/3*x*exp(16)))-
6)/ln(5)/exp(exp(exp(1/3*x*exp(16)))+x-5),x)

[Out]

-x + 2*exp(-x - exp(exp(x*exp(16)/3)) + 5)/log(5)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.90 \[ \int \frac {e^{5-e^{e^{\frac {e^{16} x}{3}}}-x} \left (-6-2 e^{16+e^{\frac {e^{16} x}{3}}+\frac {e^{16} x}{3}}-3 e^{-5+e^{e^{\frac {e^{16} x}{3}}}+x} \log (5)\right )}{3 \log (5)} \, dx=-\frac {x \log \left (5\right ) - 2 \, e^{\left (-x - e^{\left (e^{\left (\frac {1}{3} \, x e^{16}\right )}\right )} + 5\right )}}{\log \left (5\right )} \]

[In]

integrate(1/3*(-3*log(5)*exp(exp(exp(1/3*x*exp(16)))+x-5)-2*exp(16)*exp(1/3*x*exp(16))*exp(exp(1/3*x*exp(16)))
-6)/log(5)/exp(exp(exp(1/3*x*exp(16)))+x-5),x, algorithm="maxima")

[Out]

-(x*log(5) - 2*e^(-x - e^(e^(1/3*x*e^16)) + 5))/log(5)

Giac [F]

\[ \int \frac {e^{5-e^{e^{\frac {e^{16} x}{3}}}-x} \left (-6-2 e^{16+e^{\frac {e^{16} x}{3}}+\frac {e^{16} x}{3}}-3 e^{-5+e^{e^{\frac {e^{16} x}{3}}}+x} \log (5)\right )}{3 \log (5)} \, dx=\int { -\frac {{\left (3 \, e^{\left (x + e^{\left (e^{\left (\frac {1}{3} \, x e^{16}\right )}\right )} - 5\right )} \log \left (5\right ) + 2 \, e^{\left (\frac {1}{3} \, x e^{16} + e^{\left (\frac {1}{3} \, x e^{16}\right )} + 16\right )} + 6\right )} e^{\left (-x - e^{\left (e^{\left (\frac {1}{3} \, x e^{16}\right )}\right )} + 5\right )}}{3 \, \log \left (5\right )} \,d x } \]

[In]

integrate(1/3*(-3*log(5)*exp(exp(exp(1/3*x*exp(16)))+x-5)-2*exp(16)*exp(1/3*x*exp(16))*exp(exp(1/3*x*exp(16)))
-6)/log(5)/exp(exp(exp(1/3*x*exp(16)))+x-5),x, algorithm="giac")

[Out]

integrate(-1/3*(3*e^(x + e^(e^(1/3*x*e^16)) - 5)*log(5) + 2*e^(1/3*x*e^16 + e^(1/3*x*e^16) + 16) + 6)*e^(-x -
e^(e^(1/3*x*e^16)) + 5)/log(5), x)

Mupad [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {e^{5-e^{e^{\frac {e^{16} x}{3}}}-x} \left (-6-2 e^{16+e^{\frac {e^{16} x}{3}}+\frac {e^{16} x}{3}}-3 e^{-5+e^{e^{\frac {e^{16} x}{3}}}+x} \log (5)\right )}{3 \log (5)} \, dx=\frac {2\,{\mathrm {e}}^{-{\mathrm {e}}^{{\left ({\mathrm {e}}^{x\,{\mathrm {e}}^{16}}\right )}^{1/3}}}\,{\mathrm {e}}^{-x}\,{\mathrm {e}}^5}{\ln \left (5\right )}-x \]

[In]

int(-(exp(5 - exp(exp((x*exp(16))/3)) - x)*(exp(x + exp(exp((x*exp(16))/3)) - 5)*log(5) + (2*exp(exp((x*exp(16
))/3))*exp(16)*exp((x*exp(16))/3))/3 + 2))/log(5),x)

[Out]

(2*exp(-exp(exp(x*exp(16))^(1/3)))*exp(-x)*exp(5))/log(5) - x