\(\int (1+e^{e^{\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}}+\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}} (1+5 e^x)) \, dx\) [6833]

Optimal result

Integrand size = 51, antiderivative size = 19 \[ \int \left (1+e^{e^{\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}}+\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}} \left (1+5 e^x\right )\right ) \, dx=3+e^{e^{-\frac {1}{e^{16}}+5 e^x+x}}+x \]

[Out]

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

Rubi [F]

\[ \int \left (1+e^{e^{\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}}+\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}} \left (1+5 e^x\right )\right ) \, dx=\int \left (1+\exp \left (e^{\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}}+\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}\right ) \left (1+5 e^x\right )\right ) \, dx \]

[In]

Int[1 + E^(E^((-1 + 5*E^(16 + x) + E^16*x)/E^16) + (-1 + 5*E^(16 + x) + E^16*x)/E^16)*(1 + 5*E^x),x]

[Out]

x + Defer[Subst][Defer[Int][E^(-E^(-16) + 5*x + E^(-E^(-16) + 5*x)*x), x], x, E^x] + 5*Defer[Subst][Defer[Int]
[E^(-E^(-16) + 5*x + E^(-E^(-16) + 5*x)*x)*x, x], x, E^x]

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95 \[ \int \left (1+e^{e^{\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}}+\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}} \left (1+5 e^x\right )\right ) \, dx=e^{e^{-\frac {1}{e^{16}}+5 e^x+x}}+x \]

[In]

Integrate[1 + E^(E^((-1 + 5*E^(16 + x) + E^16*x)/E^16) + (-1 + 5*E^(16 + x) + E^16*x)/E^16)*(1 + 5*E^x),x]

[Out]

E^E^(-E^(-16) + 5*E^x + x) + x

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05

method	result	size
risch	\(x +{\mathrm e}^{{\mathrm e}^{\left (5 \,{\mathrm e}^{x +16}+x \,{\mathrm e}^{16}-1\right ) {\mathrm e}^{-16}}}\)	\(20\)
default	\(x +{\mathrm e}^{{\mathrm e}^{\left (5 \,{\mathrm e}^{16} {\mathrm e}^{x}+x \,{\mathrm e}^{16}-1\right ) {\mathrm e}^{-16}}}\)	\(22\)
norman	\(x +{\mathrm e}^{{\mathrm e}^{\left (5 \,{\mathrm e}^{16} {\mathrm e}^{x}+x \,{\mathrm e}^{16}-1\right ) {\mathrm e}^{-16}}}\)	\(22\)
parallelrisch	\(x +{\mathrm e}^{{\mathrm e}^{\left (5 \,{\mathrm e}^{16} {\mathrm e}^{x}+x \,{\mathrm e}^{16}-1\right ) {\mathrm e}^{-16}}}\)	\(22\)

[In]

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

[Out]

x+exp(exp((5*exp(x+16)+x*exp(16)-1)*exp(-16)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 71 vs. \(2 (15) = 30\).

Time = 0.24 (sec) , antiderivative size = 71, normalized size of antiderivative = 3.74 \[ \int \left (1+e^{e^{\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}}+\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}} \left (1+5 e^x\right )\right ) \, dx={\left (x e^{\left ({\left (x e^{16} + 5 \, e^{\left (x + 16\right )} - 1\right )} e^{\left (-16\right )}\right )} + e^{\left ({\left (x e^{16} + e^{\left ({\left (x e^{16} + 5 \, e^{\left (x + 16\right )} - 1\right )} e^{\left (-16\right )} + 16\right )} + 5 \, e^{\left (x + 16\right )} - 1\right )} e^{\left (-16\right )}\right )}\right )} e^{\left (-{\left (x e^{16} + 5 \, e^{\left (x + 16\right )} - 1\right )} e^{\left (-16\right )}\right )} \]

[In]

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

[Out]

(x*e^((x*e^16 + 5*e^(x + 16) - 1)*e^(-16)) + e^((x*e^16 + e^((x*e^16 + 5*e^(x + 16) - 1)*e^(-16) + 16) + 5*e^(
x + 16) - 1)*e^(-16)))*e^(-(x*e^16 + 5*e^(x + 16) - 1)*e^(-16))

Sympy [A] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.16 \[ \int \left (1+e^{e^{\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}}+\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}} \left (1+5 e^x\right )\right ) \, dx=x + e^{e^{\frac {x e^{16} + 5 e^{16} e^{x} - 1}{e^{16}}}} \]

[In]

integrate((5*exp(x)+1)*exp((5*exp(16)*exp(x)+x*exp(16)-1)/exp(16))*exp(exp((5*exp(16)*exp(x)+x*exp(16)-1)/exp(
16)))+1,x)

[Out]

x + exp(exp((x*exp(16) + 5*exp(16)*exp(x) - 1)*exp(-16)))

Maxima [A] (verification not implemented)

none

Time = 0.47 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \left (1+e^{e^{\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}}+\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}} \left (1+5 e^x\right )\right ) \, dx=x + e^{\left (e^{\left (x - e^{\left (-16\right )} + 5 \, e^{x}\right )}\right )} \]

[In]

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

[Out]

x + e^(e^(x - e^(-16) + 5*e^x))

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74 \[ \int \left (1+e^{e^{\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}}+\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}} \left (1+5 e^x\right )\right ) \, dx=x + e^{\left (e^{\left (x - e^{\left (-16\right )} + 5 \, e^{x}\right )}\right )} \]

[In]

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

[Out]

x + e^(e^(x - e^(-16) + 5*e^x))

Mupad [B] (verification not implemented)

Time = 12.62 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84 \[ \int \left (1+e^{e^{\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}}+\frac {-1+5 e^{16+x}+e^{16} x}{e^{16}}} \left (1+5 e^x\right )\right ) \, dx=x+{\mathrm {e}}^{{\mathrm {e}}^{-{\mathrm {e}}^{-16}}\,{\mathrm {e}}^{5\,{\mathrm {e}}^x}\,{\mathrm {e}}^x} \]

[In]

int(exp(exp(-16)*(x*exp(16) + 5*exp(16)*exp(x) - 1))*exp(exp(exp(-16)*(x*exp(16) + 5*exp(16)*exp(x) - 1)))*(5*
exp(x) + 1) + 1,x)

[Out]

x + exp(exp(-exp(-16))*exp(5*exp(x))*exp(x))