\(\int (3+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}+e^x} (e^x+4 e^{e^{e^x}+e^x+x}+2 e^{2 e^{e^x}+e^x+x})) \, dx\) [5150]

Optimal result

Integrand size = 87, antiderivative size = 24 \[ \int \left (3+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}+e^x} \left (e^x+4 e^{e^{e^x}+e^x+x}+2 e^{2 e^{e^x}+e^x+x}\right )\right ) \, dx=-5+e^{e^{e^x+\left (2+e^{e^{e^x}}\right )^2}}+3 x \]

[Out]

3*x-5+exp(exp((2+exp(exp(exp(x))))^2+exp(x)))

Rubi [F]

\[ \int \left (3+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}+e^x} \left (e^x+4 e^{e^{e^x}+e^x+x}+2 e^{2 e^{e^x}+e^x+x}\right )\right ) \, dx=\int \left (3+\exp \left (4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}+e^x\right ) \left (e^x+4 e^{e^{e^x}+e^x+x}+2 e^{2 e^{e^x}+e^x+x}\right )\right ) \, dx \]

[In]

Int[3 + E^(4 + 4*E^E^E^x + E^(2*E^E^x) + E^(4 + 4*E^E^E^x + E^(2*E^E^x) + E^x) + E^x)*(E^x + 4*E^(E^E^x + E^x
+ x) + 2*E^(2*E^E^x + E^x + x)),x]

[Out]

3*x + Defer[Subst][Defer[Int][E^(4 + 4*E^x + E^(2*x) + E^(2 + E^x)^2*x), x], x, E^E^x] + 4*Defer[Subst][Defer[
Int][E^(4 + 4*E^x + E^(2*x) + x + E^(2 + E^x)^2*x)*x, x], x, E^E^x] + 2*Defer[Subst][Defer[Int][E^(4 + 4*E^x +
E^(2*x) + 2*x + E^(2 + E^x)^2*x)*x, x], x, E^E^x]

Rubi steps \begin{align*} \text {integral}& = 3 x+\int \exp \left (4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}+e^x\right ) \left (e^x+4 e^{e^{e^x}+e^x+x}+2 e^{2 e^{e^x}+e^x+x}\right ) \, dx \\ & = 3 x+\text {Subst}\left (\int \exp \left (4+4 e^{e^x}+e^{2 e^x}+e^{4+4 e^{e^x}+e^{2 e^x}+x}+x\right ) \left (1+4 e^{e^x+x}+2 e^{2 e^x+x}\right ) \, dx,x,e^x\right ) \\ & = 3 x+\text {Subst}\left (\int e^{4+4 e^x+e^{2 x}+e^{\left (2+e^x\right )^2} x} \left (1+4 e^x x+2 e^{2 x} x\right ) \, dx,x,e^{e^x}\right ) \\ & = 3 x+\text {Subst}\left (\int \left (e^{4+4 e^x+e^{2 x}+e^{\left (2+e^x\right )^2} x}+4 e^{4+4 e^x+e^{2 x}+x+e^{\left (2+e^x\right )^2} x} x+2 e^{4+4 e^x+e^{2 x}+2 x+e^{\left (2+e^x\right )^2} x} x\right ) \, dx,x,e^{e^x}\right ) \\ & = 3 x+2 \text {Subst}\left (\int e^{4+4 e^x+e^{2 x}+2 x+e^{\left (2+e^x\right )^2} x} x \, dx,x,e^{e^x}\right )+4 \text {Subst}\left (\int e^{4+4 e^x+e^{2 x}+x+e^{\left (2+e^x\right )^2} x} x \, dx,x,e^{e^x}\right )+\text {Subst}\left (\int e^{4+4 e^x+e^{2 x}+e^{\left (2+e^x\right )^2} x} \, dx,x,e^{e^x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.29 \[ \int \left (3+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}+e^x} \left (e^x+4 e^{e^{e^x}+e^x+x}+2 e^{2 e^{e^x}+e^x+x}\right )\right ) \, dx=e^{e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}}+3 x \]

[In]

Integrate[3 + E^(4 + 4*E^E^E^x + E^(2*E^E^x) + E^(4 + 4*E^E^E^x + E^(2*E^E^x) + E^x) + E^x)*(E^x + 4*E^(E^E^x
+ E^x + x) + 2*E^(2*E^E^x + E^x + x)),x]

[Out]

E^E^(4 + 4*E^E^E^x + E^(2*E^E^x) + E^x) + 3*x

Maple [A] (verified)

Time = 1.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96

method	result	size
default	\(3 x +{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{{\mathrm e}^{x}}}+4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}}+{\mathrm e}^{x}+4}}\)	\(23\)
risch	\(3 x +{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{{\mathrm e}^{x}}}+4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}}+{\mathrm e}^{x}+4}}\)	\(23\)
parallelrisch	\(3 x +{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{2 \,{\mathrm e}^{{\mathrm e}^{x}}}+4 \,{\mathrm e}^{{\mathrm e}^{{\mathrm e}^{x}}}+{\mathrm e}^{x}+4}}\)	\(23\)

[In]

int((2*exp(x)*exp(exp(x))*exp(exp(exp(x)))^2+4*exp(x)*exp(exp(x))*exp(exp(exp(x)))+exp(x))*exp(exp(exp(exp(x))
)^2+4*exp(exp(exp(x)))+exp(x)+4)*exp(exp(exp(exp(exp(x)))^2+4*exp(exp(exp(x)))+exp(x)+4))+3,x,method=_RETURNVE
RBOSE)

[Out]

3*x+exp(exp(exp(exp(exp(x)))^2+4*exp(exp(exp(x)))+exp(x)+4))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 242 vs. \(2 (18) = 36\).

Time = 0.26 (sec) , antiderivative size = 242, normalized size of antiderivative = 10.08 \[ \int \left (3+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}+e^x} \left (e^x+4 e^{e^{e^x}+e^x+x}+2 e^{2 e^{e^x}+e^x+x}\right )\right ) \, dx={\left (3 \, x e^{\left ({\left ({\left (e^{\left (3 \, x\right )} + 4 \, e^{\left (2 \, x\right )}\right )} e^{\left (2 \, e^{x}\right )} + e^{\left (2 \, x + 2 \, e^{x} + 2 \, e^{\left (e^{x}\right )}\right )} + 4 \, e^{\left (2 \, x + 2 \, e^{x} + e^{\left (e^{x}\right )}\right )}\right )} e^{\left (-2 \, x - 2 \, e^{x}\right )}\right )} + e^{\left ({\left ({\left (e^{\left (3 \, x\right )} + 4 \, e^{\left (2 \, x\right )}\right )} e^{\left (2 \, e^{x}\right )} + e^{\left ({\left ({\left (e^{\left (3 \, x\right )} + 4 \, e^{\left (2 \, x\right )}\right )} e^{\left (2 \, e^{x}\right )} + e^{\left (2 \, x + 2 \, e^{x} + 2 \, e^{\left (e^{x}\right )}\right )} + 4 \, e^{\left (2 \, x + 2 \, e^{x} + e^{\left (e^{x}\right )}\right )}\right )} e^{\left (-2 \, x - 2 \, e^{x}\right )} + 2 \, x + 2 \, e^{x}\right )} + e^{\left (2 \, x + 2 \, e^{x} + 2 \, e^{\left (e^{x}\right )}\right )} + 4 \, e^{\left (2 \, x + 2 \, e^{x} + e^{\left (e^{x}\right )}\right )}\right )} e^{\left (-2 \, x - 2 \, e^{x}\right )}\right )}\right )} e^{\left (-{\left ({\left (e^{\left (3 \, x\right )} + 4 \, e^{\left (2 \, x\right )}\right )} e^{\left (2 \, e^{x}\right )} + e^{\left (2 \, x + 2 \, e^{x} + 2 \, e^{\left (e^{x}\right )}\right )} + 4 \, e^{\left (2 \, x + 2 \, e^{x} + e^{\left (e^{x}\right )}\right )}\right )} e^{\left (-2 \, x - 2 \, e^{x}\right )}\right )} \]

[In]

integrate((2*exp(x)*exp(exp(x))*exp(exp(exp(x)))^2+4*exp(x)*exp(exp(x))*exp(exp(exp(x)))+exp(x))*exp(exp(exp(e
xp(x)))^2+4*exp(exp(exp(x)))+exp(x)+4)*exp(exp(exp(exp(exp(x)))^2+4*exp(exp(exp(x)))+exp(x)+4))+3,x, algorithm
="fricas")

[Out]

(3*x*e^(((e^(3*x) + 4*e^(2*x))*e^(2*e^x) + e^(2*x + 2*e^x + 2*e^(e^x)) + 4*e^(2*x + 2*e^x + e^(e^x)))*e^(-2*x
- 2*e^x)) + e^(((e^(3*x) + 4*e^(2*x))*e^(2*e^x) + e^(((e^(3*x) + 4*e^(2*x))*e^(2*e^x) + e^(2*x + 2*e^x + 2*e^(
e^x)) + 4*e^(2*x + 2*e^x + e^(e^x)))*e^(-2*x - 2*e^x) + 2*x + 2*e^x) + e^(2*x + 2*e^x + 2*e^(e^x)) + 4*e^(2*x
+ 2*e^x + e^(e^x)))*e^(-2*x - 2*e^x)))*e^(-((e^(3*x) + 4*e^(2*x))*e^(2*e^x) + e^(2*x + 2*e^x + 2*e^(e^x)) + 4*
e^(2*x + 2*e^x + e^(e^x)))*e^(-2*x - 2*e^x))

Sympy [A] (verification not implemented)

Time = 1.69 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.12 \[ \int \left (3+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}+e^x} \left (e^x+4 e^{e^{e^x}+e^x+x}+2 e^{2 e^{e^x}+e^x+x}\right )\right ) \, dx=3 x + e^{e^{e^{x} + e^{2 e^{e^{x}}} + 4 e^{e^{e^{x}}} + 4}} \]

[In]

integrate((2*exp(x)*exp(exp(x))*exp(exp(exp(x)))**2+4*exp(x)*exp(exp(x))*exp(exp(exp(x)))+exp(x))*exp(exp(exp(
exp(x)))**2+4*exp(exp(exp(x)))+exp(x)+4)*exp(exp(exp(exp(exp(x)))**2+4*exp(exp(exp(x)))+exp(x)+4))+3,x)

[Out]

3*x + exp(exp(exp(x) + exp(2*exp(exp(x))) + 4*exp(exp(exp(x))) + 4))

Maxima [A] (verification not implemented)

none

Time = 0.51 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \left (3+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}+e^x} \left (e^x+4 e^{e^{e^x}+e^x+x}+2 e^{2 e^{e^x}+e^x+x}\right )\right ) \, dx=3 \, x + e^{\left (e^{\left (e^{x} + e^{\left (2 \, e^{\left (e^{x}\right )}\right )} + 4 \, e^{\left (e^{\left (e^{x}\right )}\right )} + 4\right )}\right )} \]

[In]

integrate((2*exp(x)*exp(exp(x))*exp(exp(exp(x)))^2+4*exp(x)*exp(exp(x))*exp(exp(exp(x)))+exp(x))*exp(exp(exp(e
xp(x)))^2+4*exp(exp(exp(x)))+exp(x)+4)*exp(exp(exp(exp(exp(x)))^2+4*exp(exp(exp(x)))+exp(x)+4))+3,x, algorithm
="maxima")

[Out]

3*x + e^(e^(e^x + e^(2*e^(e^x)) + 4*e^(e^(e^x)) + 4))

Giac [F]

\[ \int \left (3+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}+e^x} \left (e^x+4 e^{e^{e^x}+e^x+x}+2 e^{2 e^{e^x}+e^x+x}\right )\right ) \, dx=\int { {\left (2 \, e^{\left (x + e^{x} + 2 \, e^{\left (e^{x}\right )}\right )} + 4 \, e^{\left (x + e^{x} + e^{\left (e^{x}\right )}\right )} + e^{x}\right )} e^{\left (e^{x} + e^{\left (2 \, e^{\left (e^{x}\right )}\right )} + e^{\left (e^{x} + e^{\left (2 \, e^{\left (e^{x}\right )}\right )} + 4 \, e^{\left (e^{\left (e^{x}\right )}\right )} + 4\right )} + 4 \, e^{\left (e^{\left (e^{x}\right )}\right )} + 4\right )} + 3 \,d x } \]

[In]

integrate((2*exp(x)*exp(exp(x))*exp(exp(exp(x)))^2+4*exp(x)*exp(exp(x))*exp(exp(exp(x)))+exp(x))*exp(exp(exp(e
xp(x)))^2+4*exp(exp(exp(x)))+exp(x)+4)*exp(exp(exp(exp(exp(x)))^2+4*exp(exp(exp(x)))+exp(x)+4))+3,x, algorithm
="giac")

[Out]

integrate((2*e^(x + e^x + 2*e^(e^x)) + 4*e^(x + e^x + e^(e^x)) + e^x)*e^(e^x + e^(2*e^(e^x)) + e^(e^x + e^(2*e
^(e^x)) + 4*e^(e^(e^x)) + 4) + 4*e^(e^(e^x)) + 4) + 3, x)

Mupad [B] (verification not implemented)

Time = 11.81 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04 \[ \int \left (3+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^{4+4 e^{e^{e^x}}+e^{2 e^{e^x}}+e^x}+e^x} \left (e^x+4 e^{e^{e^x}+e^x+x}+2 e^{2 e^{e^x}+e^x+x}\right )\right ) \, dx=3\,x+{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^4\,{\mathrm {e}}^{{\mathrm {e}}^{2\,{\mathrm {e}}^{{\mathrm {e}}^x}}}\,{\mathrm {e}}^{4\,{\mathrm {e}}^{{\mathrm {e}}^{{\mathrm {e}}^x}}}} \]

[In]

int(exp(exp(exp(2*exp(exp(x))) + 4*exp(exp(exp(x))) + exp(x) + 4))*exp(exp(2*exp(exp(x))) + 4*exp(exp(exp(x)))
+ exp(x) + 4)*(exp(x) + 4*exp(exp(x))*exp(exp(exp(x)))*exp(x) + 2*exp(2*exp(exp(x)))*exp(exp(x))*exp(x)) + 3,
x)

[Out]

3*x + exp(exp(exp(x))*exp(4)*exp(exp(2*exp(exp(x))))*exp(4*exp(exp(exp(x)))))