\(\int (-1+16 e^{2 e^{4 e^{2 x}}+4 e^{2 x}+2 x}) \, dx\) [1227]

Optimal result

Integrand size = 28, antiderivative size = 20 \[ \int \left (-1+16 e^{2 e^{4 e^{2 x}}+4 e^{2 x}+2 x}\right ) \, dx=e^5+e^{2 e^{4 e^{2 x}}}-x \]

[Out]

exp(5)-x+exp(2*exp(4*exp(x)^2))

Rubi [F]

\[ \int \left (-1+16 e^{2 e^{4 e^{2 x}}+4 e^{2 x}+2 x}\right ) \, dx=\int \left (-1+16 e^{2 e^{4 e^{2 x}}+4 e^{2 x}+2 x}\right ) \, dx \]

[In]

Int[-1 + 16*E^(2*E^(4*E^(2*x)) + 4*E^(2*x) + 2*x),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = -x+16 \int e^{2 e^{4 e^{2 x}}+4 e^{2 x}+2 x} \, dx \\ & = -x+8 \text {Subst}\left (\int e^{2 \left (e^{4 x}+2 x\right )} \, dx,x,e^{2 x}\right ) \\ & = -x+4 \text {Subst}\left (\int e^{2 \left (e^{2 x}+x\right )} \, dx,x,2 e^{2 x}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \left (-1+16 e^{2 e^{4 e^{2 x}}+4 e^{2 x}+2 x}\right ) \, dx=e^{2 e^{4 e^{2 x}}}-x \]

[In]

Integrate[-1 + 16*E^(2*E^(4*E^(2*x)) + 4*E^(2*x) + 2*x),x]

[Out]

E^(2*E^(4*E^(2*x))) - x

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75

method	result	size
default	\(-x +{\mathrm e}^{2 \,{\mathrm e}^{4 \,{\mathrm e}^{2 x}}}\)	\(15\)
norman	\(-x +{\mathrm e}^{2 \,{\mathrm e}^{4 \,{\mathrm e}^{2 x}}}\)	\(15\)
risch	\(-x +{\mathrm e}^{2 \,{\mathrm e}^{4 \,{\mathrm e}^{2 x}}}\)	\(15\)
parallelrisch	\(-x +{\mathrm e}^{2 \,{\mathrm e}^{4 \,{\mathrm e}^{2 x}}}\)	\(15\)
parts	\(-x +{\mathrm e}^{2 \,{\mathrm e}^{4 \,{\mathrm e}^{2 x}}}\)	\(15\)
derivativedivides	\(-\ln \left ({\mathrm e}^{x}\right )+{\mathrm e}^{2 \,{\mathrm e}^{4 \,{\mathrm e}^{2 x}}}\)	\(17\)

[In]

int(16*exp(x)^2*exp(4*exp(x)^2)*exp(2*exp(4*exp(x)^2))-1,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (16) = 32\).

Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.45 \[ \int \left (-1+16 e^{2 e^{4 e^{2 x}}+4 e^{2 x}+2 x}\right ) \, dx=-{\left (x e^{\left (2 \, x + 4 \, e^{\left (2 \, x\right )}\right )} - e^{\left (2 \, x + 4 \, e^{\left (2 \, x\right )} + 2 \, e^{\left (4 \, e^{\left (2 \, x\right )}\right )}\right )}\right )} e^{\left (-2 \, x - 4 \, e^{\left (2 \, x\right )}\right )} \]

[In]

integrate(16*exp(x)^2*exp(4*exp(x)^2)*exp(2*exp(4*exp(x)^2))-1,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.60 \[ \int \left (-1+16 e^{2 e^{4 e^{2 x}}+4 e^{2 x}+2 x}\right ) \, dx=- x + e^{2 e^{4 e^{2 x}}} \]

[In]

integrate(16*exp(x)**2*exp(4*exp(x)**2)*exp(2*exp(4*exp(x)**2))-1,x)

[Out]

-x + exp(2*exp(4*exp(2*x)))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \left (-1+16 e^{2 e^{4 e^{2 x}}+4 e^{2 x}+2 x}\right ) \, dx=-x + e^{\left (2 \, e^{\left (4 \, e^{\left (2 \, x\right )}\right )}\right )} \]

[In]

integrate(16*exp(x)^2*exp(4*exp(x)^2)*exp(2*exp(4*exp(x)^2))-1,x, algorithm="maxima")

[Out]

-x + e^(2*e^(4*e^(2*x)))

Giac [F]

\[ \int \left (-1+16 e^{2 e^{4 e^{2 x}}+4 e^{2 x}+2 x}\right ) \, dx=\int { 16 \, e^{\left (2 \, x + 4 \, e^{\left (2 \, x\right )} + 2 \, e^{\left (4 \, e^{\left (2 \, x\right )}\right )}\right )} - 1 \,d x } \]

[In]

integrate(16*exp(x)^2*exp(4*exp(x)^2)*exp(2*exp(4*exp(x)^2))-1,x, algorithm="giac")

[Out]

integrate(16*e^(2*x + 4*e^(2*x) + 2*e^(4*e^(2*x))) - 1, x)

Mupad [B] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \left (-1+16 e^{2 e^{4 e^{2 x}}+4 e^{2 x}+2 x}\right ) \, dx={\mathrm {e}}^{2\,{\mathrm {e}}^{4\,{\mathrm {e}}^{2\,x}}}-x \]

[In]

int(16*exp(4*exp(2*x))*exp(2*x)*exp(2*exp(4*exp(2*x))) - 1,x)

[Out]

exp(2*exp(4*exp(2*x))) - x