\(\int (-e^{e^{-x}+e^x-x}+e^{e^{-x}+e^x+x}) \, dx\) [140]

Optimal result

Integrand size = 29, antiderivative size = 11 \[ \int \left (-e^{e^{-x}+e^x-x}+e^{e^{-x}+e^x+x}\right ) \, dx=e^{e^{-x}+e^x} \]

[Out]

exp(exp(x)+exp(-x))

Rubi [F]

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

[In]

Int[-E^(E^(-x) + E^x - x) + E^(E^(-x) + E^x + x),x]

[Out]

Defer[Subst][Defer[Int][E^(x^(-1) + x), x], x, E^x] - Defer[Subst][Defer[Int][E^(x^(-1) + x)/x^2, x], x, E^x]

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

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \left (-e^{e^{-x}+e^x-x}+e^{e^{-x}+e^x+x}\right ) \, dx=e^{e^{-x}+e^x} \]

[In]

Integrate[-E^(E^(-x) + E^x - x) + E^(E^(-x) + E^x + x),x]

[Out]

E^(E^(-x) + E^x)

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{-x}}\)	\(9\)
norman	\({\mathrm e}^{{\mathrm e}^{x}+{\mathrm e}^{-x}-x} {\mathrm e}^{x}\)	\(15\)

[In]

int(exp(exp(x)+exp(-x)+x)-exp(exp(x)+exp(-x)-x),x,method=_RETURNVERBOSE)

[Out]

exp(exp(x)+exp(-x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (8) = 16\).

Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.82 \[ \int \left (-e^{e^{-x}+e^x-x}+e^{e^{-x}+e^x+x}\right ) \, dx=e^{\left ({\left (x e^{x} + e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )} - x\right )} \]

[In]

integrate(exp(exp(x)+exp(-x)+x)-exp(exp(x)+exp(-x)-x),x, algorithm="fricas")

[Out]

e^((x*e^x + e^(2*x) + 1)*e^(-x) - x)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.27 \[ \int \left (-e^{e^{-x}+e^x-x}+e^{e^{-x}+e^x+x}\right ) \, dx=e^{x} e^{- x + e^{x} + e^{- x}} \]

[In]

integrate(exp(exp(x)+exp(-x)+x)-exp(exp(x)+exp(-x)-x),x)

[Out]

exp(x)*exp(-x + exp(x) + exp(-x))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \left (-e^{e^{-x}+e^x-x}+e^{e^{-x}+e^x+x}\right ) \, dx=e^{\left (e^{\left (-x\right )} + e^{x}\right )} \]

[In]

integrate(exp(exp(x)+exp(-x)+x)-exp(exp(x)+exp(-x)-x),x, algorithm="maxima")

[Out]

e^(e^(-x) + e^x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \left (-e^{e^{-x}+e^x-x}+e^{e^{-x}+e^x+x}\right ) \, dx=e^{\left ({\left (e^{\left (2 \, x\right )} + 1\right )} e^{\left (-x\right )}\right )} \]

[In]

integrate(exp(exp(x)+exp(-x)+x)-exp(exp(x)+exp(-x)-x),x, algorithm="giac")

[Out]

e^((e^(2*x) + 1)*e^(-x))

Mupad [B] (verification not implemented)

Time = 14.76 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73 \[ \int \left (-e^{e^{-x}+e^x-x}+e^{e^{-x}+e^x+x}\right ) \, dx={\mathrm {e}}^{{\mathrm {e}}^{-x}+{\mathrm {e}}^x} \]

[In]

int(exp(x + exp(-x) + exp(x)) - exp(exp(-x) - x + exp(x)),x)

[Out]

exp(exp(-x) + exp(x))