\(\int (e^{-e^x+x}+e^{e^x+x}) \, dx\) [196]

Optimal result

Integrand size = 17, antiderivative size = 6 \[ \int \left (e^{-e^x+x}+e^{e^x+x}\right ) \, dx=2 \sinh \left (e^x\right ) \]

[Out]

2*sinh(exp(x))

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(15\) vs. \(2(6)=12\).

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 2.50, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2320, 2225} \[ \int \left (e^{-e^x+x}+e^{e^x+x}\right ) \, dx=e^{e^x}-e^{-e^x} \]

[In]

Int[E^(-E^x + x) + E^(E^x + x),x]

[Out]

-E^(-E^x) + E^E^x

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(15\) vs. \(2(6)=12\).

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 2.50 \[ \int \left (e^{-e^x+x}+e^{e^x+x}\right ) \, dx=-e^{-e^x}+e^{e^x} \]

[In]

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

[Out]

-E^(-E^x) + E^E^x

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(11\) vs. \(2(5)=10\).

Time = 0.03 (sec) , antiderivative size = 12, normalized size of antiderivative = 2.00

method	result	size
default	\(-{\mathrm e}^{-{\mathrm e}^{x}}+{\mathrm e}^{{\mathrm e}^{x}}\)	\(12\)
parts	\(-{\mathrm e}^{-{\mathrm e}^{x}}+{\mathrm e}^{{\mathrm e}^{x}}\)	\(12\)
risch	\(\left ({\mathrm e}^{{\mathrm e}^{x}+x}-{\mathrm e}^{x -{\mathrm e}^{x}}\right ) {\mathrm e}^{-x}\)	\(21\)

[In]

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

[Out]

-1/exp(exp(x))+exp(exp(x))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (5) = 10\).

Time = 0.25 (sec) , antiderivative size = 27, normalized size of antiderivative = 4.50 \[ \int \left (e^{-e^x+x}+e^{e^x+x}\right ) \, dx=-{\left (e^{\left (2 \, x\right )} - e^{\left (2 \, x + 2 \, e^{x}\right )}\right )} e^{\left (-2 \, x - e^{x}\right )} \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11 vs. \(2 (5) = 10\).

Time = 0.18 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.83 \[ \int \left (e^{-e^x+x}+e^{e^x+x}\right ) \, dx=-e^{\left (-e^{x}\right )} + e^{\left (e^{x}\right )} \]

[In]

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

[Out]

-e^(-e^x) + e^(e^x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11 vs. \(2 (5) = 10\).

Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.83 \[ \int \left (e^{-e^x+x}+e^{e^x+x}\right ) \, dx=-e^{\left (-e^{x}\right )} + e^{\left (e^{x}\right )} \]

[In]

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

[Out]

-e^(-e^x) + e^(e^x)

Mupad [B] (verification not implemented)

Time = 16.61 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.83 \[ \int \left (e^{-e^x+x}+e^{e^x+x}\right ) \, dx=2\,\mathrm {sinh}\left ({\mathrm {e}}^x\right ) \]

[In]

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

[Out]

2*sinh(exp(x))