\(\int e^{-e^x} (-e^{1+x}+2 e^{e^x} x) \, dx\) [5018]

Optimal result

Integrand size = 24, antiderivative size = 16 \[ \int e^{-e^x} \left (-e^{1+x}+2 e^{e^x} x\right ) \, dx=1+e^{1-e^x}+x^2+\log (3) \]

[Out]

exp(1)/exp(exp(x))+x^2+ln(3)+1

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6820, 2320, 2225} \[ \int e^{-e^x} \left (-e^{1+x}+2 e^{e^x} x\right ) \, dx=x^2+e^{1-e^x} \]

[In]

Int[(-E^(1 + x) + 2*E^E^x*x)/E^E^x,x]

[Out]

E^(1 - E^x) + x^2

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]]]

Rule 6820

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int e^{-e^x} \left (-e^{1+x}+2 e^{e^x} x\right ) \, dx=e^{1-e^x}+x^2 \]

[In]

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

[Out]

E^(1 - E^x) + x^2

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75

method	result	size
risch	\(x^{2}+{\mathrm e}^{1-{\mathrm e}^{x}}\)	\(12\)
default	\(x^{2}+{\mathrm e} \,{\mathrm e}^{-{\mathrm e}^{x}}\)	\(13\)
parts	\(x^{2}+{\mathrm e} \,{\mathrm e}^{-{\mathrm e}^{x}}\)	\(13\)
norman	\(\left ({\mathrm e}^{{\mathrm e}^{x}} x^{2}+{\mathrm e}\right ) {\mathrm e}^{-{\mathrm e}^{x}}\)	\(17\)
parallelrisch	\(\left ({\mathrm e}^{{\mathrm e}^{x}} x^{2}+{\mathrm e}\right ) {\mathrm e}^{-{\mathrm e}^{x}}\)	\(17\)

[In]

int((2*x*exp(exp(x))-exp(1)*exp(x))/exp(exp(x)),x,method=_RETURNVERBOSE)

[Out]

x^2+exp(1-exp(x))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int e^{-e^x} \left (-e^{1+x}+2 e^{e^x} x\right ) \, dx={\left (x^{2} e^{\left (e^{x}\right )} + e\right )} e^{\left (-e^{x}\right )} \]

[In]

integrate((2*x*exp(exp(x))-exp(1)*exp(x))/exp(exp(x)),x, algorithm="fricas")

[Out]

(x^2*e^(e^x) + e)*e^(-e^x)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int e^{-e^x} \left (-e^{1+x}+2 e^{e^x} x\right ) \, dx=x^{2} + e e^{- e^{x}} \]

[In]

integrate((2*x*exp(exp(x))-exp(1)*exp(x))/exp(exp(x)),x)

[Out]

x**2 + E*exp(-exp(x))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.69 \[ \int e^{-e^x} \left (-e^{1+x}+2 e^{e^x} x\right ) \, dx=x^{2} + e^{\left (-e^{x} + 1\right )} \]

[In]

integrate((2*x*exp(exp(x))-exp(1)*exp(x))/exp(exp(x)),x, algorithm="maxima")

[Out]

x^2 + e^(-e^x + 1)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.69 \[ \int e^{-e^x} \left (-e^{1+x}+2 e^{e^x} x\right ) \, dx=x^{2} + e^{\left (-e^{x} + 1\right )} \]

[In]

integrate((2*x*exp(exp(x))-exp(1)*exp(x))/exp(exp(x)),x, algorithm="giac")

[Out]

x^2 + e^(-e^x + 1)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int e^{-e^x} \left (-e^{1+x}+2 e^{e^x} x\right ) \, dx=\mathrm {e}\,{\mathrm {e}}^{-{\mathrm {e}}^x}+x^2 \]

[In]

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

[Out]

exp(1)*exp(-exp(x)) + x^2