\(\int e^{-x} (e^x+e^{2 e^{-x} (3-15 e^x x)} (-6-30 e^x)) \, dx\) [5238]

Optimal result

Integrand size = 35, antiderivative size = 15 \[ \int e^{-x} \left (e^x+e^{2 e^{-x} \left (3-15 e^x x\right )} \left (-6-30 e^x\right )\right ) \, dx=e^{6 \left (e^{-x}-5 x\right )}+x \]

[Out]

exp(3/exp(x)-15*x)^2+x

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {6874, 2320, 2326} \[ \int e^{-x} \left (e^x+e^{2 e^{-x} \left (3-15 e^x x\right )} \left (-6-30 e^x\right )\right ) \, dx=x+e^{6 e^{-x}-30 x} \]

[In]

Int[(E^x + E^((2*(3 - 15*E^x*x))/E^x)*(-6 - 30*E^x))/E^x,x]

[Out]

E^(6/E^x - 30*x) + 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 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \left (1-6 e^{6 e^{-x}-31 x} \left (1+5 e^x\right )\right ) \, dx \\ & = x-6 \int e^{6 e^{-x}-31 x} \left (1+5 e^x\right ) \, dx \\ & = x-6 \text {Subst}\left (\int \frac {e^{6/x} (1+5 x)}{x^{32}} \, dx,x,e^x\right ) \\ & = e^{6 e^{-x}-30 x}+x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int e^{-x} \left (e^x+e^{2 e^{-x} \left (3-15 e^x x\right )} \left (-6-30 e^x\right )\right ) \, dx=e^{6 \left (e^{-x}-5 x\right )}+x \]

[In]

Integrate[(E^x + E^((2*(3 - 15*E^x*x))/E^x)*(-6 - 30*E^x))/E^x,x]

[Out]

E^(6*(E^(-x) - 5*x)) + x

Maple [A] (verified)

Time = 0.06 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00

method	result	size
default	\(x +{\mathrm e}^{6 \,{\mathrm e}^{-x}} {\mathrm e}^{-30 x}\)	\(15\)
parts	\(x +{\mathrm e}^{6 \,{\mathrm e}^{-x}} {\mathrm e}^{-30 x}\)	\(15\)
risch	\({\mathrm e}^{-6 \left (5 \,{\mathrm e}^{x} x -1\right ) {\mathrm e}^{-x}}+x\)	\(17\)
parallelrisch	\({\mathrm e}^{-6 \left (5 \,{\mathrm e}^{x} x -1\right ) {\mathrm e}^{-x}}+x\)	\(19\)
norman	\(\left ({\mathrm e}^{x} x +{\mathrm e}^{x} {\mathrm e}^{2 \left (-15 \,{\mathrm e}^{x} x +3\right ) {\mathrm e}^{-x}}\right ) {\mathrm e}^{-x}\)	\(29\)

[In]

int(((-30*exp(x)-6)*exp((-15*exp(x)*x+3)/exp(x))^2+exp(x))/exp(x),x,method=_RETURNVERBOSE)

[Out]

x+exp(1/exp(x))^6/exp(x)^30

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07 \[ \int e^{-x} \left (e^x+e^{2 e^{-x} \left (3-15 e^x x\right )} \left (-6-30 e^x\right )\right ) \, dx=x + e^{\left (-6 \, {\left (5 \, x e^{x} - 1\right )} e^{\left (-x\right )}\right )} \]

[In]

integrate(((-30*exp(x)-6)*exp((-15*exp(x)*x+3)/exp(x))^2+exp(x))/exp(x),x, algorithm="fricas")

[Out]

x + e^(-6*(5*x*e^x - 1)*e^(-x))

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int e^{-x} \left (e^x+e^{2 e^{-x} \left (3-15 e^x x\right )} \left (-6-30 e^x\right )\right ) \, dx=x + e^{2 \left (- 15 x e^{x} + 3\right ) e^{- x}} \]

[In]

integrate(((-30*exp(x)-6)*exp((-15*exp(x)*x+3)/exp(x))**2+exp(x))/exp(x),x)

[Out]

x + exp(2*(-15*x*exp(x) + 3)*exp(-x))

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int e^{-x} \left (e^x+e^{2 e^{-x} \left (3-15 e^x x\right )} \left (-6-30 e^x\right )\right ) \, dx=x + e^{\left (-30 \, x + 6 \, e^{\left (-x\right )}\right )} \]

[In]

integrate(((-30*exp(x)-6)*exp((-15*exp(x)*x+3)/exp(x))^2+exp(x))/exp(x),x, algorithm="maxima")

[Out]

x + e^(-30*x + 6*e^(-x))

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int e^{-x} \left (e^x+e^{2 e^{-x} \left (3-15 e^x x\right )} \left (-6-30 e^x\right )\right ) \, dx=x + e^{\left (-30 \, x + 6 \, e^{\left (-x\right )}\right )} \]

[In]

integrate(((-30*exp(x)-6)*exp((-15*exp(x)*x+3)/exp(x))^2+exp(x))/exp(x),x, algorithm="giac")

[Out]

x + e^(-30*x + 6*e^(-x))

Mupad [B] (verification not implemented)

Time = 10.37 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int e^{-x} \left (e^x+e^{2 e^{-x} \left (3-15 e^x x\right )} \left (-6-30 e^x\right )\right ) \, dx=x+{\mathrm {e}}^{6\,{\mathrm {e}}^{-x}-30\,x} \]

[In]

int(exp(-x)*(exp(x) - exp(-2*exp(-x)*(15*x*exp(x) - 3))*(30*exp(x) + 6)),x)

[Out]

x + exp(6*exp(-x) - 30*x)