3.4.18 \(\int e^{-x} (4 e^{4 e^{-x}}+3 e^x) \, dx\)

Optimal. Leaf size=16 \[ -3-e^{4 e^{-x}}+3 x \]

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 15, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2282, 14, 2209} \begin {gather*} 3 x-e^{4 e^{-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(4*E^(4/E^x) + 3*E^x)/E^x,x]

[Out]

-E^(4/E^x) + 3*x

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2282

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 {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {3+\frac {4 e^{4/x}}{x}}{x} \, dx,x,e^x\right )\\ &=\operatorname {Subst}\left (\int \left (\frac {4 e^{4/x}}{x^2}+\frac {3}{x}\right ) \, dx,x,e^x\right )\\ &=3 x+4 \operatorname {Subst}\left (\int \frac {e^{4/x}}{x^2} \, dx,x,e^x\right )\\ &=-e^{4 e^{-x}}+3 x\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 15, normalized size = 0.94 \begin {gather*} -e^{4 e^{-x}}+3 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4*E^(4/E^x) + 3*E^x)/E^x,x]

[Out]

-E^(4/E^x) + 3*x

________________________________________________________________________________________

fricas [A]  time = 1.07, size = 13, normalized size = 0.81 \begin {gather*} 3 \, x - e^{\left (4 \, e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(4/exp(x))+3*exp(x))/exp(x),x, algorithm="fricas")

[Out]

3*x - e^(4*e^(-x))

________________________________________________________________________________________

giac [A]  time = 0.30, size = 13, normalized size = 0.81 \begin {gather*} 3 \, x - e^{\left (4 \, e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(4/exp(x))+3*exp(x))/exp(x),x, algorithm="giac")

[Out]

3*x - e^(4*e^(-x))

________________________________________________________________________________________

maple [A]  time = 0.11, size = 14, normalized size = 0.88

method	result	size
risch	\(3 x -{\mathrm e}^{4 \,{\mathrm e}^{-x}}\)	\(14\)
derivativedivides	\(-3 \ln \left ({\mathrm e}^{-x}\right )-{\mathrm e}^{4 \,{\mathrm e}^{-x}}\)	\(18\)
default	\(-3 \ln \left ({\mathrm e}^{-x}\right )-{\mathrm e}^{4 \,{\mathrm e}^{-x}}\)	\(18\)
norman	\(\left (3 \,{\mathrm e}^{x} x -{\mathrm e}^{x} {\mathrm e}^{4 \,{\mathrm e}^{-x}}\right ) {\mathrm e}^{-x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*exp(4/exp(x))+3*exp(x))/exp(x),x,method=_RETURNVERBOSE)

[Out]

3*x-exp(4*exp(-x))

________________________________________________________________________________________

maxima [A]  time = 0.35, size = 13, normalized size = 0.81 \begin {gather*} 3 \, x - e^{\left (4 \, e^{\left (-x\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(4/exp(x))+3*exp(x))/exp(x),x, algorithm="maxima")

[Out]

3*x - e^(4*e^(-x))

________________________________________________________________________________________

mupad [B]  time = 0.34, size = 13, normalized size = 0.81 \begin {gather*} 3\,x-{\mathrm {e}}^{4\,{\mathrm {e}}^{-x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(-x)*(4*exp(4*exp(-x)) + 3*exp(x)),x)

[Out]

3*x - exp(4*exp(-x))

________________________________________________________________________________________

sympy [A]  time = 0.11, size = 8, normalized size = 0.50 \begin {gather*} 3 x - e^{4 e^{- x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*exp(4/exp(x))+3*exp(x))/exp(x),x)

[Out]

3*x - exp(4*exp(-x))

________________________________________________________________________________________