\(\int e^{-4 x} (4+e^{4 x} (2+7 e^x)) \, dx\) [3030]

Optimal result

Integrand size = 21, antiderivative size = 17 \[ \int e^{-4 x} \left (4+e^{4 x} \left (2+7 e^x\right )\right ) \, dx=2-e^{-4 x}+7 e^x+2 x \]

[Out]

2-1/exp(2*x)^2+2*x+7*exp(x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2320, 14} \[ \int e^{-4 x} \left (4+e^{4 x} \left (2+7 e^x\right )\right ) \, dx=2 x-e^{-4 x}+7 e^x \]

[In]

Int[(4 + E^(4*x)*(2 + 7*E^x))/E^(4*x),x]

[Out]

-E^(-4*x) + 7*E^x + 2*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 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}& = \text {Subst}\left (\int \frac {2+\frac {4}{x^4}+7 x}{x} \, dx,x,e^x\right ) \\ & = \text {Subst}\left (\int \left (7+\frac {4}{x^5}+\frac {2}{x}\right ) \, dx,x,e^x\right ) \\ & = -e^{-4 x}+7 e^x+2 x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int e^{-4 x} \left (4+e^{4 x} \left (2+7 e^x\right )\right ) \, dx=-e^{-4 x}+7 e^x+2 x \]

[In]

Integrate[(4 + E^(4*x)*(2 + 7*E^x))/E^(4*x),x]

[Out]

-E^(-4*x) + 7*E^x + 2*x

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88

method	result	size
default	\(2 x -{\mathrm e}^{-4 x}+7 \,{\mathrm e}^{x}\)	\(15\)
risch	\(2 x -{\mathrm e}^{-4 x}+7 \,{\mathrm e}^{x}\)	\(15\)
parts	\(2 x -{\mathrm e}^{-4 x}+7 \,{\mathrm e}^{x}\)	\(17\)
norman	\(\left (-1+7 \,{\mathrm e}^{5 x}+2 x \,{\mathrm e}^{4 x}\right ) {\mathrm e}^{-4 x}\)	\(21\)
parallelrisch	\(\left (-1+2 x \,{\mathrm e}^{4 x}+7 \,{\mathrm e}^{x} {\mathrm e}^{4 x}\right ) {\mathrm e}^{-4 x}\)	\(29\)

[In]

int(((7*exp(x)+2)*exp(2*x)^2+4)/exp(2*x)^2,x,method=_RETURNVERBOSE)

[Out]

2*x-1/exp(x)^4+7*exp(x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.18 \[ \int e^{-4 x} \left (4+e^{4 x} \left (2+7 e^x\right )\right ) \, dx={\left (2 \, x e^{\left (4 \, x\right )} + 7 \, e^{\left (5 \, x\right )} - 1\right )} e^{\left (-4 \, x\right )} \]

[In]

integrate(((7*exp(x)+2)*exp(2*x)^2+4)/exp(2*x)^2,x, algorithm="fricas")

[Out]

(2*x*e^(4*x) + 7*e^(5*x) - 1)*e^(-4*x)

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int e^{-4 x} \left (4+e^{4 x} \left (2+7 e^x\right )\right ) \, dx=2 x + 7 e^{x} - e^{- 4 x} \]

[In]

integrate(((7*exp(x)+2)*exp(2*x)**2+4)/exp(2*x)**2,x)

[Out]

2*x + 7*exp(x) - exp(-4*x)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int e^{-4 x} \left (4+e^{4 x} \left (2+7 e^x\right )\right ) \, dx=2 \, x - e^{\left (-4 \, x\right )} + 7 \, e^{x} \]

[In]

integrate(((7*exp(x)+2)*exp(2*x)^2+4)/exp(2*x)^2,x, algorithm="maxima")

[Out]

2*x - e^(-4*x) + 7*e^x

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int e^{-4 x} \left (4+e^{4 x} \left (2+7 e^x\right )\right ) \, dx=2 \, x - e^{\left (-4 \, x\right )} + 7 \, e^{x} \]

[In]

integrate(((7*exp(x)+2)*exp(2*x)^2+4)/exp(2*x)^2,x, algorithm="giac")

[Out]

2*x - e^(-4*x) + 7*e^x

Mupad [B] (verification not implemented)

Time = 9.78 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int e^{-4 x} \left (4+e^{4 x} \left (2+7 e^x\right )\right ) \, dx=2\,x-{\mathrm {e}}^{-4\,x}+7\,{\mathrm {e}}^x \]

[In]

int(exp(-4*x)*(exp(4*x)*(7*exp(x) + 2) + 4),x)

[Out]

2*x - exp(-4*x) + 7*exp(x)