3.59 \(\int e^{-x} (e^x+x) \, dx\)

Optimal. Leaf size=17 \[ -e^{-x} x+x-e^{-x} \]

[Out]

-E^(-x) + x - x/E^x

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Rubi [A]  time = 0.0313499, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {6742, 2176, 2194} \[ -e^{-x} x+x-e^{-x} \]

Antiderivative was successfully verified.

[In]

Int[(E^x + x)/E^x,x]

[Out]

-E^(-x) + x - x/E^x

Rule 6742

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

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

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]

Rubi steps

\begin{align*} \int e^{-x} \left (e^x+x\right ) \, dx &=\int \left (1+e^{-x} x\right ) \, dx\\ &=x+\int e^{-x} x \, dx\\ &=x-e^{-x} x+\int e^{-x} \, dx\\ &=-e^{-x}+x-e^{-x} x\\ \end{align*}

Mathematica [A]  time = 0.0099566, size = 13, normalized size = 0.76 \[ e^{-x} (-x-1)+x \]

Antiderivative was successfully verified.

[In]

Integrate[(E^x + x)/E^x,x]

[Out]

(-1 - x)/E^x + x

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Maple [A]  time = 0.001, size = 16, normalized size = 0.9 \begin{align*} - \left ({{\rm e}^{x}} \right ) ^{-1}+x-{\frac{x}{{{\rm e}^{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x)+x)/exp(x),x)

[Out]

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

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Maxima [A]  time = 0.932095, size = 15, normalized size = 0.88 \begin{align*} -{\left (x + 1\right )} e^{\left (-x\right )} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x + 1)*e^(-x) + x

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Fricas [A]  time = 1.87756, size = 32, normalized size = 1.88 \begin{align*}{\left (x e^{x} - x - 1\right )} e^{\left (-x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*e^x - x - 1)*e^(-x)

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Sympy [A]  time = 0.081333, size = 8, normalized size = 0.47 \begin{align*} x + \left (- x - 1\right ) e^{- x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x+exp(x))/exp(x),x)

[Out]

x + (-x - 1)*exp(-x)

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Giac [A]  time = 1.099, size = 15, normalized size = 0.88 \begin{align*} -{\left (x + 1\right )} e^{\left (-x\right )} + x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x + 1)*e^(-x) + x