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

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

[Out]

-E^(-x) - x/E^x

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Rubi [A]  time = 0.0068846, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2176, 2194} \[ -e^{-x} x-e^{-x} \]

Antiderivative was successfully verified.

[In]

Int[x/E^x,x]

[Out]

-E^(-x) - x/E^x

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} x \, dx &=-e^{-x} x+\int e^{-x} \, dx\\ &=-e^{-x}-e^{-x} x\\ \end{align*}

Mathematica [A]  time = 0.0041336, size = 11, normalized size = 0.69 \[ e^{-x} (-x-1) \]

Antiderivative was successfully verified.

[In]

Integrate[x/E^x,x]

[Out]

(-1 - x)/E^x

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Maple [A]  time = 0.001, size = 10, normalized size = 0.6 \begin{align*} -{\frac{1+x}{{{\rm e}^{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/exp(x),x)

[Out]

-(1+x)/exp(x)

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Maxima [A]  time = 0.951513, size = 12, normalized size = 0.75 \begin{align*} -{\left (x + 1\right )} e^{\left (-x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x + 1)*e^(-x)

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Fricas [A]  time = 0.516459, size = 23, normalized size = 1.44 \begin{align*} -{\left (x + 1\right )} e^{\left (-x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x + 1)*e^(-x)

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Sympy [A]  time = 0.080846, size = 7, normalized size = 0.44 \begin{align*} \left (- x - 1\right ) e^{- x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x/exp(x),x)

[Out]

(-x - 1)*exp(-x)

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Giac [A]  time = 1.08864, size = 12, normalized size = 0.75 \begin{align*} -{\left (x + 1\right )} e^{\left (-x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x + 1)*e^(-x)