3.1.75 \(\int e^{-x} x \, dx\) [75]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, 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 = {2207, 2225} \begin {gather*} -e^{-x} x-e^{-x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/E^x,x]

[Out]

-E^(-x) - x/E^x

Rule 2207

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] &&  !TrueQ[$UseGamma]

Rule 2225

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*}

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Mathematica [A]
time = 0.01, size = 11, normalized size = 0.69 \begin {gather*} e^{-x} (-1-x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/E^x,x]

[Out]

(-1 - x)/E^x

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Maple [A]
time = 0.01, size = 15, normalized size = 0.94

method	result	size
gosper	\(-\left (1+x \right ) {\mathrm e}^{-x}\)	\(10\)
norman	\(\left (-1-x \right ) {\mathrm e}^{-x}\)	\(11\)
risch	\(\left (-1-x \right ) {\mathrm e}^{-x}\)	\(11\)
meijerg	\(1-\frac {\left (2+2 x \right ) {\mathrm e}^{-x}}{2}\)	\(14\)
default	\(-{\mathrm e}^{-x}-x \,{\mathrm e}^{-x}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/exp(x),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 3.36, size = 9, normalized size = 0.56 \begin {gather*} -{\left (x + 1\right )} e^{\left (-x\right )} \end {gather*}

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 = 0.70, size = 9, normalized size = 0.56 \begin {gather*} -{\left (x + 1\right )} e^{\left (-x\right )} \end {gather*}

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 = 0.02, size = 7, normalized size = 0.44 \begin {gather*} \left (- x - 1\right ) e^{- x} \end {gather*}

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 = 0.44, size = 9, normalized size = 0.56 \begin {gather*} -{\left (x + 1\right )} e^{\left (-x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x + 1)*e^(-x)

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Mupad [B]
time = 0.02, size = 9, normalized size = 0.56 \begin {gather*} -{\mathrm {e}}^{-x}\,\left (x+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*exp(-x),x)

[Out]

-exp(-x)*(x + 1)

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