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 = 0.0134441, 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 \[ -e^{-x} x-e^{-x} \]

Antiderivative was successfully verified.

[In]  Int[x/E^x,x]

[Out]

-E^(-x) - x/E^x

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Rubi in Sympy [A]  time = 1.0391, size = 10, normalized size = 0.62 \[ - x e^{- x} - e^{- x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/exp(x),x)

[Out]

-x*exp(-x) - exp(-x)

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Mathematica [A]  time = 0.00248595, 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 = 0.003, size = 10, normalized size = 0.6 \[ -{\frac{1+x}{{{\rm e}^{x}}}} \]

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 = 1.41769, size = 12, normalized size = 0.75 \[ -{\left (x + 1\right )} e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*e^(-x),x, algorithm="maxima")

[Out]

-(x + 1)*e^(-x)

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Fricas [A]  time = 0.193265, size = 12, normalized size = 0.75 \[ -{\left (x + 1\right )} e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*e^(-x),x, algorithm="fricas")

[Out]

-(x + 1)*e^(-x)

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Sympy [A]  time = 0.070098, size = 7, normalized size = 0.44 \[ \left (- x - 1\right ) e^{- x} \]

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/XCAS [A]  time = 0.237688, size = 12, normalized size = 0.75 \[ -{\left (x + 1\right )} e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x*e^(-x),x, algorithm="giac")

[Out]

-(x + 1)*e^(-x)