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

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

[Out]

-E^(-x) + x - x/E^x

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Rubi [A]  time = 0.050053, 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 \[ -e^{-x} x+x-e^{-x} \]

Antiderivative was successfully verified.

[In]  Int[(E^x + x)/E^x,x]

[Out]

-E^(-x) + x - x/E^x

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (x + e^{x}\right ) e^{- x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((x+exp(x))/exp(x),x)

[Out]

Integral((x + exp(x))*exp(-x), x)

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Mathematica [A]  time = 0.00978476, 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 = 0.003, size = 16, normalized size = 0.9 \[ - \left ({{\rm e}^{x}} \right ) ^{-1}+x-{\frac{x}{{{\rm e}^{x}}}} \]

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 = 1.35165, size = 15, normalized size = 0.88 \[ -{\left (x + 1\right )} e^{\left (-x\right )} + x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(x + 1)*e^(-x) + x

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Fricas [A]  time = 0.220337, size = 19, normalized size = 1.12 \[{\left (x e^{x} - x - 1\right )} e^{\left (-x\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(x*e^x - x - 1)*e^(-x)

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Sympy [A]  time = 0.070081, size = 8, normalized size = 0.47 \[ x + \left (- x - 1\right ) e^{- x} \]

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/XCAS [A]  time = 0.228953, size = 15, normalized size = 0.88 \[ -{\left (x + 1\right )} e^{\left (-x\right )} + x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(x + 1)*e^(-x) + x