3.168 \(\int e^{-t} t^3 \, dt\)

Optimal. Leaf size=36 \[ -e^{-t} t^3-3 e^{-t} t^2-6 e^{-t} t-6 e^{-t} \]

[Out]

-6/E^t - (6*t)/E^t - (3*t^2)/E^t - t^3/E^t

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Rubi [A]  time = 0.0537962, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -e^{-t} t^3-3 e^{-t} t^2-6 e^{-t} t-6 e^{-t} \]

Antiderivative was successfully verified.

[In]  Int[t^3/E^t,t]

[Out]

-6/E^t - (6*t)/E^t - (3*t^2)/E^t - t^3/E^t

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Rubi in Sympy [A]  time = 2.57745, size = 27, normalized size = 0.75 \[ - t^{3} e^{- t} - 3 t^{2} e^{- t} - 6 t e^{- t} - 6 e^{- t} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(t**3/exp(t),t)

[Out]

-t**3*exp(-t) - 3*t**2*exp(-t) - 6*t*exp(-t) - 6*exp(-t)

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Mathematica [A]  time = 0.00420394, size = 21, normalized size = 0.58 \[ e^{-t} \left (-t^3-3 t^2-6 t-6\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[t^3/E^t,t]

[Out]

(-6 - 6*t - 3*t^2 - t^3)/E^t

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Maple [A]  time = 0.004, size = 20, normalized size = 0.6 \[ -{\frac{{t}^{3}+3\,{t}^{2}+6\,t+6}{{{\rm e}^{t}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(t^3/exp(t),t)

[Out]

-(t^3+3*t^2+6*t+6)/exp(t)

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Maxima [A]  time = 1.3571, size = 26, normalized size = 0.72 \[ -{\left (t^{3} + 3 \, t^{2} + 6 \, t + 6\right )} e^{\left (-t\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(t^3*e^(-t),t, algorithm="maxima")

[Out]

-(t^3 + 3*t^2 + 6*t + 6)*e^(-t)

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Fricas [A]  time = 0.198283, size = 26, normalized size = 0.72 \[ -{\left (t^{3} + 3 \, t^{2} + 6 \, t + 6\right )} e^{\left (-t\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(t^3*e^(-t),t, algorithm="fricas")

[Out]

-(t^3 + 3*t^2 + 6*t + 6)*e^(-t)

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Sympy [A]  time = 0.125162, size = 17, normalized size = 0.47 \[ \left (- t^{3} - 3 t^{2} - 6 t - 6\right ) e^{- t} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(t**3/exp(t),t)

[Out]

(-t**3 - 3*t**2 - 6*t - 6)*exp(-t)

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GIAC/XCAS [A]  time = 0.227735, size = 26, normalized size = 0.72 \[ -{\left (t^{3} + 3 \, t^{2} + 6 \, t + 6\right )} e^{\left (-t\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(t^3*e^(-t),t, algorithm="giac")

[Out]

-(t^3 + 3*t^2 + 6*t + 6)*e^(-t)