3.167 \(\int e^{-t} t^2 \, dt\)

Optimal. Leaf size=26 \[ -e^{-t} t^2-2 e^{-t} t-2 e^{-t} \]

[Out]

-2/E^t - (2*t)/E^t - t^2/E^t

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Rubi [A]  time = 0.0349204, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ -e^{-t} t^2-2 e^{-t} t-2 e^{-t} \]

Antiderivative was successfully verified.

[In]  Int[t^2/E^t,t]

[Out]

-2/E^t - (2*t)/E^t - t^2/E^t

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Rubi in Sympy [A]  time = 1.75476, size = 19, normalized size = 0.73 \[ - t^{2} e^{- t} - 2 t e^{- t} - 2 e^{- t} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(t**2/exp(t),t)

[Out]

-t**2*exp(-t) - 2*t*exp(-t) - 2*exp(-t)

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Mathematica [A]  time = 0.0034347, size = 16, normalized size = 0.62 \[ e^{-t} \left (-t^2-2 t-2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[t^2/E^t,t]

[Out]

(-2 - 2*t - t^2)/E^t

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Maple [A]  time = 0.004, size = 15, normalized size = 0.6 \[ -{\frac{{t}^{2}+2\,t+2}{{{\rm e}^{t}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(t^2/exp(t),t)

[Out]

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

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Maxima [A]  time = 1.34743, size = 19, normalized size = 0.73 \[ -{\left (t^{2} + 2 \, t + 2\right )} e^{\left (-t\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(t^2*e^(-t),t, algorithm="maxima")

[Out]

-(t^2 + 2*t + 2)*e^(-t)

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Fricas [A]  time = 0.212409, size = 19, normalized size = 0.73 \[ -{\left (t^{2} + 2 \, t + 2\right )} e^{\left (-t\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(t^2*e^(-t),t, algorithm="fricas")

[Out]

-(t^2 + 2*t + 2)*e^(-t)

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Sympy [A]  time = 0.088941, size = 12, normalized size = 0.46 \[ \left (- t^{2} - 2 t - 2\right ) e^{- t} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(t**2/exp(t),t)

[Out]

(-t**2 - 2*t - 2)*exp(-t)

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GIAC/XCAS [A]  time = 0.215646, size = 19, normalized size = 0.73 \[ -{\left (t^{2} + 2 \, t + 2\right )} e^{\left (-t\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(t^2*e^(-t),t, algorithm="giac")

[Out]

-(t^2 + 2*t + 2)*e^(-t)