Optimal. Leaf size=36 \[ -e^{-t} t^3-3 e^{-t} t^2-6 e^{-t} t-6 e^{-t} \]
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Rubi [A] time = 0.0338808, 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, Rules used = {2176, 2194} \[ -e^{-t} t^3-3 e^{-t} t^2-6 e^{-t} t-6 e^{-t} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int e^{-t} t^3 \, dt &=-e^{-t} t^3+3 \int e^{-t} t^2 \, dt\\ &=-3 e^{-t} t^2-e^{-t} t^3+6 \int e^{-t} t \, dt\\ &=-6 e^{-t} t-3 e^{-t} t^2-e^{-t} t^3+6 \int e^{-t} \, dt\\ &=-6 e^{-t}-6 e^{-t} t-3 e^{-t} t^2-e^{-t} t^3\\ \end{align*}
Mathematica [A] time = 0.0058054, size = 21, normalized size = 0.58 \[ e^{-t} \left (-t^3-3 t^2-6 t-6\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 20, normalized size = 0.6 \begin{align*} -{\frac{{t}^{3}+3\,{t}^{2}+6\,t+6}{{{\rm e}^{t}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.940438, size = 26, normalized size = 0.72 \begin{align*} -{\left (t^{3} + 3 \, t^{2} + 6 \, t + 6\right )} e^{\left (-t\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.19131, size = 45, normalized size = 1.25 \begin{align*} -{\left (t^{3} + 3 \, t^{2} + 6 \, t + 6\right )} e^{\left (-t\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.080538, size = 17, normalized size = 0.47 \begin{align*} \left (- t^{3} - 3 t^{2} - 6 t - 6\right ) e^{- t} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08778, size = 26, normalized size = 0.72 \begin{align*} -{\left (t^{3} + 3 \, t^{2} + 6 \, t + 6\right )} e^{\left (-t\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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