Optimal. Leaf size=44 \[ -\frac{1}{2} e^{-2 t} t^3-\frac{3}{4} e^{-2 t} t^2-\frac{3}{4} e^{-2 t} t-\frac{3 e^{-2 t}}{8} \]
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Rubi [A] time = 0.0385098, antiderivative size = 44, 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} \[ -\frac{1}{2} e^{-2 t} t^3-\frac{3}{4} e^{-2 t} t^2-\frac{3}{4} e^{-2 t} t-\frac{3 e^{-2 t}}{8} \]
Antiderivative was successfully verified.
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Rule 2176
Rule 2194
Rubi steps
\begin{align*} \int e^{-2 t} t^3 \, dt &=-\frac{1}{2} e^{-2 t} t^3+\frac{3}{2} \int e^{-2 t} t^2 \, dt\\ &=-\frac{3}{4} e^{-2 t} t^2-\frac{1}{2} e^{-2 t} t^3+\frac{3}{2} \int e^{-2 t} t \, dt\\ &=-\frac{3}{4} e^{-2 t} t-\frac{3}{4} e^{-2 t} t^2-\frac{1}{2} e^{-2 t} t^3+\frac{3}{4} \int e^{-2 t} \, dt\\ &=-\frac{3}{8} e^{-2 t}-\frac{3}{4} e^{-2 t} t-\frac{3}{4} e^{-2 t} t^2-\frac{1}{2} e^{-2 t} t^3\\ \end{align*}
Mathematica [A] time = 0.0087274, size = 24, normalized size = 0.55 \[ -\frac{1}{8} e^{-2 t} \left (4 t^3+6 t^2+6 t+3\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 24, normalized size = 0.6 \begin{align*} -{\frac{4\,{t}^{3}+6\,{t}^{2}+6\,t+3}{8\,{{\rm e}^{2\,t}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.93674, size = 28, normalized size = 0.64 \begin{align*} -\frac{1}{8} \,{\left (4 \, t^{3} + 6 \, t^{2} + 6 \, t + 3\right )} e^{\left (-2 \, t\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.10728, size = 55, normalized size = 1.25 \begin{align*} -\frac{1}{8} \,{\left (4 \, t^{3} + 6 \, t^{2} + 6 \, t + 3\right )} e^{\left (-2 \, t\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.082717, size = 22, normalized size = 0.5 \begin{align*} \frac{\left (- 4 t^{3} - 6 t^{2} - 6 t - 3\right ) e^{- 2 t}}{8} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.0501, size = 28, normalized size = 0.64 \begin{align*} -\frac{1}{8} \,{\left (4 \, t^{3} + 6 \, t^{2} + 6 \, t + 3\right )} e^{\left (-2 \, t\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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