Optimal. Leaf size=44 \[ -\frac {3}{8} e^{-2 t}-\frac {3}{4} e^{-2 t} t-\frac {3}{4} e^{-2 t} t^2-\frac {1}{2} e^{-2 t} t^3 \]
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Rubi [A]
time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, 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 = {2207, 2225}
\begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2225
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*}
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Mathematica [A]
time = 0.01, size = 24, normalized size = 0.55 \begin {gather*} -\frac {1}{8} e^{-2 t} \left (3+6 t+6 t^2+4 t^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Mathics [A]
time = 1.81, size = 22, normalized size = 0.50 \begin {gather*} \frac {\left (-3-6 t-6 t^2-4 t^3\right ) E^{-2 t}}{8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 41, normalized size = 0.93
method | result | size |
risch | \(\left (-\frac {1}{2} t^{3}-\frac {3}{4} t^{2}-\frac {3}{4} t -\frac {3}{8}\right ) {\mathrm e}^{-2 t}\) | \(21\) |
norman | \(\left (-\frac {1}{2} t^{3}-\frac {3}{4} t^{2}-\frac {3}{4} t -\frac {3}{8}\right ) {\mathrm e}^{-2 t}\) | \(23\) |
gosper | \(-\frac {\left (4 t^{3}+6 t^{2}+6 t +3\right ) {\mathrm e}^{-2 t}}{8}\) | \(24\) |
meijerg | \(\frac {3}{8}-\frac {\left (32 t^{3}+48 t^{2}+48 t +24\right ) {\mathrm e}^{-2 t}}{64}\) | \(24\) |
derivativedivides | \(-\frac {3 \,{\mathrm e}^{-2 t}}{8}-\frac {3 t \,{\mathrm e}^{-2 t}}{4}-\frac {3 t^{2} {\mathrm e}^{-2 t}}{4}-\frac {t^{3} {\mathrm e}^{-2 t}}{2}\) | \(41\) |
default | \(-\frac {3 \,{\mathrm e}^{-2 t}}{8}-\frac {3 t \,{\mathrm e}^{-2 t}}{4}-\frac {3 t^{2} {\mathrm e}^{-2 t}}{4}-\frac {t^{3} {\mathrm e}^{-2 t}}{2}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 21, normalized size = 0.48 \begin {gather*} -\frac {1}{8} \, {\left (4 \, t^{3} + 6 \, t^{2} + 6 \, t + 3\right )} e^{\left (-2 \, t\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 21, normalized size = 0.48 \begin {gather*} -\frac {1}{8} \, {\left (4 \, t^{3} + 6 \, t^{2} + 6 \, t + 3\right )} e^{\left (-2 \, t\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 22, normalized size = 0.50 \begin {gather*} \frac {\left (- 4 t^{3} - 6 t^{2} - 6 t - 3\right ) e^{- 2 t}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 26, normalized size = 0.59 \begin {gather*} \frac {1}{16} \left (-8 t^{3}-12 t^{2}-12 t-6\right ) \mathrm {e}^{-2 t} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 21, normalized size = 0.48 \begin {gather*} -\frac {{\mathrm {e}}^{-2\,t}\,\left (8\,t^3+12\,t^2+12\,t+6\right )}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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