Optimal. Leaf size=36 \[ -6 e^{-t}-6 e^{-t} t-3 e^{-t} t^2-e^{-t} t^3 \]
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Rubi [A]
time = 0.02, antiderivative size = 36, 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*} -e^{-t} t^3-3 e^{-t} t^2-6 e^{-t} t-6 e^{-t} \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2225
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*}
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Mathematica [A]
time = 0.01, size = 21, normalized size = 0.58 \begin {gather*} e^{-t} \left (-6-6 t-3 t^2-t^3\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 33, normalized size = 0.92
method | result | size |
gosper | \(-\left (t^{3}+3 t^{2}+6 t +6\right ) {\mathrm e}^{-t}\) | \(20\) |
norman | \(\left (-t^{3}-3 t^{2}-6 t -6\right ) {\mathrm e}^{-t}\) | \(21\) |
risch | \(\left (-t^{3}-3 t^{2}-6 t -6\right ) {\mathrm e}^{-t}\) | \(21\) |
meijerg | \(6-\frac {\left (4 t^{3}+12 t^{2}+24 t +24\right ) {\mathrm e}^{-t}}{4}\) | \(24\) |
default | \(-6 \,{\mathrm e}^{-t}-6 t \,{\mathrm e}^{-t}-3 t^{2} {\mathrm e}^{-t}-t^{3} {\mathrm e}^{-t}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.54, size = 19, normalized size = 0.53 \begin {gather*} -{\left (t^{3} + 3 \, t^{2} + 6 \, t + 6\right )} e^{\left (-t\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.72, size = 19, normalized size = 0.53 \begin {gather*} -{\left (t^{3} + 3 \, t^{2} + 6 \, t + 6\right )} e^{\left (-t\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.02, size = 17, normalized size = 0.47 \begin {gather*} \left (- t^{3} - 3 t^{2} - 6 t - 6\right ) e^{- t} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 19, normalized size = 0.53 \begin {gather*} -{\left (t^{3} + 3 \, t^{2} + 6 \, t + 6\right )} e^{\left (-t\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.02, size = 19, normalized size = 0.53 \begin {gather*} -{\mathrm {e}}^{-t}\,\left (t^3+3\,t^2+6\,t+6\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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