Optimal. Leaf size=26 \[ -2 e^{-t}-2 e^{-t} t-e^{-t} t^2 \]
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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 3, 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^2-2 e^{-t} t-2 e^{-t} \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2225
Rubi steps
\begin {align*} \int e^{-t} t^2 \, dt &=-e^{-t} t^2+2 \int e^{-t} t \, dt\\ &=-2 e^{-t} t-e^{-t} t^2+2 \int e^{-t} \, dt\\ &=-2 e^{-t}-2 e^{-t} t-e^{-t} t^2\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 16, normalized size = 0.62 \begin {gather*} e^{-t} \left (-2-2 t-t^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 24, normalized size = 0.92
method | result | size |
gosper | \(-\left (t^{2}+2 t +2\right ) {\mathrm e}^{-t}\) | \(15\) |
norman | \(\left (-t^{2}-2 t -2\right ) {\mathrm e}^{-t}\) | \(16\) |
risch | \(\left (-t^{2}-2 t -2\right ) {\mathrm e}^{-t}\) | \(16\) |
meijerg | \(2-\frac {\left (3 t^{2}+6 t +6\right ) {\mathrm e}^{-t}}{3}\) | \(19\) |
default | \(-2 \,{\mathrm e}^{-t}-2 t \,{\mathrm e}^{-t}-t^{2} {\mathrm e}^{-t}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.96, size = 14, normalized size = 0.54 \begin {gather*} -{\left (t^{2} + 2 \, t + 2\right )} e^{\left (-t\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.77, size = 14, normalized size = 0.54 \begin {gather*} -{\left (t^{2} + 2 \, t + 2\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 = 12, normalized size = 0.46 \begin {gather*} \left (- t^{2} - 2 t - 2\right ) e^{- t} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 14, normalized size = 0.54 \begin {gather*} -{\left (t^{2} + 2 \, t + 2\right )} e^{\left (-t\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.03, size = 14, normalized size = 0.54 \begin {gather*} -{\mathrm {e}}^{-t}\,\left (t^2+2\,t+2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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