Optimal. Leaf size=14 \[ \frac {80}{27}-e^{-2 x} x^2 \]
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Rubi [A]
time = 0.04, antiderivative size = 10, normalized size of antiderivative = 0.71, number of steps
used = 8, number of rules used = 4, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {1607, 2227,
2207, 2225} \begin {gather*} -e^{-2 x} x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 1607
Rule 2207
Rule 2225
Rule 2227
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int e^{-2 x} x (-2+2 x) \, dx\\ &=\int \left (-2 e^{-2 x} x+2 e^{-2 x} x^2\right ) \, dx\\ &=-\left (2 \int e^{-2 x} x \, dx\right )+2 \int e^{-2 x} x^2 \, dx\\ &=e^{-2 x} x-e^{-2 x} x^2+2 \int e^{-2 x} x \, dx-\int e^{-2 x} \, dx\\ &=\frac {e^{-2 x}}{2}-e^{-2 x} x^2+\int e^{-2 x} \, dx\\ &=-e^{-2 x} x^2\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 10, normalized size = 0.71 \begin {gather*} -e^{-2 x} x^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 10, normalized size = 0.71
method | result | size |
gosper | \(-x^{2} {\mathrm e}^{-2 x}\) | \(10\) |
default | \(-x^{2} {\mathrm e}^{-2 x}\) | \(10\) |
norman | \(-x^{2} {\mathrm e}^{-2 x}\) | \(10\) |
risch | \(-x^{2} {\mathrm e}^{-2 x}\) | \(10\) |
meijerg | \(-\frac {\left (12 x^{2}+12 x +6\right ) {\mathrm e}^{-2 x}}{12}+\frac {\left (4 x +2\right ) {\mathrm e}^{-2 x}}{4}\) | \(29\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs.
\(2 (11) = 22\).
time = 0.27, size = 28, normalized size = 2.00 \begin {gather*} -\frac {1}{2} \, {\left (2 \, x^{2} + 2 \, x + 1\right )} e^{\left (-2 \, x\right )} + \frac {1}{2} \, {\left (2 \, x + 1\right )} e^{\left (-2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 9, normalized size = 0.64 \begin {gather*} -x^{2} e^{\left (-2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 8, normalized size = 0.57 \begin {gather*} - x^{2} e^{- 2 x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 9, normalized size = 0.64 \begin {gather*} -x^{2} e^{\left (-2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.39, size = 9, normalized size = 0.64 \begin {gather*} -x^2\,{\mathrm {e}}^{-2\,x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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