Optimal. Leaf size=14 \[ -2 e^{2 x} x (2+x \log (4)) \]
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Rubi [A]
time = 0.05, antiderivative size = 21, normalized size of antiderivative = 1.50, number of steps
used = 12, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2227, 2225,
2207} \begin {gather*} -2 e^{2 x} x^2 \log (4)-4 e^{2 x} x \end {gather*}
Antiderivative was successfully verified.
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Rule 2207
Rule 2225
Rule 2227
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-4 e^{2 x}-8 e^{2 x} x-4 e^{2 x} x (1+x) \log (4)\right ) \, dx\\ &=-\left (4 \int e^{2 x} \, dx\right )-8 \int e^{2 x} x \, dx-(4 \log (4)) \int e^{2 x} x (1+x) \, dx\\ &=-2 e^{2 x}-4 e^{2 x} x+4 \int e^{2 x} \, dx-(4 \log (4)) \int \left (e^{2 x} x+e^{2 x} x^2\right ) \, dx\\ &=-4 e^{2 x} x-(4 \log (4)) \int e^{2 x} x \, dx-(4 \log (4)) \int e^{2 x} x^2 \, dx\\ &=-4 e^{2 x} x-2 e^{2 x} x \log (4)-2 e^{2 x} x^2 \log (4)+(2 \log (4)) \int e^{2 x} \, dx+(4 \log (4)) \int e^{2 x} x \, dx\\ &=-4 e^{2 x} x+e^{2 x} \log (4)-2 e^{2 x} x^2 \log (4)-(2 \log (4)) \int e^{2 x} \, dx\\ &=-4 e^{2 x} x-2 e^{2 x} x^2 \log (4)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 18, normalized size = 1.29 \begin {gather*} -4 e^{2 x} \left (x+\frac {1}{2} x^2 \log (4)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 20, normalized size = 1.43
method | result | size |
gosper | \(-4 \,{\mathrm e}^{2 x} x \left (x \ln \left (2\right )+1\right )\) | \(14\) |
risch | \(\left (-4 x^{2} \ln \left (2\right )-4 x \right ) {\mathrm e}^{2 x}\) | \(17\) |
default | \(-4 x \,{\mathrm e}^{2 x}-4 x^{2} \ln \left (2\right ) {\mathrm e}^{2 x}\) | \(20\) |
norman | \(-4 x \,{\mathrm e}^{2 x}-4 x^{2} \ln \left (2\right ) {\mathrm e}^{2 x}\) | \(20\) |
meijerg | \(2-2 \,{\mathrm e}^{2 x}+\ln \left (2\right ) \left (2-\frac {\left (12 x^{2}-12 x +6\right ) {\mathrm e}^{2 x}}{3}\right )+\frac {\left (-8 \ln \left (2\right )-8\right ) \left (1-\frac {\left (-4 x +2\right ) {\mathrm e}^{2 x}}{2}\right )}{4}\) | \(51\) |
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. 49 vs.
\(2 (13) = 26\).
time = 0.26, size = 49, normalized size = 3.50 \begin {gather*} -2 \, {\left (2 \, x^{2} - 2 \, x + 1\right )} e^{\left (2 \, x\right )} \log \left (2\right ) - 2 \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} \log \left (2\right ) - 2 \, {\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} - 2 \, e^{\left (2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 14, normalized size = 1.00 \begin {gather*} -4 \, {\left (x^{2} \log \left (2\right ) + x\right )} e^{\left (2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 17, normalized size = 1.21 \begin {gather*} \left (- 4 x^{2} \log {\left (2 \right )} - 4 x\right ) 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 = 14, normalized size = 1.00 \begin {gather*} -4 \, {\left (x^{2} \log \left (2\right ) + x\right )} e^{\left (2 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.05, size = 13, normalized size = 0.93 \begin {gather*} -4\,x\,{\mathrm {e}}^{2\,x}\,\left (x\,\ln \left (2\right )+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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