Optimal. Leaf size=23 \[ \frac{1}{2} e^{2 x} \log \left (e^x\right )-\frac{e^{2 x}}{4} \]
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Rubi [A] time = 0.012565, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2194, 2554, 12} \[ \frac{1}{2} e^{2 x} \log \left (e^x\right )-\frac{e^{2 x}}{4} \]
Antiderivative was successfully verified.
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Rule 2194
Rule 2554
Rule 12
Rubi steps
\begin{align*} \int e^{2 x} \log \left (e^x\right ) \, dx &=\frac{1}{2} e^{2 x} \log \left (e^x\right )-\int \frac{e^{2 x}}{2} \, dx\\ &=\frac{1}{2} e^{2 x} \log \left (e^x\right )-\frac{1}{2} \int e^{2 x} \, dx\\ &=-\frac{e^{2 x}}{4}+\frac{1}{2} e^{2 x} \log \left (e^x\right )\\ \end{align*}
Mathematica [A] time = 0.004753, size = 17, normalized size = 0.74 \[ \frac{1}{4} e^{2 x} \left (2 \log \left (e^x\right )-1\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 28, normalized size = 1.2 \begin{align*}{\frac{{{\rm e}^{2\,x}}x}{2}}-{\frac{{{\rm e}^{2\,x}}}{4}}+{\frac{{{\rm e}^{2\,x}} \left ( \ln \left ({{\rm e}^{x}} \right ) -x \right ) }{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.951424, size = 15, normalized size = 0.65 \begin{align*} \frac{1}{4} \,{\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91165, size = 31, normalized size = 1.35 \begin{align*} \frac{1}{4} \,{\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.088962, size = 10, normalized size = 0.43 \begin{align*} \frac{\left (2 x - 1\right ) e^{2 x}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07369, size = 15, normalized size = 0.65 \begin{align*} \frac{1}{4} \,{\left (2 \, x - 1\right )} e^{\left (2 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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