Optimal. Leaf size=23 \[ -\frac {e^{2 x}}{4}+\frac {1}{2} e^{2 x} \log \left (e^x\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2225, 2634, 12}
\begin {gather*} \frac {1}{2} e^{2 x} \log \left (e^x\right )-\frac {e^{2 x}}{4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2225
Rule 2634
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*}
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Mathematica [A]
time = 0.01, size = 17, normalized size = 0.74 \begin {gather*} \frac {1}{4} e^{2 x} \left (-1+2 \log \left (e^x\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.01, size = 28, normalized size = 1.22
method | result | size |
norman | \(-\frac {{\mathrm e}^{2 x}}{4}+\frac {{\mathrm e}^{2 x} \ln \left ({\mathrm e}^{x}\right )}{2}\) | \(17\) |
risch | \(-\frac {{\mathrm e}^{2 x}}{4}+\frac {{\mathrm e}^{2 x} \ln \left ({\mathrm e}^{x}\right )}{2}\) | \(17\) |
default | \(\frac {{\mathrm e}^{2 x} x}{2}-\frac {{\mathrm e}^{2 x}}{4}+\frac {{\mathrm e}^{2 x} \left (\ln \left ({\mathrm e}^{x}\right )-x \right )}{2}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.94, size = 11, normalized size = 0.48 \begin {gather*} \frac {1}{4} \, {\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 = 0.47, size = 11, normalized size = 0.48 \begin {gather*} \frac {1}{4} \, {\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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Sympy [A]
time = 0.03, size = 10, normalized size = 0.43 \begin {gather*} \frac {\left (2 x - 1\right ) e^{2 x}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.61, size = 11, normalized size = 0.48 \begin {gather*} \frac {1}{4} \, {\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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Mupad [B]
time = 0.04, size = 11, normalized size = 0.48 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}\,\left (2\,x-1\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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