Optimal. Leaf size=27 \[ -\frac {3}{13} e^{2 x} \cos (3 x)+\frac {2}{13} e^{2 x} \sin (3 x) \]
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Rubi [A]
time = 0.01, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4517}
\begin {gather*} \frac {2}{13} e^{2 x} \sin (3 x)-\frac {3}{13} e^{2 x} \cos (3 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 4517
Rubi steps
\begin {align*} \int e^{2 x} \sin (3 x) \, dx &=-\frac {3}{13} e^{2 x} \cos (3 x)+\frac {2}{13} e^{2 x} \sin (3 x)\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 22, normalized size = 0.81 \begin {gather*} \frac {1}{13} e^{2 x} (-3 \cos (3 x)+2 \sin (3 x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 22, normalized size = 0.81
method | result | size |
default | \(-\frac {3 \,{\mathrm e}^{2 x} \cos \left (3 x \right )}{13}+\frac {2 \,{\mathrm e}^{2 x} \sin \left (3 x \right )}{13}\) | \(22\) |
risch | \(-\frac {3 \,{\mathrm e}^{\left (2+3 i\right ) x}}{26}-\frac {i {\mathrm e}^{\left (2+3 i\right ) x}}{13}-\frac {3 \,{\mathrm e}^{\left (2-3 i\right ) x}}{26}+\frac {i {\mathrm e}^{\left (2-3 i\right ) x}}{13}\) | \(36\) |
norman | \(\frac {\frac {4 \,{\mathrm e}^{2 x} \tan \left (\frac {3 x}{2}\right )}{13}+\frac {3 \,{\mathrm e}^{2 x} \left (\tan ^{2}\left (\frac {3 x}{2}\right )\right )}{13}-\frac {3 \,{\mathrm e}^{2 x}}{13}}{1+\tan ^{2}\left (\frac {3 x}{2}\right )}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.61, size = 19, normalized size = 0.70 \begin {gather*} -\frac {1}{13} \, {\left (3 \, \cos \left (3 \, x\right ) - 2 \, \sin \left (3 \, x\right )\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 = 0.80, size = 21, normalized size = 0.78 \begin {gather*} -\frac {3}{13} \, \cos \left (3 \, x\right ) e^{\left (2 \, x\right )} + \frac {2}{13} \, e^{\left (2 \, x\right )} \sin \left (3 \, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 26, normalized size = 0.96 \begin {gather*} \frac {2 e^{2 x} \sin {\left (3 x \right )}}{13} - \frac {3 e^{2 x} \cos {\left (3 x \right )}}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.79, size = 19, normalized size = 0.70 \begin {gather*} -\frac {1}{13} \, {\left (3 \, \cos \left (3 \, x\right ) - 2 \, \sin \left (3 \, x\right )\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 = 19, normalized size = 0.70 \begin {gather*} -\frac {{\mathrm {e}}^{2\,x}\,\left (3\,\cos \left (3\,x\right )-2\,\sin \left (3\,x\right )\right )}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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