Optimal. Leaf size=27 \[ \frac {2}{13} e^{2 x} \cos (3 x)+\frac {3}{13} e^{2 x} \sin (3 x) \]
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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 = {4518}
\begin {gather*} \frac {3}{13} e^{2 x} \sin (3 x)+\frac {2}{13} e^{2 x} \cos (3 x) \end {gather*}
Antiderivative was successfully verified.
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Rule 4518
Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\frac {2}{13} e^{2 x} \cos (3 x)+\frac {3}{13} e^{2 x} \sin (3 x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 22, normalized size = 0.81 \begin {gather*} \frac {1}{13} e^{2 x} (2 \cos (3 x)+3 \sin (3 x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 22, normalized size = 0.81
method | result | size |
parallelrisch | \(\frac {{\mathrm e}^{2 x} \left (2 \cos \left (3 x \right )+3 \sin \left (3 x \right )\right )}{13}\) | \(20\) |
default | \(\frac {2 \,{\mathrm e}^{2 x} \cos \left (3 x \right )}{13}+\frac {3 \,{\mathrm e}^{2 x} \sin \left (3 x \right )}{13}\) | \(22\) |
risch | \(\frac {{\mathrm e}^{\left (2+3 i\right ) x}}{13}-\frac {3 i {\mathrm e}^{\left (2+3 i\right ) x}}{26}+\frac {{\mathrm e}^{\left (2-3 i\right ) x}}{13}+\frac {3 i {\mathrm e}^{\left (2-3 i\right ) x}}{26}\) | \(36\) |
norman | \(\frac {\frac {6 \,{\mathrm e}^{2 x} \tan \left (\frac {3 x}{2}\right )}{13}-\frac {2 \,{\mathrm e}^{2 x} \left (\tan ^{2}\left (\frac {3 x}{2}\right )\right )}{13}+\frac {2 \,{\mathrm e}^{2 x}}{13}}{1+\tan ^{2}\left (\frac {3 x}{2}\right )}\) | \(41\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{13} \, {\left (2 \, \cos \left (3 \, x\right ) + 3 \, \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 = 0.58, size = 21, normalized size = 0.78 \begin {gather*} \frac {2}{13} \, \cos \left (3 \, x\right ) e^{\left (2 \, x\right )} + \frac {3}{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 = 0.10, size = 26, normalized size = 0.96 \begin {gather*} \frac {3 e^{2 x} \sin {\left (3 x \right )}}{13} + \frac {2 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 = 0.45, size = 19, normalized size = 0.70 \begin {gather*} \frac {1}{13} \, {\left (2 \, \cos \left (3 \, x\right ) + 3 \, \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 = 0.03, size = 19, normalized size = 0.70 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}\,\left (2\,\cos \left (3\,x\right )+3\,\sin \left (3\,x\right )\right )}{13} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F] Failed to verify
time = 1.00, size = 19, normalized size = 0.70 \begin {gather*} \frac {{\mathrm e}^{2 x} \left (13 \cos \left (3 x \right )-6 \sin \left (3 x \right )\right )}{13} \end {gather*}
Warning: Unable to verify antiderivative.
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