Optimal. Leaf size=27 \[ \frac {2}{13} e^{-3 x} \sin (2 x)-\frac {3}{13} e^{-3 x} \cos (2 x) \]
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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 = {4433} \[ \frac {2}{13} e^{-3 x} \sin (2 x)-\frac {3}{13} e^{-3 x} \cos (2 x) \]
Antiderivative was successfully verified.
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Rule 4433
Rubi steps
\begin {align*} \int e^{-3 x} \cos (2 x) \, dx &=-\frac {3}{13} e^{-3 x} \cos (2 x)+\frac {2}{13} e^{-3 x} \sin (2 x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 22, normalized size = 0.81 \[ \frac {1}{13} e^{-3 x} (2 \sin (2 x)-3 \cos (2 x)) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int e^{-3 x} \cos (2 x) \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 1.19, size = 21, normalized size = 0.78 \[ -\frac {3}{13} \, \cos \left (2 \, x\right ) e^{\left (-3 \, x\right )} + \frac {2}{13} \, e^{\left (-3 \, x\right )} \sin \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 19, normalized size = 0.70 \[ -\frac {1}{13} \, {\left (3 \, \cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )\right )} e^{\left (-3 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 22, normalized size = 0.81
method | result | size |
default | \(-\frac {3 \,{\mathrm e}^{-3 x} \cos \left (2 x \right )}{13}+\frac {2 \,{\mathrm e}^{-3 x} \sin \left (2 x \right )}{13}\) | \(22\) |
norman | \(\frac {\left (-\frac {3}{13}+\frac {3 \left (\tan ^{2}\relax (x )\right )}{13}+\frac {4 \tan \relax (x )}{13}\right ) {\mathrm e}^{-3 x}}{1+\tan ^{2}\relax (x )}\) | \(28\) |
risch | \(-\frac {3 \,{\mathrm e}^{\left (-3+2 i\right ) x}}{26}-\frac {i {\mathrm e}^{\left (-3+2 i\right ) x}}{13}-\frac {3 \,{\mathrm e}^{\left (-3-2 i\right ) x}}{26}+\frac {i {\mathrm e}^{\left (-3-2 i\right ) x}}{13}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 19, normalized size = 0.70 \[ -\frac {1}{13} \, {\left (3 \, \cos \left (2 \, x\right ) - 2 \, \sin \left (2 \, x\right )\right )} e^{\left (-3 \, x\right )} \]
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 \[ -\frac {{\mathrm {e}}^{-3\,x}\,\left (3\,\cos \left (2\,x\right )-2\,\sin \left (2\,x\right )\right )}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.47, size = 26, normalized size = 0.96 \[ \frac {2 e^{- 3 x} \sin {\left (2 x \right )}}{13} - \frac {3 e^{- 3 x} \cos {\left (2 x \right )}}{13} \]
Verification of antiderivative is not currently implemented for this CAS.
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