Integrand size = 8, antiderivative size = 23 \[ \int e^{-3 x} \cos (x) \, dx=-\frac {3}{10} e^{-3 x} \cos (x)+\frac {1}{10} e^{-3 x} \sin (x) \]
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Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4518} \[ \int e^{-3 x} \cos (x) \, dx=\frac {1}{10} e^{-3 x} \sin (x)-\frac {3}{10} e^{-3 x} \cos (x) \]
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Rule 4518
Rubi steps \begin{align*} \text {integral}& = -\frac {3}{10} e^{-3 x} \cos (x)+\frac {1}{10} e^{-3 x} \sin (x) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70 \[ \int e^{-3 x} \cos (x) \, dx=\frac {1}{10} e^{-3 x} (-3 \cos (x)+\sin (x)) \]
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Time = 0.30 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61
method | result | size |
parallelrisch | \(\frac {\left (-3 \cos \left (x \right )+\sin \left (x \right )\right ) {\mathrm e}^{-3 x}}{10}\) | \(14\) |
default | \(-\frac {3 \,{\mathrm e}^{-3 x} \cos \left (x \right )}{10}+\frac {{\mathrm e}^{-3 x} \sin \left (x \right )}{10}\) | \(18\) |
norman | \(\frac {\left (-\frac {3}{10}+\frac {3 \tan \left (\frac {x}{2}\right )^{2}}{10}+\frac {\tan \left (\frac {x}{2}\right )}{5}\right ) {\mathrm e}^{-3 x}}{1+\tan \left (\frac {x}{2}\right )^{2}}\) | \(34\) |
risch | \(-\frac {3 \,{\mathrm e}^{\left (-3+i\right ) x}}{20}-\frac {i {\mathrm e}^{\left (-3+i\right ) x}}{20}-\frac {3 \,{\mathrm e}^{\left (-3-i\right ) x}}{20}+\frac {i {\mathrm e}^{\left (-3-i\right ) x}}{20}\) | \(36\) |
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none
Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int e^{-3 x} \cos (x) \, dx=-\frac {3}{10} \, \cos \left (x\right ) e^{\left (-3 \, x\right )} + \frac {1}{10} \, e^{\left (-3 \, x\right )} \sin \left (x\right ) \]
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Time = 0.18 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int e^{-3 x} \cos (x) \, dx=\frac {e^{- 3 x} \sin {\left (x \right )}}{10} - \frac {3 e^{- 3 x} \cos {\left (x \right )}}{10} \]
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none
Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int e^{-3 x} \cos (x) \, dx=-\frac {1}{10} \, {\left (3 \, \cos \left (x\right ) - \sin \left (x\right )\right )} e^{\left (-3 \, x\right )} \]
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none
Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int e^{-3 x} \cos (x) \, dx=-\frac {1}{10} \, {\left (3 \, \cos \left (x\right ) - \sin \left (x\right )\right )} e^{\left (-3 \, x\right )} \]
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Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int e^{-3 x} \cos (x) \, dx=-\frac {{\mathrm {e}}^{-3\,x}\,\left (3\,\cos \left (x\right )-\sin \left (x\right )\right )}{10} \]
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