Integrand size = 8, antiderivative size = 23 \[ \int e^{-x} \sin (x) \, dx=-\frac {1}{2} e^{-x} \cos (x)-\frac {1}{2} e^{-x} \sin (x) \]
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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 = {4517} \[ \int e^{-x} \sin (x) \, dx=-\frac {1}{2} e^{-x} \sin (x)-\frac {1}{2} e^{-x} \cos (x) \]
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Rule 4517
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2} e^{-x} \cos (x)-\frac {1}{2} e^{-x} \sin (x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61 \[ \int e^{-x} \sin (x) \, dx=-\frac {1}{2} e^{-x} (\cos (x)+\sin (x)) \]
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Time = 0.12 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.52
method | result | size |
parallelrisch | \(-\frac {\left (\cos \left (x \right )+\sin \left (x \right )\right ) {\mathrm e}^{-x}}{2}\) | \(12\) |
default | \(-\frac {{\mathrm e}^{-x} \cos \left (x \right )}{2}-\frac {{\mathrm e}^{-x} \sin \left (x \right )}{2}\) | \(18\) |
norman | \(\frac {\left (-\frac {1}{2}+\frac {\left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{2}-\tan \left (\frac {x}{2}\right )\right ) {\mathrm e}^{-x}}{1+\tan ^{2}\left (\frac {x}{2}\right )}\) | \(32\) |
risch | \(-\frac {{\mathrm e}^{\left (-1+i\right ) x}}{4}+\frac {i {\mathrm e}^{\left (-1+i\right ) x}}{4}-\frac {{\mathrm e}^{\left (-1-i\right ) x}}{4}-\frac {i {\mathrm e}^{\left (-1-i\right ) x}}{4}\) | \(36\) |
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none
Time = 0.25 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int e^{-x} \sin (x) \, dx=-\frac {1}{2} \, \cos \left (x\right ) e^{\left (-x\right )} - \frac {1}{2} \, e^{\left (-x\right )} \sin \left (x\right ) \]
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Time = 0.16 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int e^{-x} \sin (x) \, dx=- \frac {e^{- x} \sin {\left (x \right )}}{2} - \frac {e^{- x} \cos {\left (x \right )}}{2} \]
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none
Time = 0.21 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.48 \[ \int e^{-x} \sin (x) \, dx=-\frac {1}{2} \, {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} e^{\left (-x\right )} \]
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none
Time = 0.28 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.48 \[ \int e^{-x} \sin (x) \, dx=-\frac {1}{2} \, {\left (\cos \left (x\right ) + \sin \left (x\right )\right )} e^{\left (-x\right )} \]
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Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.48 \[ \int e^{-x} \sin (x) \, dx=-\frac {{\mathrm {e}}^{-x}\,\left (\cos \left (x\right )+\sin \left (x\right )\right )}{2} \]
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