Integrand size = 12, antiderivative size = 10 \[ \int e^{\cos (x) \sin (x)} \cos (2 x) \, dx=e^{\frac {1}{2} \sin (2 x)} \]
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Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4441, 2225} \[ \int e^{\cos (x) \sin (x)} \cos (2 x) \, dx=e^{\frac {1}{2} \sin (2 x)} \]
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Rule 2225
Rule 4441
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int e^{x/2} \, dx,x,\sin (2 x)\right ) \\ & = e^{\frac {1}{2} \sin (2 x)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int e^{\cos (x) \sin (x)} \cos (2 x) \, dx=e^{\cos (x) \sin (x)} \]
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Time = 208.58 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \({\mathrm e}^{\cos \left (x \right ) \sin \left (x \right )}\) | \(7\) |
default | \({\mathrm e}^{\cos \left (x \right ) \sin \left (x \right )}\) | \(7\) |
risch | \({\mathrm e}^{\cos \left (x \right ) \sin \left (x \right )}\) | \(7\) |
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none
Time = 0.25 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.60 \[ \int e^{\cos (x) \sin (x)} \cos (2 x) \, dx=e^{\left (\cos \left (x\right ) \sin \left (x\right )\right )} \]
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Timed out. \[ \int e^{\cos (x) \sin (x)} \cos (2 x) \, dx=\text {Timed out} \]
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none
Time = 0.69 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int e^{\cos (x) \sin (x)} \cos (2 x) \, dx=e^{\left (\frac {1}{2} \, \sin \left (2 \, x\right )\right )} \]
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none
Time = 0.28 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int e^{\cos (x) \sin (x)} \cos (2 x) \, dx=e^{\left (\frac {\tan \left (x\right )}{\tan \left (x\right )^{2} + 1}\right )} \]
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Time = 26.28 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.70 \[ \int e^{\cos (x) \sin (x)} \cos (2 x) \, dx={\mathrm {e}}^{\frac {\sin \left (2\,x\right )}{2}} \]
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