Integrand size = 26, antiderivative size = 30 \[ \int e^{-1+3 x} \cos \left (e^{-1+3 x}\right ) \sin \left (1+e^{-1+3 x}\right ) \, dx=-\frac {1}{12} \cos \left (1+2 e^{-1+3 x}\right )+\frac {1}{6} e^{-1+3 x} \sin (1) \]
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Time = 0.04 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {2320, 4670, 2718} \[ \int e^{-1+3 x} \cos \left (e^{-1+3 x}\right ) \sin \left (1+e^{-1+3 x}\right ) \, dx=\frac {1}{6} e^{3 x-1} \sin (1)-\frac {1}{12} \cos \left (2 e^{3 x-1}+1\right ) \]
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Rule 2320
Rule 2718
Rule 4670
Rubi steps \begin{align*} \text {integral}& = \frac {1}{3} \text {Subst}\left (\int \cos (x) \sin (1+x) \, dx,x,e^{-1+3 x}\right ) \\ & = \frac {1}{3} \text {Subst}\left (\int \left (\frac {\sin (1)}{2}+\frac {1}{2} \sin (1+2 x)\right ) \, dx,x,e^{-1+3 x}\right ) \\ & = \frac {1}{6} e^{-1+3 x} \sin (1)+\frac {1}{6} \text {Subst}\left (\int \sin (1+2 x) \, dx,x,e^{-1+3 x}\right ) \\ & = -\frac {1}{12} \cos \left (1+2 e^{-1+3 x}\right )+\frac {1}{6} e^{-1+3 x} \sin (1) \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int e^{-1+3 x} \cos \left (e^{-1+3 x}\right ) \sin \left (1+e^{-1+3 x}\right ) \, dx=-\frac {1}{12} \cos \left (1+2 e^{-1+3 x}\right )+\frac {1}{6} e^{-1+3 x} \sin (1) \]
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Time = 0.88 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(-\frac {\cos \left (1+2 \,{\mathrm e}^{-1+3 x}\right )}{12}+\frac {{\mathrm e}^{-1+3 x} \sin \left (1\right )}{6}\) | \(25\) |
default | \(-\frac {\cos \left (1+2 \,{\mathrm e}^{-1+3 x}\right )}{12}+\frac {{\mathrm e}^{-1+3 x} \sin \left (1\right )}{6}\) | \(25\) |
risch | \(-\frac {\cos \left (1+2 \,{\mathrm e}^{-1+3 x}\right )}{12}+\frac {{\mathrm e}^{-1+3 x} \sin \left (1\right )}{6}\) | \(25\) |
parallelrisch | \(\frac {{\mathrm e}^{-1+3 x} \sin \left (1\right )}{6}-\frac {\cos \left (1+2 \,{\mathrm e}^{-1+3 x}\right )}{12}-\frac {\cos \left (1\right )}{12}+\frac {1}{6}\) | \(30\) |
norman | \(\frac {\frac {2 \tan \left (\frac {{\mathrm e}^{-1+3 x}}{2}\right ) \tan \left (\frac {1}{2}+\frac {{\mathrm e}^{-1+3 x}}{2}\right )}{3}-\frac {{\mathrm e}^{-1+3 x} \tan \left (\frac {{\mathrm e}^{-1+3 x}}{2}\right )}{3}+\frac {{\mathrm e}^{-1+3 x} \tan \left (\frac {1}{2}+\frac {{\mathrm e}^{-1+3 x}}{2}\right )}{3}+\frac {{\mathrm e}^{-1+3 x} \tan \left (\frac {{\mathrm e}^{-1+3 x}}{2}\right ) \tan \left (\frac {1}{2}+\frac {{\mathrm e}^{-1+3 x}}{2}\right )^{2}}{3}-\frac {{\mathrm e}^{-1+3 x} \tan \left (\frac {{\mathrm e}^{-1+3 x}}{2}\right )^{2} \tan \left (\frac {1}{2}+\frac {{\mathrm e}^{-1+3 x}}{2}\right )}{3}}{\left (1+\tan \left (\frac {{\mathrm e}^{-1+3 x}}{2}\right )^{2}\right ) \left (1+\tan \left (\frac {1}{2}+\frac {{\mathrm e}^{-1+3 x}}{2}\right )^{2}\right )}\) | \(153\) |
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Time = 0.25 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.40 \[ \int e^{-1+3 x} \cos \left (e^{-1+3 x}\right ) \sin \left (1+e^{-1+3 x}\right ) \, dx=-\frac {1}{6} \, \cos \left (1\right ) \cos \left (e^{\left (3 \, x - 1\right )}\right )^{2} + \frac {1}{6} \, \cos \left (e^{\left (3 \, x - 1\right )}\right ) \sin \left (1\right ) \sin \left (e^{\left (3 \, x - 1\right )}\right ) + \frac {1}{6} \, e^{\left (3 \, x - 1\right )} \sin \left (1\right ) \]
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Timed out. \[ \int e^{-1+3 x} \cos \left (e^{-1+3 x}\right ) \sin \left (1+e^{-1+3 x}\right ) \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int e^{-1+3 x} \cos \left (e^{-1+3 x}\right ) \sin \left (1+e^{-1+3 x}\right ) \, dx=\frac {1}{6} \, e^{\left (3 \, x - 1\right )} \sin \left (1\right ) - \frac {1}{12} \, \cos \left (2 \, e^{\left (3 \, x - 1\right )} + 1\right ) \]
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Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int e^{-1+3 x} \cos \left (e^{-1+3 x}\right ) \sin \left (1+e^{-1+3 x}\right ) \, dx=\frac {1}{6} \, e^{\left (3 \, x - 1\right )} \sin \left (1\right ) - \frac {1}{12} \, \cos \left (2 \, e^{\left (3 \, x - 1\right )} + 1\right ) \]
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Time = 0.49 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int e^{-1+3 x} \cos \left (e^{-1+3 x}\right ) \sin \left (1+e^{-1+3 x}\right ) \, dx=\frac {{\mathrm {e}}^{3\,x-1}\,\sin \left (1\right )}{6}-\frac {\cos \left (2\,{\mathrm {e}}^{3\,x-1}+1\right )}{12} \]
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