Integrand size = 12, antiderivative size = 21 \[ \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx=2 e^{\cos (x)} \cos \left (\frac {x}{2}+\sin (x)\right ) \sin \left (\frac {x}{2}\right ) \]
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\[ \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx=\int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx=2 e^{\cos (x)} \cos \left (\frac {x}{2}+\sin (x)\right ) \sin \left (\frac {x}{2}\right ) \]
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Result contains complex when optimal does not.
Time = 2.17 (sec) , antiderivative size = 52, normalized size of antiderivative = 2.48
method | result | size |
risch | \(-\frac {i {\mathrm e}^{i x} {\mathrm e}^{{\mathrm e}^{i x}}}{2}+\frac {i {\mathrm e}^{{\mathrm e}^{i x}}}{2}+\frac {i {\mathrm e}^{-i x} {\mathrm e}^{{\mathrm e}^{-i x}}}{2}-\frac {i {\mathrm e}^{{\mathrm e}^{-i x}}}{2}\) | \(52\) |
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Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (16) = 32\).
Time = 0.26 (sec) , antiderivative size = 84, normalized size of antiderivative = 4.00 \[ \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx={\left (2 \, \cos \left (x\right ) - 1\right )} \cos \left (\frac {2 \, {\left (x \tan \left (\frac {1}{2} \, x\right )^{2} + x + \tan \left (\frac {1}{2} \, x\right )\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) e^{\cos \left (x\right )} \sin \left (x\right ) - {\left (2 \, \cos \left (x\right )^{2} - \cos \left (x\right ) - 1\right )} e^{\cos \left (x\right )} \sin \left (\frac {2 \, {\left (x \tan \left (\frac {1}{2} \, x\right )^{2} + x + \tan \left (\frac {1}{2} \, x\right )\right )}}{\tan \left (\frac {1}{2} \, x\right )^{2} + 1}\right ) \]
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\[ \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx=\int e^{\cos {\left (x \right )}} \cos {\left (2 x + \sin {\left (x \right )} \right )}\, dx \]
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\[ \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx=\int { \cos \left (2 \, x + \sin \left (x\right )\right ) e^{\cos \left (x\right )} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (16) = 32\).
Time = 0.28 (sec) , antiderivative size = 136, normalized size of antiderivative = 6.48 \[ \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx=\frac {\cos \left (x\right )^{3} e^{\cos \left (x\right )} \sin \left (2 \, x + \sin \left (x\right )\right ) - \cos \left (2 \, x + \sin \left (x\right )\right ) \cos \left (x\right )^{2} e^{\cos \left (x\right )} \sin \left (x\right ) + \cos \left (x\right ) e^{\cos \left (x\right )} \sin \left (2 \, x + \sin \left (x\right )\right ) \sin \left (x\right )^{2} - \cos \left (2 \, x + \sin \left (x\right )\right ) e^{\cos \left (x\right )} \sin \left (x\right )^{3} - \cos \left (x\right )^{2} e^{\cos \left (x\right )} \sin \left (2 \, x + \sin \left (x\right )\right ) + 2 \, \cos \left (2 \, x + \sin \left (x\right )\right ) \cos \left (x\right ) e^{\cos \left (x\right )} \sin \left (x\right ) + e^{\cos \left (x\right )} \sin \left (2 \, x + \sin \left (x\right )\right ) \sin \left (x\right )^{2}}{\cos \left (x\right )^{4} + 2 \, \cos \left (x\right )^{2} \sin \left (x\right )^{2} + \sin \left (x\right )^{4}} \]
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Time = 17.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx={\mathrm {e}}^{\cos \left (x\right )}\,\left (\sin \left (x+\sin \left (x\right )\right )-\sin \left (\sin \left (x\right )\right )\right ) \]
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