Optimal. Leaf size=21 \[ 2 e^{\cos (x)} \cos \left (\frac {x}{2}+\sin (x)\right ) \sin \left (\frac {x}{2}\right ) \]
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Rubi [F]
time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {Integral} &=\int e^{\cos (x)} \cos (2 x+\sin (x)) \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 21, normalized size = 1.00 \begin {gather*} 2 e^{\cos (x)} \cos \left (\frac {x}{2}+\sin (x)\right ) \sin \left (\frac {x}{2}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 1.80, size = 52, normalized size = 2.48
method | result | size |
risch | \(-\frac {i {\mathrm e}^{{\mathrm e}^{i x}} {\mathrm e}^{i x}}{2}+\frac {i {\mathrm e}^{{\mathrm e}^{i x}}}{2}+\frac {i {\mathrm e}^{{\mathrm e}^{-i x}} {\mathrm e}^{-i x}}{2}-\frac {i {\mathrm e}^{{\mathrm e}^{-i x}}}{2}\) | \(52\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 84 vs.
\(2 (16) = 32\).
time = 0.62, size = 84, normalized size = 4.00 \begin {gather*} {\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int e^{\cos {\left (x \right )}} \cos {\left (2 x + \sin {\left (x \right )} \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 136 vs.
\(2 (16) = 32\).
time = 0.55, size = 136, normalized size = 6.48 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.36, size = 15, normalized size = 0.71 \begin {gather*} {\mathrm {e}}^{\cos \left (x\right )}\,\left (\sin \left (x+\sin \left (x\right )\right )-\sin \left (\sin \left (x\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Chatgpt [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {not solved} \end {gather*}
Warning: Unable to verify antiderivative.
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