Optimal. Leaf size=20 \[ \frac {x}{4}-\frac {1}{4} \cos \left (2 x+\frac {\pi }{6}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {4574, 2638} \[ \frac {x}{4}-\frac {1}{4} \cos \left (2 x+\frac {\pi }{6}\right ) \]
Antiderivative was successfully verified.
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Rule 2638
Rule 4574
Rubi steps
\begin {align*} \int \cos (x) \sin \left (\frac {\pi }{6}+x\right ) \, dx &=\int \left (\frac {1}{4}+\frac {1}{2} \sin \left (\frac {\pi }{6}+2 x\right )\right ) \, dx\\ &=\frac {x}{4}+\frac {1}{2} \int \sin \left (\frac {\pi }{6}+2 x\right ) \, dx\\ &=\frac {x}{4}-\frac {1}{4} \cos \left (\frac {\pi }{6}+2 x\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.00 \[ \frac {x}{4}-\frac {1}{4} \cos \left (2 x+\frac {\pi }{6}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 31, normalized size = 1.55 \[ -\frac {1}{4} \, \sqrt {3} \cos \left (\frac {1}{6} \, \pi + x\right )^{2} - \frac {1}{4} \, \cos \left (\frac {1}{6} \, \pi + x\right ) \sin \left (\frac {1}{6} \, \pi + x\right ) + \frac {1}{4} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.89, size = 14, normalized size = 0.70 \[ \frac {1}{4} \, x - \frac {1}{4} \, \cos \left (\frac {1}{6} \, \pi + 2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 15, normalized size = 0.75 \[ \frac {x}{4}-\frac {\cos \left (2 x +\frac {\pi }{6}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 14, normalized size = 0.70 \[ \frac {1}{4} \, x - \frac {1}{4} \, \cos \left (\frac {1}{6} \, \pi + 2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 18, normalized size = 0.90 \[ \frac {x\,\sin \left (\frac {\Pi }{6}\right )}{2}-\frac {\cos \left (\frac {\Pi }{6}+2\,x\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.56, size = 37, normalized size = 1.85 \[ - \frac {x \sin {\relax (x )} \cos {\left (x + \frac {\pi }{6} \right )}}{2} + \frac {x \sin {\left (x + \frac {\pi }{6} \right )} \cos {\relax (x )}}{2} - \frac {\cos {\relax (x )} \cos {\left (x + \frac {\pi }{6} \right )}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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