Optimal. Leaf size=20 \[ \frac {x}{4}-\frac {1}{4} \cos \left (\frac {\pi }{6}+2 x\right ) \]
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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 = {4670, 2718}
\begin {gather*} \frac {x}{4}-\frac {1}{4} \cos \left (2 x+\frac {\pi }{6}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2718
Rule 4670
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 \begin {gather*} \frac {x}{4}-\frac {1}{4} \cos \left (\frac {\pi }{6}+2 x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 15, normalized size = 0.75
method | result | size |
default | \(\frac {x}{4}-\frac {\cos \left (\frac {\pi }{6}+2 x \right )}{4}\) | \(15\) |
risch | \(\frac {x}{4}-\frac {\sqrt {3}\, \cos \left (2 x \right )}{8}+\frac {\sin \left (2 x \right )}{8}\) | \(20\) |
norman | \(\frac {x \tan \left (\frac {\pi }{12}+\frac {x}{2}\right )+x \tan \left (\frac {x}{2}\right ) \left (\tan ^{2}\left (\frac {\pi }{12}+\frac {x}{2}\right )\right )+2 \tan \left (\frac {x}{2}\right ) \tan \left (\frac {\pi }{12}+\frac {x}{2}\right )-x \tan \left (\frac {x}{2}\right )-x \left (\tan ^{2}\left (\frac {x}{2}\right )\right ) \tan \left (\frac {\pi }{12}+\frac {x}{2}\right )}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right ) \left (1+\tan ^{2}\left (\frac {\pi }{12}+\frac {x}{2}\right )\right )}\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.50, size = 14, normalized size = 0.70 \begin {gather*} \frac {1}{4} \, x - \frac {1}{4} \, \cos \left (\frac {1}{6} \, \pi + 2 \, x\right ) \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. 31 vs.
\(2 (14) = 28\).
time = 0.66, size = 31, normalized size = 1.55 \begin {gather*} -\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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 37 vs.
\(2 (12) = 24\).
time = 0.13, size = 37, normalized size = 1.85 \begin {gather*} - \frac {x \sin {\left (x \right )} \cos {\left (x + \frac {\pi }{6} \right )}}{2} + \frac {x \sin {\left (x + \frac {\pi }{6} \right )} \cos {\left (x \right )}}{2} - \frac {\cos {\left (x \right )} \cos {\left (x + \frac {\pi }{6} \right )}}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.64, size = 14, normalized size = 0.70 \begin {gather*} \frac {1}{4} \, x - \frac {1}{4} \, \cos \left (\frac {1}{6} \, \pi + 2 \, x\right ) \end {gather*}
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 \begin {gather*} \frac {x\,\sin \left (\frac {\Pi }{6}\right )}{2}-\frac {\cos \left (\frac {\Pi }{6}+2\,x\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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