Optimal. Leaf size=20 \[ \frac {3 x}{8}+\frac {\cos (x)}{2}-\frac {1}{8} \sin (x) \cos (x) \]
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Rubi [B] time = 0.02, antiderivative size = 64, normalized size of antiderivative = 3.20, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2635, 8} \[ \frac {3 x}{8}+\frac {1}{2} \sin \left (\frac {x}{2}+\frac {\pi }{4}\right ) \cos ^3\left (\frac {x}{2}+\frac {\pi }{4}\right )+\frac {3}{4} \sin \left (\frac {x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {x}{2}+\frac {\pi }{4}\right ) \]
Antiderivative was successfully verified.
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Rule 8
Rule 2635
Rubi steps
\begin {align*} \int \cos ^4\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx &=\frac {1}{2} \cos ^3\left (\frac {\pi }{4}+\frac {x}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {x}{2}\right )+\frac {3}{4} \int \sin ^2\left (\frac {\pi }{4}-\frac {x}{2}\right ) \, dx\\ &=\frac {3}{4} \cos \left (\frac {\pi }{4}+\frac {x}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {x}{2}\right )+\frac {1}{2} \cos ^3\left (\frac {\pi }{4}+\frac {x}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {x}{2}\right )+\frac {3 \int 1 \, dx}{8}\\ &=\frac {3 x}{8}+\frac {3}{4} \cos \left (\frac {\pi }{4}+\frac {x}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {x}{2}\right )+\frac {1}{2} \cos ^3\left (\frac {\pi }{4}+\frac {x}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {x}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 21, normalized size = 1.05 \[ \frac {1}{16} (6 x+8 \cos (x)-2 \sin (x) \cos (x)+3 \pi ) \]
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos ^4\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [B] time = 0.62, size = 37, normalized size = 1.85 \[ \frac {1}{4} \, {\left (2 \, \cos \left (\frac {1}{4} \, \pi + \frac {1}{2} \, x\right )^{3} + 3 \, \cos \left (\frac {1}{4} \, \pi + \frac {1}{2} \, x\right )\right )} \sin \left (\frac {1}{4} \, \pi + \frac {1}{2} \, x\right ) + \frac {3}{8} \, x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.59, size = 14, normalized size = 0.70 \[ \frac {3}{8} \, x + \frac {1}{2} \, \cos \relax (x) - \frac {1}{16} \, \sin \left (2 \, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 15, normalized size = 0.75
method | result | size |
risch | \(\frac {3 x}{8}+\frac {\cos \relax (x )}{2}-\frac {\sin \left (2 x \right )}{16}\) | \(15\) |
derivativedivides | \(\frac {\left (\cos ^{3}\left (\frac {\pi }{4}+\frac {x}{2}\right )+\frac {3 \cos \left (\frac {\pi }{4}+\frac {x}{2}\right )}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {x}{2}\right )}{2}+\frac {3 \pi }{16}+\frac {3 x}{8}\) | \(39\) |
default | \(\frac {\left (\cos ^{3}\left (\frac {\pi }{4}+\frac {x}{2}\right )+\frac {3 \cos \left (\frac {\pi }{4}+\frac {x}{2}\right )}{2}\right ) \sin \left (\frac {\pi }{4}+\frac {x}{2}\right )}{2}+\frac {3 \pi }{16}+\frac {3 x}{8}\) | \(39\) |
norman | \(\frac {\frac {3 x}{8}-\frac {3 \left (\tan ^{3}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{2}+\frac {3 \left (\tan ^{5}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{2}-\frac {5 \left (\tan ^{7}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{2}+\frac {3 x \left (\tan ^{2}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{2}+\frac {9 x \left (\tan ^{4}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{4}+\frac {3 x \left (\tan ^{6}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{2}+\frac {3 x \left (\tan ^{8}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{8}+\frac {5 \tan \left (\frac {\pi }{8}+\frac {x}{4}\right )}{2}}{\left (1+\tan ^{2}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )^{4}}\) | \(118\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 23, normalized size = 1.15 \[ \frac {3}{16} \, \pi + \frac {3}{8} \, x + \frac {1}{16} \, \sin \left (\pi + 2 \, x\right ) + \frac {1}{2} \, \sin \left (\frac {1}{2} \, \pi + x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 20, normalized size = 1.00 \[ \frac {3\,x}{8}+\frac {\sin \left (\Pi +2\,x\right )}{16}+\frac {\sin \left (\frac {\Pi }{2}+x\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.65, size = 99, normalized size = 4.95 \[ \frac {3 x \sin ^{4}{\left (\frac {x}{2} + \frac {\pi }{4} \right )}}{8} + \frac {3 x \sin ^{2}{\left (\frac {x}{2} + \frac {\pi }{4} \right )} \cos ^{2}{\left (\frac {x}{2} + \frac {\pi }{4} \right )}}{4} + \frac {3 x \cos ^{4}{\left (\frac {x}{2} + \frac {\pi }{4} \right )}}{8} + \frac {3 \sin ^{3}{\left (\frac {x}{2} + \frac {\pi }{4} \right )} \cos {\left (\frac {x}{2} + \frac {\pi }{4} \right )}}{4} + \frac {5 \sin {\left (\frac {x}{2} + \frac {\pi }{4} \right )} \cos ^{3}{\left (\frac {x}{2} + \frac {\pi }{4} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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