Optimal. Leaf size=20 \[ \frac {3 x}{8}+\frac {\cos (x)}{2}-\frac {1}{8} \cos (x) \sin (x) \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(20)=40\).
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 = {2715, 8}
\begin {gather*} \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 ) \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2715
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 \begin {gather*} \frac {1}{16} (3 \pi +6 x+8 \cos (x)-2 \cos (x) \sin (x)) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(38\) vs.
\(2(14)=28\).
time = 0.09, size = 39, normalized size = 1.95
method | result | size |
risch | \(\frac {3 x}{8}+\frac {\cos \left (x \right )}{2}-\frac {\sin \left (2 x \right )}{16}\) | \(15\) |
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 = 1.93, size = 23, normalized size = 1.15 \begin {gather*} \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 ) \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. 37 vs.
\(2 (14) = 28\).
time = 0.98, size = 37, normalized size = 1.85 \begin {gather*} \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 \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. 99 vs.
\(2 (17) = 34\).
time = 0.15, size = 99, normalized size = 4.95 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.24, size = 14, normalized size = 0.70 \begin {gather*} \frac {3}{8} \, x + \frac {1}{2} \, \cos \left (x\right ) - \frac {1}{16} \, \sin \left (2 \, x\right ) \end {gather*}
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 \begin {gather*} \frac {3\,x}{8}+\frac {\sin \left (\Pi +2\,x\right )}{16}+\frac {\sin \left (\frac {\Pi }{2}+x\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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