Integrand size = 14, antiderivative size = 20 \[ \int \cos ^4\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx=\frac {3 x}{8}+\frac {\cos (x)}{2}-\frac {1}{8} \cos (x) \sin (x) \]
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Leaf count is larger than twice the leaf count of optimal. \(64\) vs. \(2(20)=40\).
Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 3.20, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2715, 8} \[ \int \cos ^4\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx=\frac {3 x}{8}+\frac {1}{2} \sin \left (\frac {x}{2}+\frac {\pi }{4}\right ) \cos ^3\left (\frac {x}{2}+\frac {\pi }{4}\right )+\frac {3}{4} \sin \left (\frac {x}{2}+\frac {\pi }{4}\right ) \cos \left (\frac {x}{2}+\frac {\pi }{4}\right ) \]
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Rule 8
Rule 2715
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \cos ^4\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx=\frac {1}{16} (3 \pi +6 x+8 \cos (x)-2 \cos (x) \sin (x)) \]
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Time = 0.44 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.75
method | result | size |
risch | \(\frac {3 x}{8}+\frac {\cos \left (x \right )}{2}-\frac {\sin \left (2 x \right )}{16}\) | \(15\) |
parallelrisch | \(\frac {3 x}{8}+\frac {\cos \left (x \right )}{2}-\frac {\sin \left (2 x \right )}{16}\) | \(15\) |
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 | \(\frac {\frac {3 x}{8}-\frac {3 \left (\tan ^{3}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{2}+\frac {3 \left (\tan ^{5}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{2}-\frac {5 \left (\tan ^{7}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{2}+\frac {3 x \left (\tan ^{2}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{2}+\frac {9 x \left (\tan ^{4}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{4}+\frac {3 x \left (\tan ^{6}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{2}+\frac {3 x \left (\tan ^{8}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )}{8}+\frac {5 \tan \left (\frac {\pi }{8}+\frac {x}{4}\right )}{2}}{\left (1+\tan ^{2}\left (\frac {\pi }{8}+\frac {x}{4}\right )\right )^{4}}\) | \(118\) |
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Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (14) = 28\).
Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.85 \[ \int \cos ^4\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx=\frac {1}{4} \, {\left (2 \, \cos \left (\frac {1}{4} \, \pi + \frac {1}{2} \, x\right )^{3} + 3 \, \cos \left (\frac {1}{4} \, \pi + \frac {1}{2} \, x\right )\right )} \sin \left (\frac {1}{4} \, \pi + \frac {1}{2} \, x\right ) + \frac {3}{8} \, x \]
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Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (17) = 34\).
Time = 0.13 (sec) , antiderivative size = 99, normalized size of antiderivative = 4.95 \[ \int \cos ^4\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx=\frac {3 x \sin ^{4}{\left (\frac {x}{2} + \frac {\pi }{4} \right )}}{8} + \frac {3 x \sin ^{2}{\left (\frac {x}{2} + \frac {\pi }{4} \right )} \cos ^{2}{\left (\frac {x}{2} + \frac {\pi }{4} \right )}}{4} + \frac {3 x \cos ^{4}{\left (\frac {x}{2} + \frac {\pi }{4} \right )}}{8} + \frac {3 \sin ^{3}{\left (\frac {x}{2} + \frac {\pi }{4} \right )} \cos {\left (\frac {x}{2} + \frac {\pi }{4} \right )}}{4} + \frac {5 \sin {\left (\frac {x}{2} + \frac {\pi }{4} \right )} \cos ^{3}{\left (\frac {x}{2} + \frac {\pi }{4} \right )}}{4} \]
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Time = 0.20 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \cos ^4\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx=\frac {3}{16} \, \pi + \frac {3}{8} \, x + \frac {1}{16} \, \sin \left (\pi + 2 \, x\right ) + \frac {1}{2} \, \sin \left (\frac {1}{2} \, \pi + x\right ) \]
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Time = 0.27 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \cos ^4\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx=\frac {3}{8} \, x + \frac {1}{2} \, \cos \left (x\right ) - \frac {1}{16} \, \sin \left (2 \, x\right ) \]
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Time = 0.32 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \cos ^4\left (\frac {\pi }{4}+\frac {x}{2}\right ) \, dx=\frac {3\,x}{8}+\frac {\sin \left (\Pi +2\,x\right )}{16}+\frac {\sin \left (\frac {\Pi }{2}+x\right )}{2} \]
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