Integrand size = 17, antiderivative size = 70 \[ \int (4-3 \cos (x)) \left (1-\frac {\sin (x)}{2}\right )^4 \, dx=\frac {227 x}{32}+10 \cos (x)-3 \cos ^2(x)-\frac {2 \cos ^3(x)}{3}-3 \sin (x)-\frac {99}{32} \cos (x) \sin (x)-\frac {3 \sin ^3(x)}{2}-\frac {1}{16} \cos (x) \sin ^3(x)+\frac {3 \sin ^4(x)}{8}-\frac {3 \sin ^5(x)}{80} \]
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Time = 0.13 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {4486, 2717, 2747, 2748, 2715, 8, 655} \[ \int (4-3 \cos (x)) \left (1-\frac {\sin (x)}{2}\right )^4 \, dx=\frac {227 x}{32}-\frac {3}{80} \sin ^5(x)+\frac {3 \sin ^4(x)}{8}-\frac {3 \sin ^3(x)}{2}-3 \sin (x)-\frac {2 \cos ^3(x)}{3}-3 \cos ^2(x)+10 \cos (x)-\frac {1}{16} \sin ^3(x) \cos (x)-\frac {99}{32} \sin (x) \cos (x) \]
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Rule 8
Rule 655
Rule 2715
Rule 2717
Rule 2747
Rule 2748
Rule 4486
Rubi steps \begin{align*} \text {integral}& = \int \left (4-3 \cos (x)+2 (-4+3 \cos (x)) \sin (x)-\frac {3}{2} (-4+3 \cos (x)) \sin ^2(x)+\frac {1}{2} (-4+3 \cos (x)) \sin ^3(x)-\frac {1}{16} (-4+3 \cos (x)) \sin ^4(x)\right ) \, dx \\ & = 4 x-\frac {1}{16} \int (-4+3 \cos (x)) \sin ^4(x) \, dx+\frac {1}{2} \int (-4+3 \cos (x)) \sin ^3(x) \, dx-\frac {3}{2} \int (-4+3 \cos (x)) \sin ^2(x) \, dx+2 \int (-4+3 \cos (x)) \sin (x) \, dx-3 \int \cos (x) \, dx \\ & = 4 x-3 \sin (x)-\frac {3 \sin ^3(x)}{2}-\frac {3 \sin ^5(x)}{80}-\frac {1}{54} \text {Subst}\left (\int (-4+x) \left (9-x^2\right ) \, dx,x,3 \cos (x)\right )+\frac {1}{4} \int \sin ^4(x) \, dx-\frac {2}{3} \text {Subst}(\int (-4+x) \, dx,x,3 \cos (x))+6 \int \sin ^2(x) \, dx \\ & = 4 x+8 \cos (x)-3 \cos ^2(x)-3 \sin (x)-3 \cos (x) \sin (x)-\frac {3 \sin ^3(x)}{2}-\frac {1}{16} \cos (x) \sin ^3(x)+\frac {3 \sin ^4(x)}{8}-\frac {3 \sin ^5(x)}{80}+\frac {2}{27} \text {Subst}\left (\int \left (9-x^2\right ) \, dx,x,3 \cos (x)\right )+\frac {3}{16} \int \sin ^2(x) \, dx+3 \int 1 \, dx \\ & = 7 x+10 \cos (x)-3 \cos ^2(x)-\frac {2 \cos ^3(x)}{3}-3 \sin (x)-\frac {99}{32} \cos (x) \sin (x)-\frac {3 \sin ^3(x)}{2}-\frac {1}{16} \cos (x) \sin ^3(x)+\frac {3 \sin ^4(x)}{8}-\frac {3 \sin ^5(x)}{80}+\frac {3 \int 1 \, dx}{32} \\ & = \frac {227 x}{32}+10 \cos (x)-3 \cos ^2(x)-\frac {2 \cos ^3(x)}{3}-3 \sin (x)-\frac {99}{32} \cos (x) \sin (x)-\frac {3 \sin ^3(x)}{2}-\frac {1}{16} \cos (x) \sin ^3(x)+\frac {3 \sin ^4(x)}{8}-\frac {3 \sin ^5(x)}{80} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.06 \[ \int (4-3 \cos (x)) \left (1-\frac {\sin (x)}{2}\right )^4 \, dx=\frac {227 x}{32}+\frac {19 \cos (x)}{2}-\frac {27}{16} \cos (2 x)-\frac {1}{6} \cos (3 x)+\frac {3}{64} \cos (4 x)-\frac {531 \sin (x)}{128}-\frac {25}{16} \sin (2 x)+\frac {99}{256} \sin (3 x)+\frac {1}{128} \sin (4 x)-\frac {3 \sin (5 x)}{1280} \]
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Time = 0.69 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.66
method | result | size |
parts | \(\frac {227 x}{32}-\frac {3 \left (\sin \left (x \right )-2\right )^{5}}{80}-3 \cos \left (x \right ) \sin \left (x \right )+\frac {2 \left (2+\sin ^{2}\left (x \right )\right ) \cos \left (x \right )}{3}-\frac {\left (\sin ^{3}\left (x \right )+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{16}+8 \cos \left (x \right )\) | \(46\) |
risch | \(\frac {227 x}{32}+\frac {19 \cos \left (x \right )}{2}-\frac {531 \sin \left (x \right )}{128}-\frac {3 \sin \left (5 x \right )}{1280}+\frac {3 \cos \left (4 x \right )}{64}+\frac {\sin \left (4 x \right )}{128}-\frac {\cos \left (3 x \right )}{6}+\frac {99 \sin \left (3 x \right )}{256}-\frac {27 \cos \left (2 x \right )}{16}-\frac {25 \sin \left (2 x \right )}{16}\) | \(55\) |
parallelrisch | \(-\frac {409}{960}+\frac {227 x}{32}-\frac {25 \sin \left (2 x \right )}{16}-\frac {3 \sin \left (5 x \right )}{1280}+\frac {99 \sin \left (3 x \right )}{256}+\frac {\sin \left (4 x \right )}{128}-\frac {531 \sin \left (x \right )}{128}+\frac {3 \cos \left (4 x \right )}{64}-\frac {27 \cos \left (2 x \right )}{16}-\frac {\cos \left (3 x \right )}{6}+\frac {19 \cos \left (x \right )}{2}\) | \(56\) |
default | \(-\frac {\left (\sin ^{3}\left (x \right )+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{16}+\frac {227 x}{32}+\frac {2 \left (2+\sin ^{2}\left (x \right )\right ) \cos \left (x \right )}{3}-3 \cos \left (x \right ) \sin \left (x \right )+8 \cos \left (x \right )-\frac {3 \left (\sin ^{5}\left (x \right )\right )}{80}+\frac {3 \left (\sin ^{4}\left (x \right )\right )}{8}-\frac {3 \left (\sin ^{3}\left (x \right )\right )}{2}-3 \left (\cos ^{2}\left (x \right )\right )-3 \sin \left (x \right )\) | \(66\) |
norman | \(\frac {28 \left (\tan ^{8}\left (\frac {x}{2}\right )\right )+114 \left (\tan ^{6}\left (\frac {x}{2}\right )\right )+\frac {268 \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{3}+\frac {470 \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{3}+\frac {227 x}{32}-\frac {391 \left (\tan ^{3}\left (\frac {x}{2}\right )\right )}{8}-\frac {306 \left (\tan ^{5}\left (\frac {x}{2}\right )\right )}{5}-\frac {185 \left (\tan ^{7}\left (\frac {x}{2}\right )\right )}{8}+\frac {3 \left (\tan ^{9}\left (\frac {x}{2}\right )\right )}{16}+\frac {1135 x \left (\tan ^{2}\left (\frac {x}{2}\right )\right )}{32}+\frac {1135 x \left (\tan ^{4}\left (\frac {x}{2}\right )\right )}{16}+\frac {1135 x \left (\tan ^{6}\left (\frac {x}{2}\right )\right )}{16}+\frac {1135 x \left (\tan ^{8}\left (\frac {x}{2}\right )\right )}{32}+\frac {227 x \left (\tan ^{10}\left (\frac {x}{2}\right )\right )}{32}-\frac {195 \tan \left (\frac {x}{2}\right )}{16}+\frac {56}{3}}{\left (1+\tan ^{2}\left (\frac {x}{2}\right )\right )^{5}}\) | \(132\) |
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Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.77 \[ \int (4-3 \cos (x)) \left (1-\frac {\sin (x)}{2}\right )^4 \, dx=\frac {3}{8} \, \cos \left (x\right )^{4} - \frac {2}{3} \, \cos \left (x\right )^{3} - \frac {15}{4} \, \cos \left (x\right )^{2} - \frac {1}{160} \, {\left (6 \, \cos \left (x\right )^{4} - 10 \, \cos \left (x\right )^{3} - 252 \, \cos \left (x\right )^{2} + 505 \, \cos \left (x\right ) + 726\right )} \sin \left (x\right ) + \frac {227}{32} \, x + 10 \, \cos \left (x\right ) \]
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Time = 0.20 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.11 \[ \int (4-3 \cos (x)) \left (1-\frac {\sin (x)}{2}\right )^4 \, dx=\frac {3 x \sin ^{4}{\left (x \right )}}{32} + \frac {3 x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{16} + 3 x \sin ^{2}{\left (x \right )} + \frac {3 x \cos ^{4}{\left (x \right )}}{32} + 3 x \cos ^{2}{\left (x \right )} + 4 x - \frac {3 \sin ^{5}{\left (x \right )}}{80} + \frac {3 \sin ^{4}{\left (x \right )}}{8} - \frac {5 \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{32} - \frac {3 \sin ^{3}{\left (x \right )}}{2} + 2 \sin ^{2}{\left (x \right )} \cos {\left (x \right )} - \frac {3 \sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{32} - 3 \sin {\left (x \right )} \cos {\left (x \right )} - 3 \sin {\left (x \right )} + \frac {4 \cos ^{3}{\left (x \right )}}{3} - 3 \cos ^{2}{\left (x \right )} + 8 \cos {\left (x \right )} \]
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Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.77 \[ \int (4-3 \cos (x)) \left (1-\frac {\sin (x)}{2}\right )^4 \, dx=-\frac {3}{80} \, \sin \left (x\right )^{5} + \frac {3}{8} \, \sin \left (x\right )^{4} - \frac {2}{3} \, \cos \left (x\right )^{3} - \frac {3}{2} \, \sin \left (x\right )^{3} - 3 \, \cos \left (x\right )^{2} + \frac {227}{32} \, x + 10 \, \cos \left (x\right ) + \frac {1}{128} \, \sin \left (4 \, x\right ) - \frac {25}{16} \, \sin \left (2 \, x\right ) - 3 \, \sin \left (x\right ) \]
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Time = 0.29 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.77 \[ \int (4-3 \cos (x)) \left (1-\frac {\sin (x)}{2}\right )^4 \, dx=\frac {227}{32} \, x + \frac {3}{64} \, \cos \left (4 \, x\right ) - \frac {1}{6} \, \cos \left (3 \, x\right ) - \frac {27}{16} \, \cos \left (2 \, x\right ) + \frac {19}{2} \, \cos \left (x\right ) - \frac {3}{1280} \, \sin \left (5 \, x\right ) + \frac {1}{128} \, \sin \left (4 \, x\right ) + \frac {99}{256} \, \sin \left (3 \, x\right ) - \frac {25}{16} \, \sin \left (2 \, x\right ) - \frac {531}{128} \, \sin \left (x\right ) \]
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Time = 0.43 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.34 \[ \int (4-3 \cos (x)) \left (1-\frac {\sin (x)}{2}\right )^4 \, dx=-\frac {6\,\sin \left (\frac {x}{2}\right )\,{\cos \left (\frac {x}{2}\right )}^9}{5}+6\,{\cos \left (\frac {x}{2}\right )}^8+\frac {17\,\sin \left (\frac {x}{2}\right )\,{\cos \left (\frac {x}{2}\right )}^7}{5}-\frac {52\,{\cos \left (\frac {x}{2}\right )}^6}{3}+\frac {93\,\sin \left (\frac {x}{2}\right )\,{\cos \left (\frac {x}{2}\right )}^5}{10}+2\,{\cos \left (\frac {x}{2}\right )}^4-\frac {191\,\sin \left (\frac {x}{2}\right )\,{\cos \left (\frac {x}{2}\right )}^3}{8}+28\,{\cos \left (\frac {x}{2}\right )}^2+\frac {3\,\sin \left (\frac {x}{2}\right )\,\cos \left (\frac {x}{2}\right )}{16}+\frac {227\,x}{32} \]
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