Integrand size = 17, antiderivative size = 70 \[ \int (4-3 \cos (x)) \left (1-\frac {\sin (x)}{2}\right )^4 \, dx=\frac {227 x}{32}+10 \cos (x)-3 \cos ^2(x)-\frac {2 \cos ^3(x)}{3}-3 \sin (x)-\frac {99}{32} \cos (x) \sin (x)-\frac {3 \sin ^3(x)}{2}-\frac {1}{16} \cos (x) \sin ^3(x)+\frac {3 \sin ^4(x)}{8}-\frac {3 \sin ^5(x)}{80} \] Output:
227/32*x+10*cos(x)-3*cos(x)^2-2/3*cos(x)^3-3*sin(x)-99/32*cos(x)*sin(x)-3/ 2*sin(x)^3-1/16*cos(x)*sin(x)^3+3/8*sin(x)^4-3/80*sin(x)^5
Time = 0.20 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.06 \[ \int (4-3 \cos (x)) \left (1-\frac {\sin (x)}{2}\right )^4 \, dx=\frac {227 x}{32}+\frac {19 \cos (x)}{2}-\frac {27}{16} \cos (2 x)-\frac {1}{6} \cos (3 x)+\frac {3}{64} \cos (4 x)-\frac {531 \sin (x)}{128}-\frac {25}{16} \sin (2 x)+\frac {99}{256} \sin (3 x)+\frac {1}{128} \sin (4 x)-\frac {3 \sin (5 x)}{1280} \] Input:
Integrate[(4 - 3*Cos[x])*(1 - Sin[x]/2)^4,x]
Output:
(227*x)/32 + (19*Cos[x])/2 - (27*Cos[2*x])/16 - Cos[3*x]/6 + (3*Cos[4*x])/ 64 - (531*Sin[x])/128 - (25*Sin[2*x])/16 + (99*Sin[3*x])/256 + Sin[4*x]/12 8 - (3*Sin[5*x])/1280
Time = 0.33 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.09, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3042, 4901, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \left (1-\frac {\sin (x)}{2}\right )^4 (4-3 \cos (x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (1-\frac {\sin (x)}{2}\right )^4 (4-3 \cos (x))dx\) |
\(\Big \downarrow \) 4901 |
\(\displaystyle \int \left (-3 \cos (x)-\frac {1}{16} \sin ^4(x) (3 \cos (x)-4)+\frac {1}{2} \sin ^3(x) (3 \cos (x)-4)-\frac {3}{2} \sin ^2(x) (3 \cos (x)-4)+2 \sin (x) (3 \cos (x)-4)+4\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {227 x}{32}-\frac {3}{80} \sin ^5(x)+\frac {3 \sin ^4(x)}{8}-\frac {3 \sin ^3(x)}{2}-3 \sin (x)-\frac {2 \cos ^3(x)}{3}-\frac {1}{3} (4-3 \cos (x))^2+2 \cos (x)-\frac {1}{16} \sin ^3(x) \cos (x)-\frac {99}{32} \sin (x) \cos (x)\) |
Input:
Int[(4 - 3*Cos[x])*(1 - Sin[x]/2)^4,x]
Output:
(227*x)/32 - (4 - 3*Cos[x])^2/3 + 2*Cos[x] - (2*Cos[x]^3)/3 - 3*Sin[x] - ( 99*Cos[x]*Sin[x])/32 - (3*Sin[x]^3)/2 - (Cos[x]*Sin[x]^3)/16 + (3*Sin[x]^4 )/8 - (3*Sin[x]^5)/80
Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /; !InertTrigFreeQ[u]
Time = 97.49 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.66
method | result | size |
parts | \(\frac {227 x}{32}-\frac {3 \left (-2+\sin \left (x \right )\right )^{5}}{80}-3 \cos \left (x \right ) \sin \left (x \right )+\frac {2 \left (2+\sin \left (x \right )^{2}\right ) \cos \left (x \right )}{3}-\frac {\left (\sin \left (x \right )^{3}+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{16}+8 \cos \left (x \right )\) | \(46\) |
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 | \(-\frac {409}{960}+\frac {227 x}{32}+\frac {\sin \left (4 x \right )}{128}+\frac {99 \sin \left (3 x \right )}{256}-\frac {25 \sin \left (2 x \right )}{16}-\frac {531 \sin \left (x \right )}{128}-\frac {3 \sin \left (5 x \right )}{1280}+\frac {3 \cos \left (4 x \right )}{64}-\frac {\cos \left (3 x \right )}{6}-\frac {27 \cos \left (2 x \right )}{16}+\frac {19 \cos \left (x \right )}{2}\) | \(56\) |
default | \(\frac {227 x}{32}+8 \cos \left (x \right )-3 \cos \left (x \right ) \sin \left (x \right )+\frac {2 \left (2+\sin \left (x \right )^{2}\right ) \cos \left (x \right )}{3}-\frac {\left (\sin \left (x \right )^{3}+\frac {3 \sin \left (x \right )}{2}\right ) \cos \left (x \right )}{16}-3 \sin \left (x \right )-3 \cos \left (x \right )^{2}-\frac {3 \sin \left (x \right )^{3}}{2}+\frac {3 \sin \left (x \right )^{4}}{8}-\frac {3 \sin \left (x \right )^{5}}{80}\) | \(66\) |
norman | \(\frac {28 \tan \left (\frac {x}{2}\right )^{8}+114 \tan \left (\frac {x}{2}\right )^{6}+\frac {268 \tan \left (\frac {x}{2}\right )^{2}}{3}+\frac {470 \tan \left (\frac {x}{2}\right )^{4}}{3}+\frac {227 x}{32}-\frac {391 \tan \left (\frac {x}{2}\right )^{3}}{8}-\frac {306 \tan \left (\frac {x}{2}\right )^{5}}{5}-\frac {185 \tan \left (\frac {x}{2}\right )^{7}}{8}+\frac {3 \tan \left (\frac {x}{2}\right )^{9}}{16}+\frac {1135 x \tan \left (\frac {x}{2}\right )^{2}}{32}+\frac {1135 x \tan \left (\frac {x}{2}\right )^{4}}{16}+\frac {1135 x \tan \left (\frac {x}{2}\right )^{6}}{16}+\frac {1135 x \tan \left (\frac {x}{2}\right )^{8}}{32}+\frac {227 x \tan \left (\frac {x}{2}\right )^{10}}{32}-\frac {195 \tan \left (\frac {x}{2}\right )}{16}+\frac {56}{3}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{5}}\) | \(132\) |
orering | \(\text {Expression too large to display}\) | \(1213\) |
Input:
int((4-3*cos(x))*(1-1/2*sin(x))^4,x,method=_RETURNVERBOSE)
Output:
227/32*x-3/80*(-2+sin(x))^5-3*cos(x)*sin(x)+2/3*(2+sin(x)^2)*cos(x)-1/16*( sin(x)^3+3/2*sin(x))*cos(x)+8*cos(x)
Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.77 \[ \int (4-3 \cos (x)) \left (1-\frac {\sin (x)}{2}\right )^4 \, dx=\frac {3}{8} \, \cos \left (x\right )^{4} - \frac {2}{3} \, \cos \left (x\right )^{3} - \frac {15}{4} \, \cos \left (x\right )^{2} - \frac {1}{160} \, {\left (6 \, \cos \left (x\right )^{4} - 10 \, \cos \left (x\right )^{3} - 252 \, \cos \left (x\right )^{2} + 505 \, \cos \left (x\right ) + 726\right )} \sin \left (x\right ) + \frac {227}{32} \, x + 10 \, \cos \left (x\right ) \] Input:
integrate((4-3*cos(x))*(1-1/2*sin(x))^4,x, algorithm="fricas")
Output:
3/8*cos(x)^4 - 2/3*cos(x)^3 - 15/4*cos(x)^2 - 1/160*(6*cos(x)^4 - 10*cos(x )^3 - 252*cos(x)^2 + 505*cos(x) + 726)*sin(x) + 227/32*x + 10*cos(x)
Time = 0.19 (sec) , antiderivative size = 148, normalized size of antiderivative = 2.11 \[ \int (4-3 \cos (x)) \left (1-\frac {\sin (x)}{2}\right )^4 \, dx=\frac {3 x \sin ^{4}{\left (x \right )}}{32} + \frac {3 x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{16} + 3 x \sin ^{2}{\left (x \right )} + \frac {3 x \cos ^{4}{\left (x \right )}}{32} + 3 x \cos ^{2}{\left (x \right )} + 4 x - \frac {3 \sin ^{5}{\left (x \right )}}{80} + \frac {3 \sin ^{4}{\left (x \right )}}{8} - \frac {5 \sin ^{3}{\left (x \right )} \cos {\left (x \right )}}{32} - \frac {3 \sin ^{3}{\left (x \right )}}{2} + 2 \sin ^{2}{\left (x \right )} \cos {\left (x \right )} - \frac {3 \sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{32} - 3 \sin {\left (x \right )} \cos {\left (x \right )} - 3 \sin {\left (x \right )} + \frac {4 \cos ^{3}{\left (x \right )}}{3} - 3 \cos ^{2}{\left (x \right )} + 8 \cos {\left (x \right )} \] Input:
integrate((4-3*cos(x))*(1-1/2*sin(x))**4,x)
Output:
3*x*sin(x)**4/32 + 3*x*sin(x)**2*cos(x)**2/16 + 3*x*sin(x)**2 + 3*x*cos(x) **4/32 + 3*x*cos(x)**2 + 4*x - 3*sin(x)**5/80 + 3*sin(x)**4/8 - 5*sin(x)** 3*cos(x)/32 - 3*sin(x)**3/2 + 2*sin(x)**2*cos(x) - 3*sin(x)*cos(x)**3/32 - 3*sin(x)*cos(x) - 3*sin(x) + 4*cos(x)**3/3 - 3*cos(x)**2 + 8*cos(x)
Time = 0.03 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.77 \[ \int (4-3 \cos (x)) \left (1-\frac {\sin (x)}{2}\right )^4 \, dx=-\frac {3}{80} \, \sin \left (x\right )^{5} + \frac {3}{8} \, \sin \left (x\right )^{4} - \frac {2}{3} \, \cos \left (x\right )^{3} - \frac {3}{2} \, \sin \left (x\right )^{3} - 3 \, \cos \left (x\right )^{2} + \frac {227}{32} \, x + 10 \, \cos \left (x\right ) + \frac {1}{128} \, \sin \left (4 \, x\right ) - \frac {25}{16} \, \sin \left (2 \, x\right ) - 3 \, \sin \left (x\right ) \] Input:
integrate((4-3*cos(x))*(1-1/2*sin(x))^4,x, algorithm="maxima")
Output:
-3/80*sin(x)^5 + 3/8*sin(x)^4 - 2/3*cos(x)^3 - 3/2*sin(x)^3 - 3*cos(x)^2 + 227/32*x + 10*cos(x) + 1/128*sin(4*x) - 25/16*sin(2*x) - 3*sin(x)
Time = 0.13 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.77 \[ \int (4-3 \cos (x)) \left (1-\frac {\sin (x)}{2}\right )^4 \, dx=\frac {227}{32} \, x + \frac {3}{64} \, \cos \left (4 \, x\right ) - \frac {1}{6} \, \cos \left (3 \, x\right ) - \frac {27}{16} \, \cos \left (2 \, x\right ) + \frac {19}{2} \, \cos \left (x\right ) - \frac {3}{1280} \, \sin \left (5 \, x\right ) + \frac {1}{128} \, \sin \left (4 \, x\right ) + \frac {99}{256} \, \sin \left (3 \, x\right ) - \frac {25}{16} \, \sin \left (2 \, x\right ) - \frac {531}{128} \, \sin \left (x\right ) \] Input:
integrate((4-3*cos(x))*(1-1/2*sin(x))^4,x, algorithm="giac")
Output:
227/32*x + 3/64*cos(4*x) - 1/6*cos(3*x) - 27/16*cos(2*x) + 19/2*cos(x) - 3 /1280*sin(5*x) + 1/128*sin(4*x) + 99/256*sin(3*x) - 25/16*sin(2*x) - 531/1 28*sin(x)
Time = 0.23 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.34 \[ \int (4-3 \cos (x)) \left (1-\frac {\sin (x)}{2}\right )^4 \, dx=-\frac {6\,\sin \left (\frac {x}{2}\right )\,{\cos \left (\frac {x}{2}\right )}^9}{5}+6\,{\cos \left (\frac {x}{2}\right )}^8+\frac {17\,\sin \left (\frac {x}{2}\right )\,{\cos \left (\frac {x}{2}\right )}^7}{5}-\frac {52\,{\cos \left (\frac {x}{2}\right )}^6}{3}+\frac {93\,\sin \left (\frac {x}{2}\right )\,{\cos \left (\frac {x}{2}\right )}^5}{10}+2\,{\cos \left (\frac {x}{2}\right )}^4-\frac {191\,\sin \left (\frac {x}{2}\right )\,{\cos \left (\frac {x}{2}\right )}^3}{8}+28\,{\cos \left (\frac {x}{2}\right )}^2+\frac {3\,\sin \left (\frac {x}{2}\right )\,\cos \left (\frac {x}{2}\right )}{16}+\frac {227\,x}{32} \] Input:
int(-(3*cos(x) - 4)*(sin(x)/2 - 1)^4,x)
Output:
(227*x)/32 - (191*cos(x/2)^3*sin(x/2))/8 + (93*cos(x/2)^5*sin(x/2))/10 + ( 17*cos(x/2)^7*sin(x/2))/5 - (6*cos(x/2)^9*sin(x/2))/5 + 28*cos(x/2)^2 + 2* cos(x/2)^4 - (52*cos(x/2)^6)/3 + 6*cos(x/2)^8 + (3*cos(x/2)*sin(x/2))/16
Time = 0.16 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.84 \[ \int (4-3 \cos (x)) \left (1-\frac {\sin (x)}{2}\right )^4 \, dx=-\frac {\cos \left (x \right ) \sin \left (x \right )^{3}}{16}+\frac {2 \cos \left (x \right ) \sin \left (x \right )^{2}}{3}-\frac {99 \cos \left (x \right ) \sin \left (x \right )}{32}+\frac {28 \cos \left (x \right )}{3}-\frac {3 \sin \left (x \right )^{5}}{80}+\frac {3 \sin \left (x \right )^{4}}{8}-\frac {3 \sin \left (x \right )^{3}}{2}+3 \sin \left (x \right )^{2}-3 \sin \left (x \right )+\frac {227 x}{32}-\frac {28}{3} \] Input:
int((4-3*cos(x))*(1-1/2*sin(x))^4,x)
Output:
( - 30*cos(x)*sin(x)**3 + 320*cos(x)*sin(x)**2 - 1485*cos(x)*sin(x) + 4480 *cos(x) - 18*sin(x)**5 + 180*sin(x)**4 - 720*sin(x)**3 + 1440*sin(x)**2 - 1440*sin(x) + 3405*x - 4480)/480