Integrand size = 13, antiderivative size = 84 \[ \int (1-\cos (x))^5 \cos (5 x) \, dx=-\frac {x}{32}+\frac {5 \sin (x)}{16}-\frac {45}{64} \sin (2 x)+\frac {5}{4} \sin (3 x)-\frac {105}{64} \sin (4 x)+\frac {63}{40} \sin (5 x)-\frac {35}{32} \sin (6 x)+\frac {15}{28} \sin (7 x)-\frac {45}{256} \sin (8 x)+\frac {5}{144} \sin (9 x)-\frac {1}{320} \sin (10 x) \]
-1/32*x+5/16*sin(x)-45/64*sin(2*x)+5/4*sin(3*x)-105/64*sin(4*x)+63/40*sin( 5*x)-35/32*sin(6*x)+15/28*sin(7*x)-45/256*sin(8*x)+5/144*sin(9*x)-1/320*si n(10*x)
Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00 \[ \int (1-\cos (x))^5 \cos (5 x) \, dx=-\frac {x}{32}+\frac {5 \sin (x)}{16}-\frac {45}{64} \sin (2 x)+\frac {5}{4} \sin (3 x)-\frac {105}{64} \sin (4 x)+\frac {63}{40} \sin (5 x)-\frac {35}{32} \sin (6 x)+\frac {15}{28} \sin (7 x)-\frac {45}{256} \sin (8 x)+\frac {5}{144} \sin (9 x)-\frac {1}{320} \sin (10 x) \]
-1/32*x + (5*Sin[x])/16 - (45*Sin[2*x])/64 + (5*Sin[3*x])/4 - (105*Sin[4*x ])/64 + (63*Sin[5*x])/40 - (35*Sin[6*x])/32 + (15*Sin[7*x])/28 - (45*Sin[8 *x])/256 + (5*Sin[9*x])/144 - Sin[10*x]/320
Time = 0.38 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, 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.
\(\displaystyle \int (1-\cos (x))^5 \cos (5 x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (1-\cos (x))^5 \cos (5 x)dx\) |
\(\Big \downarrow \) 4901 |
\(\displaystyle \int \left (-\cos (5 x) \cos ^5(x)+5 \cos (5 x) \cos ^4(x)-10 \cos (5 x) \cos ^3(x)+10 \cos (5 x) \cos ^2(x)-5 \cos (5 x) \cos (x)+\cos (5 x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {x}{32}+\frac {5 \sin (x)}{16}-\frac {45}{64} \sin (2 x)+\frac {5}{4} \sin (3 x)-\frac {105}{64} \sin (4 x)+\frac {63}{40} \sin (5 x)-\frac {35}{32} \sin (6 x)+\frac {15}{28} \sin (7 x)-\frac {45}{256} \sin (8 x)+\frac {5}{144} \sin (9 x)-\frac {1}{320} \sin (10 x)\) |
-1/32*x + (5*Sin[x])/16 - (45*Sin[2*x])/64 + (5*Sin[3*x])/4 - (105*Sin[4*x ])/64 + (63*Sin[5*x])/40 - (35*Sin[6*x])/32 + (15*Sin[7*x])/28 - (45*Sin[8 *x])/256 + (5*Sin[9*x])/144 - Sin[10*x]/320
3.3.60.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = ExpandTrig[u, x]}, Int[v, x] /; SumQ[v]] /; !InertTrigFreeQ[u]
Time = 1.30 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.75
method | result | size |
default | \(-\frac {x}{32}+\frac {5 \sin \left (x \right )}{16}-\frac {45 \sin \left (2 x \right )}{64}+\frac {5 \sin \left (3 x \right )}{4}-\frac {105 \sin \left (4 x \right )}{64}+\frac {63 \sin \left (5 x \right )}{40}-\frac {35 \sin \left (6 x \right )}{32}+\frac {15 \sin \left (7 x \right )}{28}-\frac {45 \sin \left (8 x \right )}{256}+\frac {5 \sin \left (9 x \right )}{144}-\frac {\sin \left (10 x \right )}{320}\) | \(63\) |
risch | \(-\frac {x}{32}+\frac {5 \sin \left (x \right )}{16}-\frac {45 \sin \left (2 x \right )}{64}+\frac {5 \sin \left (3 x \right )}{4}-\frac {105 \sin \left (4 x \right )}{64}+\frac {63 \sin \left (5 x \right )}{40}-\frac {35 \sin \left (6 x \right )}{32}+\frac {15 \sin \left (7 x \right )}{28}-\frac {45 \sin \left (8 x \right )}{256}+\frac {5 \sin \left (9 x \right )}{144}-\frac {\sin \left (10 x \right )}{320}\) | \(63\) |
parallelrisch | \(-\frac {x}{32}+\frac {5 \sin \left (x \right )}{16}-\frac {45 \sin \left (2 x \right )}{64}+\frac {5 \sin \left (3 x \right )}{4}-\frac {105 \sin \left (4 x \right )}{64}+\frac {63 \sin \left (5 x \right )}{40}-\frac {35 \sin \left (6 x \right )}{32}+\frac {15 \sin \left (7 x \right )}{28}-\frac {45 \sin \left (8 x \right )}{256}+\frac {5 \sin \left (9 x \right )}{144}-\frac {\sin \left (10 x \right )}{320}\) | \(63\) |
parts | \(-\frac {x}{32}+\frac {5 \sin \left (x \right )}{16}-\frac {45 \sin \left (2 x \right )}{64}+\frac {5 \sin \left (3 x \right )}{4}-\frac {105 \sin \left (4 x \right )}{64}+\frac {63 \sin \left (5 x \right )}{40}-\frac {35 \sin \left (6 x \right )}{32}+\frac {15 \sin \left (7 x \right )}{28}-\frac {45 \sin \left (8 x \right )}{256}+\frac {5 \sin \left (9 x \right )}{144}-\frac {\sin \left (10 x \right )}{320}\) | \(63\) |
-1/32*x+5/16*sin(x)-45/64*sin(2*x)+5/4*sin(3*x)-105/64*sin(4*x)+63/40*sin( 5*x)-35/32*sin(6*x)+15/28*sin(7*x)-45/256*sin(8*x)+5/144*sin(9*x)-1/320*si n(10*x)
Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.74 \[ \int (1-\cos (x))^5 \cos (5 x) \, dx=-\frac {1}{10080} \, {\left (16128 \, \cos \left (x\right )^{9} - 89600 \, \cos \left (x\right )^{8} + 194544 \, \cos \left (x\right )^{7} - 188800 \, \cos \left (x\right )^{6} + 33768 \, \cos \left (x\right )^{5} + 93984 \, \cos \left (x\right )^{4} - 83790 \, \cos \left (x\right )^{3} + 24512 \, \cos \left (x\right )^{2} + 315 \, \cos \left (x\right ) - 1376\right )} \sin \left (x\right ) - \frac {1}{32} \, x \]
-1/10080*(16128*cos(x)^9 - 89600*cos(x)^8 + 194544*cos(x)^7 - 188800*cos(x )^6 + 33768*cos(x)^5 + 93984*cos(x)^4 - 83790*cos(x)^3 + 24512*cos(x)^2 + 315*cos(x) - 1376)*sin(x) - 1/32*x
Leaf count of result is larger than twice the leaf count of optimal. 394 vs. \(2 (83) = 166\).
Time = 4.27 (sec) , antiderivative size = 394, normalized size of antiderivative = 4.69 \[ \int (1-\cos (x))^5 \cos (5 x) \, dx=- \frac {x \sin ^{5}{\left (x \right )} \sin {\left (5 x \right )}}{32} - \frac {5 x \sin ^{4}{\left (x \right )} \cos {\left (x \right )} \cos {\left (5 x \right )}}{32} + \frac {5 x \sin ^{3}{\left (x \right )} \sin {\left (5 x \right )} \cos ^{2}{\left (x \right )}}{16} + \frac {5 x \sin ^{2}{\left (x \right )} \cos ^{3}{\left (x \right )} \cos {\left (5 x \right )}}{16} - \frac {5 x \sin {\left (x \right )} \sin {\left (5 x \right )} \cos ^{4}{\left (x \right )}}{32} - \frac {x \cos ^{5}{\left (x \right )} \cos {\left (5 x \right )}}{32} - \frac {\sin ^{5}{\left (x \right )} \cos {\left (5 x \right )}}{64} + \frac {3 \sin ^{4}{\left (x \right )} \sin {\left (5 x \right )} \cos {\left (x \right )}}{64} + \frac {8 \sin ^{4}{\left (x \right )} \sin {\left (5 x \right )}}{63} + \frac {40 \sin ^{3}{\left (x \right )} \cos {\left (x \right )} \cos {\left (5 x \right )}}{63} - \frac {5 \sin ^{3}{\left (x \right )} \cos {\left (5 x \right )}}{32} + \frac {\sin ^{2}{\left (x \right )} \sin {\left (5 x \right )} \cos ^{3}{\left (x \right )}}{6} - \frac {4 \sin ^{2}{\left (x \right )} \sin {\left (5 x \right )} \cos ^{2}{\left (x \right )}}{3} + \frac {25 \sin ^{2}{\left (x \right )} \sin {\left (5 x \right )} \cos {\left (x \right )}}{32} - \frac {4 \sin ^{2}{\left (x \right )} \sin {\left (5 x \right )}}{21} + \frac {55 \sin {\left (x \right )} \cos ^{4}{\left (x \right )} \cos {\left (5 x \right )}}{192} - \frac {100 \sin {\left (x \right )} \cos ^{3}{\left (x \right )} \cos {\left (5 x \right )}}{63} + \frac {55 \sin {\left (x \right )} \cos ^{2}{\left (x \right )} \cos {\left (5 x \right )}}{32} - \frac {20 \sin {\left (x \right )} \cos {\left (x \right )} \cos {\left (5 x \right )}}{21} + \frac {5 \sin {\left (x \right )} \cos {\left (5 x \right )}}{24} - \frac {241 \sin {\left (5 x \right )} \cos ^{5}{\left (x \right )}}{960} + \frac {83 \sin {\left (5 x \right )} \cos ^{4}{\left (x \right )}}{63} - \frac {75 \sin {\left (5 x \right )} \cos ^{3}{\left (x \right )}}{32} + \frac {46 \sin {\left (5 x \right )} \cos ^{2}{\left (x \right )}}{21} - \frac {25 \sin {\left (5 x \right )} \cos {\left (x \right )}}{24} + \frac {\sin {\left (5 x \right )}}{5} \]
-x*sin(x)**5*sin(5*x)/32 - 5*x*sin(x)**4*cos(x)*cos(5*x)/32 + 5*x*sin(x)** 3*sin(5*x)*cos(x)**2/16 + 5*x*sin(x)**2*cos(x)**3*cos(5*x)/16 - 5*x*sin(x) *sin(5*x)*cos(x)**4/32 - x*cos(x)**5*cos(5*x)/32 - sin(x)**5*cos(5*x)/64 + 3*sin(x)**4*sin(5*x)*cos(x)/64 + 8*sin(x)**4*sin(5*x)/63 + 40*sin(x)**3*c os(x)*cos(5*x)/63 - 5*sin(x)**3*cos(5*x)/32 + sin(x)**2*sin(5*x)*cos(x)**3 /6 - 4*sin(x)**2*sin(5*x)*cos(x)**2/3 + 25*sin(x)**2*sin(5*x)*cos(x)/32 - 4*sin(x)**2*sin(5*x)/21 + 55*sin(x)*cos(x)**4*cos(5*x)/192 - 100*sin(x)*co s(x)**3*cos(5*x)/63 + 55*sin(x)*cos(x)**2*cos(5*x)/32 - 20*sin(x)*cos(x)*c os(5*x)/21 + 5*sin(x)*cos(5*x)/24 - 241*sin(5*x)*cos(x)**5/960 + 83*sin(5* x)*cos(x)**4/63 - 75*sin(5*x)*cos(x)**3/32 + 46*sin(5*x)*cos(x)**2/21 - 25 *sin(5*x)*cos(x)/24 + sin(5*x)/5
Time = 0.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.79 \[ \int (1-\cos (x))^5 \cos (5 x) \, dx=\frac {80}{9} \, \sin \left (x\right )^{9} - \frac {380}{7} \, \sin \left (x\right )^{7} - \frac {1}{20} \, \sin \left (2 \, x\right )^{5} + \frac {501}{5} \, \sin \left (x\right )^{5} + \frac {71}{16} \, \sin \left (2 \, x\right )^{3} - \frac {212}{3} \, \sin \left (x\right )^{3} - \frac {1}{32} \, x - \frac {45}{256} \, \sin \left (8 \, x\right ) - \frac {105}{64} \, \sin \left (4 \, x\right ) - 4 \, \sin \left (2 \, x\right ) + 16 \, \sin \left (x\right ) \]
80/9*sin(x)^9 - 380/7*sin(x)^7 - 1/20*sin(2*x)^5 + 501/5*sin(x)^5 + 71/16* sin(2*x)^3 - 212/3*sin(x)^3 - 1/32*x - 45/256*sin(8*x) - 105/64*sin(4*x) - 4*sin(2*x) + 16*sin(x)
Time = 0.26 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.74 \[ \int (1-\cos (x))^5 \cos (5 x) \, dx=-\frac {1}{32} \, x - \frac {1}{320} \, \sin \left (10 \, x\right ) + \frac {5}{144} \, \sin \left (9 \, x\right ) - \frac {45}{256} \, \sin \left (8 \, x\right ) + \frac {15}{28} \, \sin \left (7 \, x\right ) - \frac {35}{32} \, \sin \left (6 \, x\right ) + \frac {63}{40} \, \sin \left (5 \, x\right ) - \frac {105}{64} \, \sin \left (4 \, x\right ) + \frac {5}{4} \, \sin \left (3 \, x\right ) - \frac {45}{64} \, \sin \left (2 \, x\right ) + \frac {5}{16} \, \sin \left (x\right ) \]
-1/32*x - 1/320*sin(10*x) + 5/144*sin(9*x) - 45/256*sin(8*x) + 15/28*sin(7 *x) - 35/32*sin(6*x) + 63/40*sin(5*x) - 105/64*sin(4*x) + 5/4*sin(3*x) - 4 5/64*sin(2*x) + 5/16*sin(x)
Time = 16.54 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.93 \[ \int (1-\cos (x))^5 \cos (5 x) \, dx=-\frac {8\,\sin \left (x\right )\,{\cos \left (x\right )}^9}{5}+\frac {80\,\sin \left (x\right )\,{\cos \left (x\right )}^8}{9}-\frac {193\,\sin \left (x\right )\,{\cos \left (x\right )}^7}{10}+\frac {1180\,\sin \left (x\right )\,{\cos \left (x\right )}^6}{63}-\frac {67\,\sin \left (x\right )\,{\cos \left (x\right )}^5}{20}-\frac {979\,\sin \left (x\right )\,{\cos \left (x\right )}^4}{105}+\frac {133\,\sin \left (x\right )\,{\cos \left (x\right )}^3}{16}-\frac {766\,\sin \left (x\right )\,{\cos \left (x\right )}^2}{315}-\frac {\sin \left (x\right )\,\cos \left (x\right )}{32}-\frac {x}{32}+\frac {43\,\sin \left (x\right )}{315} \]