Integrand size = 9, antiderivative size = 90 \[ \int \cos ^8(x) \sin ^8(x) \, dx=\frac {35 x}{32768}+\frac {35 \cos (x) \sin (x)}{32768}+\frac {35 \cos ^3(x) \sin (x)}{49152}+\frac {7 \cos ^5(x) \sin (x)}{12288}+\frac {\cos ^7(x) \sin (x)}{2048}-\frac {1}{256} \cos ^9(x) \sin (x)-\frac {5}{384} \cos ^9(x) \sin ^3(x)-\frac {1}{32} \cos ^9(x) \sin ^5(x)-\frac {1}{16} \cos ^9(x) \sin ^7(x) \]
35/32768*x+35/32768*cos(x)*sin(x)+35/49152*cos(x)^3*sin(x)+7/12288*cos(x)^ 5*sin(x)+1/2048*cos(x)^7*sin(x)-1/256*cos(x)^9*sin(x)-5/384*cos(x)^9*sin(x )^3-1/32*cos(x)^9*sin(x)^5-1/16*cos(x)^9*sin(x)^7
Time = 0.04 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.42 \[ \int \cos ^8(x) \sin ^8(x) \, dx=\frac {35 x}{32768}-\frac {7 \sin (4 x)}{16384}+\frac {7 \sin (8 x)}{65536}-\frac {\sin (12 x)}{49152}+\frac {\sin (16 x)}{524288} \]
Time = 0.63 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.39, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.889, Rules used = {3042, 3048, 3042, 3048, 3042, 3048, 3042, 3048, 3042, 3115, 3042, 3115, 3042, 3115, 3042, 3115, 24}
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.
| \(\displaystyle \int \sin ^8(x) \cos ^8(x) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (x)^8 \cos (x)^8dx\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {7}{16} \int \cos ^8(x) \sin ^6(x)dx-\frac {1}{16} \sin ^7(x) \cos ^9(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{16} \int \cos (x)^8 \sin (x)^6dx-\frac {1}{16} \sin ^7(x) \cos ^9(x)\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {7}{16} \left (\frac {5}{14} \int \cos ^8(x) \sin ^4(x)dx-\frac {1}{14} \sin ^5(x) \cos ^9(x)\right )-\frac {1}{16} \sin ^7(x) \cos ^9(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{16} \left (\frac {5}{14} \int \cos (x)^8 \sin (x)^4dx-\frac {1}{14} \sin ^5(x) \cos ^9(x)\right )-\frac {1}{16} \sin ^7(x) \cos ^9(x)\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {7}{16} \left (\frac {5}{14} \left (\frac {1}{4} \int \cos ^8(x) \sin ^2(x)dx-\frac {1}{12} \sin ^3(x) \cos ^9(x)\right )-\frac {1}{14} \sin ^5(x) \cos ^9(x)\right )-\frac {1}{16} \sin ^7(x) \cos ^9(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{16} \left (\frac {5}{14} \left (\frac {1}{4} \int \cos (x)^8 \sin (x)^2dx-\frac {1}{12} \sin ^3(x) \cos ^9(x)\right )-\frac {1}{14} \sin ^5(x) \cos ^9(x)\right )-\frac {1}{16} \sin ^7(x) \cos ^9(x)\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle \frac {7}{16} \left (\frac {5}{14} \left (\frac {1}{4} \left (\frac {1}{10} \int \cos ^8(x)dx-\frac {1}{10} \sin (x) \cos ^9(x)\right )-\frac {1}{12} \sin ^3(x) \cos ^9(x)\right )-\frac {1}{14} \sin ^5(x) \cos ^9(x)\right )-\frac {1}{16} \sin ^7(x) \cos ^9(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{16} \left (\frac {5}{14} \left (\frac {1}{4} \left (\frac {1}{10} \int \sin \left (x+\frac {\pi }{2}\right )^8dx-\frac {1}{10} \sin (x) \cos ^9(x)\right )-\frac {1}{12} \sin ^3(x) \cos ^9(x)\right )-\frac {1}{14} \sin ^5(x) \cos ^9(x)\right )-\frac {1}{16} \sin ^7(x) \cos ^9(x)\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {7}{16} \left (\frac {5}{14} \left (\frac {1}{4} \left (\frac {1}{10} \left (\frac {7}{8} \int \cos ^6(x)dx+\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin (x) \cos ^9(x)\right )-\frac {1}{12} \sin ^3(x) \cos ^9(x)\right )-\frac {1}{14} \sin ^5(x) \cos ^9(x)\right )-\frac {1}{16} \sin ^7(x) \cos ^9(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{16} \left (\frac {5}{14} \left (\frac {1}{4} \left (\frac {1}{10} \left (\frac {7}{8} \int \sin \left (x+\frac {\pi }{2}\right )^6dx+\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin (x) \cos ^9(x)\right )-\frac {1}{12} \sin ^3(x) \cos ^9(x)\right )-\frac {1}{14} \sin ^5(x) \cos ^9(x)\right )-\frac {1}{16} \sin ^7(x) \cos ^9(x)\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {7}{16} \left (\frac {5}{14} \left (\frac {1}{4} \left (\frac {1}{10} \left (\frac {7}{8} \left (\frac {5}{6} \int \cos ^4(x)dx+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin (x) \cos ^9(x)\right )-\frac {1}{12} \sin ^3(x) \cos ^9(x)\right )-\frac {1}{14} \sin ^5(x) \cos ^9(x)\right )-\frac {1}{16} \sin ^7(x) \cos ^9(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{16} \left (\frac {5}{14} \left (\frac {1}{4} \left (\frac {1}{10} \left (\frac {7}{8} \left (\frac {5}{6} \int \sin \left (x+\frac {\pi }{2}\right )^4dx+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin (x) \cos ^9(x)\right )-\frac {1}{12} \sin ^3(x) \cos ^9(x)\right )-\frac {1}{14} \sin ^5(x) \cos ^9(x)\right )-\frac {1}{16} \sin ^7(x) \cos ^9(x)\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {7}{16} \left (\frac {5}{14} \left (\frac {1}{4} \left (\frac {1}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(x)dx+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin (x) \cos ^9(x)\right )-\frac {1}{12} \sin ^3(x) \cos ^9(x)\right )-\frac {1}{14} \sin ^5(x) \cos ^9(x)\right )-\frac {1}{16} \sin ^7(x) \cos ^9(x)\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {7}{16} \left (\frac {5}{14} \left (\frac {1}{4} \left (\frac {1}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (x+\frac {\pi }{2}\right )^2dx+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin (x) \cos ^9(x)\right )-\frac {1}{12} \sin ^3(x) \cos ^9(x)\right )-\frac {1}{14} \sin ^5(x) \cos ^9(x)\right )-\frac {1}{16} \sin ^7(x) \cos ^9(x)\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {7}{16} \left (\frac {5}{14} \left (\frac {1}{4} \left (\frac {1}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sin (x) \cos (x)\right )+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )+\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin (x) \cos ^9(x)\right )-\frac {1}{12} \sin ^3(x) \cos ^9(x)\right )-\frac {1}{14} \sin ^5(x) \cos ^9(x)\right )-\frac {1}{16} \sin ^7(x) \cos ^9(x)\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {7}{16} \left (\frac {5}{14} \left (\frac {1}{4} \left (\frac {1}{10} \left (\frac {1}{8} \sin (x) \cos ^7(x)+\frac {7}{8} \left (\frac {1}{6} \sin (x) \cos ^5(x)+\frac {5}{6} \left (\frac {1}{4} \sin (x) \cos ^3(x)+\frac {3}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )\right )\right )\right )-\frac {1}{10} \sin (x) \cos ^9(x)\right )-\frac {1}{12} \sin ^3(x) \cos ^9(x)\right )-\frac {1}{14} \sin ^5(x) \cos ^9(x)\right )-\frac {1}{16} \sin ^7(x) \cos ^9(x)\) |
-1/16*(Cos[x]^9*Sin[x]^7) + (7*(-1/14*(Cos[x]^9*Sin[x]^5) + (5*(-1/12*(Cos [x]^9*Sin[x]^3) + (-1/10*(Cos[x]^9*Sin[x]) + ((Cos[x]^7*Sin[x])/8 + (7*((C os[x]^5*Sin[x])/6 + (5*((Cos[x]^3*Sin[x])/4 + (3*(x/2 + (Cos[x]*Sin[x])/2) )/4))/6))/8)/10)/4))/14))/16
3.4.51.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Time = 0.54 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.32
| method | result | size |
| risch | \(\frac {35 x}{32768}+\frac {\sin \left (16 x \right )}{524288}-\frac {\sin \left (12 x \right )}{49152}+\frac {7 \sin \left (8 x \right )}{65536}-\frac {7 \sin \left (4 x \right )}{16384}\) | \(29\) |
| parallelrisch | \(\frac {35 x}{32768}+\frac {\sin \left (16 x \right )}{524288}-\frac {\sin \left (12 x \right )}{49152}+\frac {7 \sin \left (8 x \right )}{65536}-\frac {7 \sin \left (4 x \right )}{16384}\) | \(29\) |
| default | \(-\frac {\left (\cos ^{9}\left (x \right )\right ) \left (\sin ^{7}\left (x \right )\right )}{16}-\frac {\left (\cos ^{9}\left (x \right )\right ) \left (\sin ^{5}\left (x \right )\right )}{32}-\frac {5 \left (\sin ^{3}\left (x \right )\right ) \left (\cos ^{9}\left (x \right )\right )}{384}-\frac {\left (\cos ^{9}\left (x \right )\right ) \sin \left (x \right )}{256}+\frac {\left (\cos ^{7}\left (x \right )+\frac {7 \left (\cos ^{5}\left (x \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (x \right )\right )}{24}+\frac {35 \cos \left (x \right )}{16}\right ) \sin \left (x \right )}{2048}+\frac {35 x}{32768}\) | \(68\) |
Time = 0.28 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.61 \[ \int \cos ^8(x) \sin ^8(x) \, dx=\frac {1}{98304} \, {\left (6144 \, \cos \left (x\right )^{15} - 21504 \, \cos \left (x\right )^{13} + 25856 \, \cos \left (x\right )^{11} - 10880 \, \cos \left (x\right )^{9} + 48 \, \cos \left (x\right )^{7} + 56 \, \cos \left (x\right )^{5} + 70 \, \cos \left (x\right )^{3} + 105 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {35}{32768} \, x \]
1/98304*(6144*cos(x)^15 - 21504*cos(x)^13 + 25856*cos(x)^11 - 10880*cos(x) ^9 + 48*cos(x)^7 + 56*cos(x)^5 + 70*cos(x)^3 + 105*cos(x))*sin(x) + 35/327 68*x
Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.68 \[ \int \cos ^8(x) \sin ^8(x) \, dx=\frac {35 x}{32768} - \frac {\sin ^{7}{\left (2 x \right )} \cos {\left (2 x \right )}}{4096} - \frac {7 \sin ^{5}{\left (2 x \right )} \cos {\left (2 x \right )}}{24576} - \frac {35 \sin ^{3}{\left (2 x \right )} \cos {\left (2 x \right )}}{98304} - \frac {35 \sin {\left (2 x \right )} \cos {\left (2 x \right )}}{65536} \]
35*x/32768 - sin(2*x)**7*cos(2*x)/4096 - 7*sin(2*x)**5*cos(2*x)/24576 - 35 *sin(2*x)**3*cos(2*x)/98304 - 35*sin(2*x)*cos(2*x)/65536
Time = 0.18 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.33 \[ \int \cos ^8(x) \sin ^8(x) \, dx=\frac {1}{12288} \, \sin \left (4 \, x\right )^{3} + \frac {35}{32768} \, x + \frac {1}{524288} \, \sin \left (16 \, x\right ) + \frac {7}{65536} \, \sin \left (8 \, x\right ) - \frac {1}{2048} \, \sin \left (4 \, x\right ) \]
Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.31 \[ \int \cos ^8(x) \sin ^8(x) \, dx=\frac {35}{32768} \, x + \frac {1}{524288} \, \sin \left (16 \, x\right ) - \frac {1}{49152} \, \sin \left (12 \, x\right ) + \frac {7}{65536} \, \sin \left (8 \, x\right ) - \frac {7}{16384} \, \sin \left (4 \, x\right ) \]
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.62 \[ \int \cos ^8(x) \sin ^8(x) \, dx=\left (\frac {{\cos \left (x\right )}^7}{16}+\frac {{\cos \left (x\right )}^5}{32}+\frac {5\,{\cos \left (x\right )}^3}{384}+\frac {\cos \left (x\right )}{256}\right )\,{\sin \left (x\right )}^9+\frac {35\,x}{32768}-\frac {7\,\sin \left (2\,x\right )}{8192}+\frac {7\,\sin \left (4\,x\right )}{32768}-\frac {\sin \left (6\,x\right )}{24576}+\frac {\sin \left (8\,x\right )}{262144} \]