Integrand size = 9, antiderivative size = 68 \[ \int \cos ^6(x) \sin ^6(x) \, dx=\frac {5 x}{1024}+\frac {5 \cos (x) \sin (x)}{1024}+\frac {5 \cos ^3(x) \sin (x)}{1536}+\frac {1}{384} \cos ^5(x) \sin (x)-\frac {1}{64} \cos ^7(x) \sin (x)-\frac {1}{24} \cos ^7(x) \sin ^3(x)-\frac {1}{12} \cos ^7(x) \sin ^5(x) \]
5/1024*x+5/1024*cos(x)*sin(x)+5/1536*cos(x)^3*sin(x)+1/384*cos(x)^5*sin(x) -1/64*cos(x)^7*sin(x)-1/24*cos(x)^7*sin(x)^3-1/12*cos(x)^7*sin(x)^5
Time = 0.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.44 \[ \int \cos ^6(x) \sin ^6(x) \, dx=\frac {5 x}{1024}-\frac {15 \sin (4 x)}{8192}+\frac {3 \sin (8 x)}{8192}-\frac {\sin (12 x)}{24576} \]
Time = 0.47 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.37, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.444, Rules used = {3042, 3048, 3042, 3048, 3042, 3048, 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.
\(\displaystyle \int \sin ^6(x) \cos ^6(x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (x)^6 \cos (x)^6dx\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle \frac {5}{12} \int \cos ^6(x) \sin ^4(x)dx-\frac {1}{12} \sin ^5(x) \cos ^7(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{12} \int \cos (x)^6 \sin (x)^4dx-\frac {1}{12} \sin ^5(x) \cos ^7(x)\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle \frac {5}{12} \left (\frac {3}{10} \int \cos ^6(x) \sin ^2(x)dx-\frac {1}{10} \sin ^3(x) \cos ^7(x)\right )-\frac {1}{12} \sin ^5(x) \cos ^7(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{12} \left (\frac {3}{10} \int \cos (x)^6 \sin (x)^2dx-\frac {1}{10} \sin ^3(x) \cos ^7(x)\right )-\frac {1}{12} \sin ^5(x) \cos ^7(x)\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle \frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \int \cos ^6(x)dx-\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin ^3(x) \cos ^7(x)\right )-\frac {1}{12} \sin ^5(x) \cos ^7(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \int \sin \left (x+\frac {\pi }{2}\right )^6dx-\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin ^3(x) \cos ^7(x)\right )-\frac {1}{12} \sin ^5(x) \cos ^7(x)\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \cos ^4(x)dx+\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin ^3(x) \cos ^7(x)\right )-\frac {1}{12} \sin ^5(x) \cos ^7(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \sin \left (x+\frac {\pi }{2}\right )^4dx+\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin ^3(x) \cos ^7(x)\right )-\frac {1}{12} \sin ^5(x) \cos ^7(x)\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(x)dx+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin ^3(x) \cos ^7(x)\right )-\frac {1}{12} \sin ^5(x) \cos ^7(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (x+\frac {\pi }{2}\right )^2dx+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin ^3(x) \cos ^7(x)\right )-\frac {1}{12} \sin ^5(x) \cos ^7(x)\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sin (x) \cos (x)\right )+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin ^3(x) \cos ^7(x)\right )-\frac {1}{12} \sin ^5(x) \cos ^7(x)\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {5}{12} \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {1}{6} \sin (x) \cos ^5(x)+\frac {5}{6} \left (\frac {1}{4} \sin (x) \cos ^3(x)+\frac {3}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )\right )\right )-\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin ^3(x) \cos ^7(x)\right )-\frac {1}{12} \sin ^5(x) \cos ^7(x)\) |
-1/12*(Cos[x]^7*Sin[x]^5) + (5*(-1/10*(Cos[x]^7*Sin[x]^3) + (3*(-1/8*(Cos[ x]^7*Sin[x]) + ((Cos[x]^5*Sin[x])/6 + (5*((Cos[x]^3*Sin[x])/4 + (3*(x/2 + (Cos[x]*Sin[x])/2))/4))/6)/8))/10))/12
3.4.50.3.1 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 = 0.48 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.34
method | result | size |
risch | \(\frac {5 x}{1024}-\frac {\sin \left (12 x \right )}{24576}+\frac {3 \sin \left (8 x \right )}{8192}-\frac {15 \sin \left (4 x \right )}{8192}\) | \(23\) |
parallelrisch | \(\frac {5 x}{1024}-\frac {\sin \left (12 x \right )}{24576}+\frac {3 \sin \left (8 x \right )}{8192}-\frac {15 \sin \left (4 x \right )}{8192}\) | \(23\) |
default | \(-\frac {\left (\sin ^{5}\left (x \right )\right ) \left (\cos ^{7}\left (x \right )\right )}{12}-\frac {\left (\cos ^{7}\left (x \right )\right ) \left (\sin ^{3}\left (x \right )\right )}{24}-\frac {\left (\cos ^{7}\left (x \right )\right ) \sin \left (x \right )}{64}+\frac {\left (\cos ^{5}\left (x \right )+\frac {5 \left (\cos ^{3}\left (x \right )\right )}{4}+\frac {15 \cos \left (x \right )}{8}\right ) \sin \left (x \right )}{384}+\frac {5 x}{1024}\) | \(52\) |
Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.63 \[ \int \cos ^6(x) \sin ^6(x) \, dx=-\frac {1}{3072} \, {\left (256 \, \cos \left (x\right )^{11} - 640 \, \cos \left (x\right )^{9} + 432 \, \cos \left (x\right )^{7} - 8 \, \cos \left (x\right )^{5} - 10 \, \cos \left (x\right )^{3} - 15 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {5}{1024} \, x \]
-1/3072*(256*cos(x)^11 - 640*cos(x)^9 + 432*cos(x)^7 - 8*cos(x)^5 - 10*cos (x)^3 - 15*cos(x))*sin(x) + 5/1024*x
Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.68 \[ \int \cos ^6(x) \sin ^6(x) \, dx=\frac {5 x}{1024} - \frac {\sin ^{5}{\left (2 x \right )} \cos {\left (2 x \right )}}{768} - \frac {5 \sin ^{3}{\left (2 x \right )} \cos {\left (2 x \right )}}{3072} - \frac {5 \sin {\left (2 x \right )} \cos {\left (2 x \right )}}{2048} \]
5*x/1024 - sin(2*x)**5*cos(2*x)/768 - 5*sin(2*x)**3*cos(2*x)/3072 - 5*sin( 2*x)*cos(2*x)/2048
Time = 0.19 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.35 \[ \int \cos ^6(x) \sin ^6(x) \, dx=\frac {1}{6144} \, \sin \left (4 \, x\right )^{3} + \frac {5}{1024} \, x + \frac {3}{8192} \, \sin \left (8 \, x\right ) - \frac {1}{512} \, \sin \left (4 \, x\right ) \]
Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.32 \[ \int \cos ^6(x) \sin ^6(x) \, dx=\frac {5}{1024} \, x - \frac {1}{24576} \, \sin \left (12 \, x\right ) + \frac {3}{8192} \, \sin \left (8 \, x\right ) - \frac {15}{8192} \, \sin \left (4 \, x\right ) \]
Time = 0.03 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.65 \[ \int \cos ^6(x) \sin ^6(x) \, dx=\left (\frac {{\cos \left (x\right )}^5}{12}+\frac {{\cos \left (x\right )}^3}{24}+\frac {\cos \left (x\right )}{64}\right )\,{\sin \left (x\right )}^7+\frac {5\,x}{1024}-\frac {15\,\sin \left (2\,x\right )}{4096}+\frac {3\,\sin \left (4\,x\right )}{4096}-\frac {\sin \left (6\,x\right )}{12288} \]
(5*x)/1024 - (15*sin(2*x))/4096 + (3*sin(4*x))/4096 - sin(6*x)/12288 + sin (x)^7*(cos(x)/64 + cos(x)^3/24 + cos(x)^5/12)
Time = 0.00 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.74 \[ \int \cos ^6(x) \sin ^6(x) \, dx=\frac {\cos \left (x \right ) \sin \left (x \right )^{11}}{12}-\frac {5 \cos \left (x \right ) \sin \left (x \right )^{9}}{24}+\frac {9 \cos \left (x \right ) \sin \left (x \right )^{7}}{64}-\frac {\cos \left (x \right ) \sin \left (x \right )^{5}}{384}-\frac {5 \cos \left (x \right ) \sin \left (x \right )^{3}}{1536}-\frac {5 \cos \left (x \right ) \sin \left (x \right )}{1024}+\frac {5 x}{1024} \]