Integrand size = 9, antiderivative size = 56 \[ \int \cos ^6(x) \sin ^4(x) \, dx=\frac {3 x}{256}+\frac {3}{256} \cos (x) \sin (x)+\frac {1}{128} \cos ^3(x) \sin (x)+\frac {1}{160} \cos ^5(x) \sin (x)-\frac {3}{80} \cos ^7(x) \sin (x)-\frac {1}{10} \cos ^7(x) \sin ^3(x) \] Output:
3/256*x+3/256*cos(x)*sin(x)+1/128*cos(x)^3*sin(x)+1/160*cos(x)^5*sin(x)-3/ 80*cos(x)^7*sin(x)-1/10*cos(x)^7*sin(x)^3
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int \cos ^6(x) \sin ^4(x) \, dx=\frac {3 x}{256}+\frac {1}{512} \sin (2 x)-\frac {1}{256} \sin (4 x)-\frac {\sin (6 x)}{1024}+\frac {\sin (8 x)}{2048}+\frac {\sin (10 x)}{5120} \] Input:
Integrate[Cos[x]^6*Sin[x]^4,x]
Output:
(3*x)/256 + Sin[2*x]/512 - Sin[4*x]/256 - Sin[6*x]/1024 + Sin[8*x]/2048 + Sin[10*x]/5120
Time = 0.40 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.36, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.222, Rules used = {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.
\(\displaystyle \int \sin ^4(x) \cos ^6(x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (x)^4 \cos (x)^6dx\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle \frac {3}{10} \int \cos ^6(x) \sin ^2(x)dx-\frac {1}{10} \sin ^3(x) \cos ^7(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{10} \int \cos (x)^6 \sin (x)^2dx-\frac {1}{10} \sin ^3(x) \cos ^7(x)\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle \frac {3}{10} \left (\frac {1}{8} \int \cos ^6(x)dx-\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin ^3(x) \cos ^7(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{10} \left (\frac {1}{8} \int \sin \left (x+\frac {\pi }{2}\right )^6dx-\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin ^3(x) \cos ^7(x)\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \cos ^4(x)dx+\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin ^3(x) \cos ^7(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \sin \left (x+\frac {\pi }{2}\right )^4dx+\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin ^3(x) \cos ^7(x)\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(x)dx+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin ^3(x) \cos ^7(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (x+\frac {\pi }{2}\right )^2dx+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin ^3(x) \cos ^7(x)\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sin (x) \cos (x)\right )+\frac {1}{4} \sin (x) \cos ^3(x)\right )+\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin ^3(x) \cos ^7(x)\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {3}{10} \left (\frac {1}{8} \left (\frac {1}{6} \sin (x) \cos ^5(x)+\frac {5}{6} \left (\frac {1}{4} \sin (x) \cos ^3(x)+\frac {3}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )\right )\right )-\frac {1}{8} \sin (x) \cos ^7(x)\right )-\frac {1}{10} \sin ^3(x) \cos ^7(x)\) |
Input:
Int[Cos[x]^6*Sin[x]^4,x]
Output:
-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))/1 0
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 = 8.77 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.62
method | result | size |
risch | \(\frac {3 x}{256}+\frac {\sin \left (10 x \right )}{5120}+\frac {\sin \left (8 x \right )}{2048}-\frac {\sin \left (6 x \right )}{1024}-\frac {\sin \left (4 x \right )}{256}+\frac {\sin \left (2 x \right )}{512}\) | \(35\) |
parallelrisch | \(\frac {3 x}{256}+\frac {\sin \left (10 x \right )}{5120}+\frac {\sin \left (8 x \right )}{2048}-\frac {\sin \left (6 x \right )}{1024}-\frac {\sin \left (4 x \right )}{256}+\frac {\sin \left (2 x \right )}{512}\) | \(35\) |
default | \(-\frac {\cos \left (x \right )^{7} \sin \left (x \right )^{3}}{10}-\frac {3 \sin \left (x \right ) \cos \left (x \right )^{7}}{80}+\frac {\left (\cos \left (x \right )^{5}+\frac {5 \cos \left (x \right )^{3}}{4}+\frac {15 \cos \left (x \right )}{8}\right ) \sin \left (x \right )}{160}+\frac {3 x}{256}\) | \(42\) |
orering | \(x \cos \left (x \right )^{6} \sin \left (x \right )^{4}+\frac {\cos \left (x \right )^{5} \sin \left (x \right )^{5}}{10}-\frac {7 \cos \left (x \right )^{7} \sin \left (x \right )^{3}}{128}+\frac {5269 x \left (30 \cos \left (x \right )^{4} \sin \left (x \right )^{6}-58 \cos \left (x \right )^{6} \sin \left (x \right )^{4}+12 \cos \left (x \right )^{8} \sin \left (x \right )^{2}\right )}{14400}+\frac {7 \cos \left (x \right )^{3} \sin \left (x \right )^{7}}{128}-\frac {3 \cos \left (x \right )^{9} \sin \left (x \right )}{256}+\frac {1529 x \left (360 \cos \left (x \right )^{2} \sin \left (x \right )^{8}-3480 \cos \left (x \right )^{4} \sin \left (x \right )^{6}+4936 \cos \left (x \right )^{6} \sin \left (x \right )^{4}-1200 \cos \left (x \right )^{8} \sin \left (x \right )^{2}+24 \cos \left (x \right )^{10}\right )}{46080}+\frac {3 \cos \left (x \right ) \sin \left (x \right )^{9}}{256}+\frac {341 x \left (720 \sin \left (x \right )^{10}-56880 \cos \left (x \right )^{2} \sin \left (x \right )^{8}+370080 \cos \left (x \right )^{4} \sin \left (x \right )^{6}-457888 \cos \left (x \right )^{6} \sin \left (x \right )^{4}+111792 \cos \left (x \right )^{8} \sin \left (x \right )^{2}-2640 \cos \left (x \right )^{10}\right )}{307200}+\frac {11 x \left (-120960 \sin \left (x \right )^{10}+6894720 \cos \left (x \right )^{2} \sin \left (x \right )^{8}-38386560 \cos \left (x \right )^{4} \sin \left (x \right )^{6}+43920256 \cos \left (x \right )^{6} \sin \left (x \right )^{4}-10427520 \cos \left (x \right )^{8} \sin \left (x \right )^{2}+249984 \cos \left (x \right )^{10}\right )}{737280}+\frac {x \left (14999040 \sin \left (x \right )^{10}-761103360 \cos \left (x \right )^{2} \sin \left (x \right )^{8}+3930132480 \cos \left (x \right )^{4} \sin \left (x \right )^{6}-4282912768 \cos \left (x \right )^{6} \sin \left (x \right )^{4}+987497472 \cos \left (x \right )^{8} \sin \left (x \right )^{2}-23354880 \cos \left (x \right )^{10}\right )}{14745600}\) | \(310\) |
Input:
int(cos(x)^6*sin(x)^4,x,method=_RETURNVERBOSE)
Output:
3/256*x+1/5120*sin(10*x)+1/2048*sin(8*x)-1/1024*sin(6*x)-1/256*sin(4*x)+1/ 512*sin(2*x)
Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.66 \[ \int \cos ^6(x) \sin ^4(x) \, dx=\frac {1}{1280} \, {\left (128 \, \cos \left (x\right )^{9} - 176 \, \cos \left (x\right )^{7} + 8 \, \cos \left (x\right )^{5} + 10 \, \cos \left (x\right )^{3} + 15 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {3}{256} \, x \] Input:
integrate(cos(x)^6*sin(x)^4,x, algorithm="fricas")
Output:
1/1280*(128*cos(x)^9 - 176*cos(x)^7 + 8*cos(x)^5 + 10*cos(x)^3 + 15*cos(x) )*sin(x) + 3/256*x
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \cos ^6(x) \sin ^4(x) \, dx=\frac {3 x}{256} + \frac {\sin {\left (x \right )} \cos ^{9}{\left (x \right )}}{10} - \frac {11 \sin {\left (x \right )} \cos ^{7}{\left (x \right )}}{80} + \frac {\sin {\left (x \right )} \cos ^{5}{\left (x \right )}}{160} + \frac {\sin {\left (x \right )} \cos ^{3}{\left (x \right )}}{128} + \frac {3 \sin {\left (x \right )} \cos {\left (x \right )}}{256} \] Input:
integrate(cos(x)**6*sin(x)**4,x)
Output:
3*x/256 + sin(x)*cos(x)**9/10 - 11*sin(x)*cos(x)**7/80 + sin(x)*cos(x)**5/ 160 + sin(x)*cos(x)**3/128 + 3*sin(x)*cos(x)/256
Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.43 \[ \int \cos ^6(x) \sin ^4(x) \, dx=\frac {1}{320} \, \sin \left (2 \, x\right )^{5} + \frac {3}{256} \, x + \frac {1}{2048} \, \sin \left (8 \, x\right ) - \frac {1}{256} \, \sin \left (4 \, x\right ) \] Input:
integrate(cos(x)^6*sin(x)^4,x, algorithm="maxima")
Output:
1/320*sin(2*x)^5 + 3/256*x + 1/2048*sin(8*x) - 1/256*sin(4*x)
Time = 0.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.61 \[ \int \cos ^6(x) \sin ^4(x) \, dx=\frac {3}{256} \, x + \frac {1}{5120} \, \sin \left (10 \, x\right ) + \frac {1}{2048} \, \sin \left (8 \, x\right ) - \frac {1}{1024} \, \sin \left (6 \, x\right ) - \frac {1}{256} \, \sin \left (4 \, x\right ) + \frac {1}{512} \, \sin \left (2 \, x\right ) \] Input:
integrate(cos(x)^6*sin(x)^4,x, algorithm="giac")
Output:
3/256*x + 1/5120*sin(10*x) + 1/2048*sin(8*x) - 1/1024*sin(6*x) - 1/256*sin (4*x) + 1/512*sin(2*x)
Time = 0.02 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.68 \[ \int \cos ^6(x) \sin ^4(x) \, dx=\left (\frac {{\cos \left (x\right )}^5}{10}+\frac {{\cos \left (x\right )}^3}{16}+\frac {\cos \left (x\right )}{32}\right )\,{\sin \left (x\right )}^5+\frac {3\,x}{256}-\frac {\sin \left (2\,x\right )}{128}+\frac {\sin \left (4\,x\right )}{1024} \] Input:
int(cos(x)^6*sin(x)^4,x)
Output:
(3*x)/256 - sin(2*x)/128 + sin(4*x)/1024 + sin(x)^5*(cos(x)/32 + cos(x)^3/ 16 + cos(x)^5/10)
Time = 0.16 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.75 \[ \int \cos ^6(x) \sin ^4(x) \, dx=\frac {\cos \left (x \right ) \sin \left (x \right )^{9}}{10}-\frac {21 \cos \left (x \right ) \sin \left (x \right )^{7}}{80}+\frac {31 \cos \left (x \right ) \sin \left (x \right )^{5}}{160}-\frac {\cos \left (x \right ) \sin \left (x \right )^{3}}{128}-\frac {3 \cos \left (x \right ) \sin \left (x \right )}{256}+\frac {3 x}{256} \] Input:
int(cos(x)^6*sin(x)^4,x)
Output:
(128*cos(x)*sin(x)**9 - 336*cos(x)*sin(x)**7 + 248*cos(x)*sin(x)**5 - 10*c os(x)*sin(x)**3 - 15*cos(x)*sin(x) + 15*x)/1280