Integrand size = 9, antiderivative size = 46 \[ \int \cos ^4(x) \sin ^4(x) \, dx=\frac {3 x}{128}+\frac {3}{128} \cos (x) \sin (x)+\frac {1}{64} \cos ^3(x) \sin (x)-\frac {1}{16} \cos ^5(x) \sin (x)-\frac {1}{8} \cos ^5(x) \sin ^3(x) \] Output:
3/128*x+3/128*cos(x)*sin(x)+1/64*cos(x)^3*sin(x)-1/16*cos(x)^5*sin(x)-1/8* cos(x)^5*sin(x)^3
Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.48 \[ \int \cos ^4(x) \sin ^4(x) \, dx=\frac {3 x}{128}-\frac {1}{128} \sin (4 x)+\frac {\sin (8 x)}{1024} \] Input:
Integrate[Cos[x]^4*Sin[x]^4,x]
Output:
(3*x)/128 - Sin[4*x]/128 + Sin[8*x]/1024
Time = 0.33 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.33, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3048, 3042, 3048, 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.
\(\displaystyle \int \sin ^4(x) \cos ^4(x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (x)^4 \cos (x)^4dx\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle \frac {3}{8} \int \cos ^4(x) \sin ^2(x)dx-\frac {1}{8} \sin ^3(x) \cos ^5(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{8} \int \cos (x)^4 \sin (x)^2dx-\frac {1}{8} \sin ^3(x) \cos ^5(x)\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle \frac {3}{8} \left (\frac {1}{6} \int \cos ^4(x)dx-\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin ^3(x) \cos ^5(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{8} \left (\frac {1}{6} \int \sin \left (x+\frac {\pi }{2}\right )^4dx-\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin ^3(x) \cos ^5(x)\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {3}{8} \left (\frac {1}{6} \left (\frac {3}{4} \int \cos ^2(x)dx+\frac {1}{4} \sin (x) \cos ^3(x)\right )-\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin ^3(x) \cos ^5(x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {3}{8} \left (\frac {1}{6} \left (\frac {3}{4} \int \sin \left (x+\frac {\pi }{2}\right )^2dx+\frac {1}{4} \sin (x) \cos ^3(x)\right )-\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin ^3(x) \cos ^5(x)\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle \frac {3}{8} \left (\frac {1}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sin (x) \cos (x)\right )+\frac {1}{4} \sin (x) \cos ^3(x)\right )-\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin ^3(x) \cos ^5(x)\) |
\(\Big \downarrow \) 24 |
\(\displaystyle \frac {3}{8} \left (\frac {1}{6} \left (\frac {1}{4} \sin (x) \cos ^3(x)+\frac {3}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )\right )-\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin ^3(x) \cos ^5(x)\) |
Input:
Int[Cos[x]^4*Sin[x]^4,x]
Output:
-1/8*(Cos[x]^5*Sin[x]^3) + (3*(-1/6*(Cos[x]^5*Sin[x]) + ((Cos[x]^3*Sin[x]) /4 + (3*(x/2 + (Cos[x]*Sin[x])/2))/4)/6))/8
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 = 2.48 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.37
method | result | size |
risch | \(\frac {3 x}{128}+\frac {\sin \left (8 x \right )}{1024}-\frac {\sin \left (4 x \right )}{128}\) | \(17\) |
parallelrisch | \(\frac {3 x}{128}+\frac {\sin \left (8 x \right )}{1024}-\frac {\sin \left (4 x \right )}{128}\) | \(17\) |
default | \(-\frac {\cos \left (x \right )^{5} \sin \left (x \right )^{3}}{8}-\frac {\sin \left (x \right ) \cos \left (x \right )^{5}}{16}+\frac {\left (\cos \left (x \right )^{3}+\frac {3 \cos \left (x \right )}{2}\right ) \sin \left (x \right )}{64}+\frac {3 x}{128}\) | \(36\) |
orering | \(x \cos \left (x \right )^{4} \sin \left (x \right )^{4}+\frac {11 \cos \left (x \right )^{3} \sin \left (x \right )^{5}}{128}-\frac {11 \cos \left (x \right )^{5} \sin \left (x \right )^{3}}{128}+\frac {5 x \left (12 \cos \left (x \right )^{2} \sin \left (x \right )^{6}-40 \cos \left (x \right )^{4} \sin \left (x \right )^{4}+12 \cos \left (x \right )^{6} \sin \left (x \right )^{2}\right )}{64}+\frac {3 \cos \left (x \right ) \sin \left (x \right )^{7}}{128}-\frac {3 \sin \left (x \right ) \cos \left (x \right )^{7}}{128}+\frac {x \left (24 \sin \left (x \right )^{8}-864 \cos \left (x \right )^{2} \sin \left (x \right )^{6}+2320 \cos \left (x \right )^{4} \sin \left (x \right )^{4}-864 \cos \left (x \right )^{6} \sin \left (x \right )^{2}+24 \cos \left (x \right )^{8}\right )}{1024}\) | \(128\) |
norman | \(\frac {\frac {3 x}{128}-\frac {23 \tan \left (\frac {x}{2}\right )^{3}}{64}+\frac {333 \tan \left (\frac {x}{2}\right )^{5}}{64}-\frac {671 \tan \left (\frac {x}{2}\right )^{7}}{64}+\frac {671 \tan \left (\frac {x}{2}\right )^{9}}{64}-\frac {333 \tan \left (\frac {x}{2}\right )^{11}}{64}+\frac {23 \tan \left (\frac {x}{2}\right )^{13}}{64}+\frac {3 \tan \left (\frac {x}{2}\right )^{15}}{64}+\frac {3 x \tan \left (\frac {x}{2}\right )^{2}}{16}+\frac {21 x \tan \left (\frac {x}{2}\right )^{4}}{32}+\frac {21 x \tan \left (\frac {x}{2}\right )^{6}}{16}+\frac {105 x \tan \left (\frac {x}{2}\right )^{8}}{64}+\frac {21 x \tan \left (\frac {x}{2}\right )^{10}}{16}+\frac {21 x \tan \left (\frac {x}{2}\right )^{12}}{32}+\frac {3 x \tan \left (\frac {x}{2}\right )^{14}}{16}+\frac {3 x \tan \left (\frac {x}{2}\right )^{16}}{128}-\frac {3 \tan \left (\frac {x}{2}\right )}{64}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{8}}\) | \(150\) |
Input:
int(cos(x)^4*sin(x)^4,x,method=_RETURNVERBOSE)
Output:
3/128*x+1/1024*sin(8*x)-1/128*sin(4*x)
Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.67 \[ \int \cos ^4(x) \sin ^4(x) \, dx=\frac {1}{128} \, {\left (16 \, \cos \left (x\right )^{7} - 24 \, \cos \left (x\right )^{5} + 2 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {3}{128} \, x \] Input:
integrate(cos(x)^4*sin(x)^4,x, algorithm="fricas")
Output:
1/128*(16*cos(x)^7 - 24*cos(x)^5 + 2*cos(x)^3 + 3*cos(x))*sin(x) + 3/128*x
Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.67 \[ \int \cos ^4(x) \sin ^4(x) \, dx=\frac {3 x}{128} - \frac {\sin ^{3}{\left (2 x \right )} \cos {\left (2 x \right )}}{128} - \frac {3 \sin {\left (2 x \right )} \cos {\left (2 x \right )}}{256} \] Input:
integrate(cos(x)**4*sin(x)**4,x)
Output:
3*x/128 - sin(2*x)**3*cos(2*x)/128 - 3*sin(2*x)*cos(2*x)/256
Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.35 \[ \int \cos ^4(x) \sin ^4(x) \, dx=\frac {3}{128} \, x + \frac {1}{1024} \, \sin \left (8 \, x\right ) - \frac {1}{128} \, \sin \left (4 \, x\right ) \] Input:
integrate(cos(x)^4*sin(x)^4,x, algorithm="maxima")
Output:
3/128*x + 1/1024*sin(8*x) - 1/128*sin(4*x)
Time = 0.12 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.35 \[ \int \cos ^4(x) \sin ^4(x) \, dx=\frac {3}{128} \, x + \frac {1}{1024} \, \sin \left (8 \, x\right ) - \frac {1}{128} \, \sin \left (4 \, x\right ) \] Input:
integrate(cos(x)^4*sin(x)^4,x, algorithm="giac")
Output:
3/128*x + 1/1024*sin(8*x) - 1/128*sin(4*x)
Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.70 \[ \int \cos ^4(x) \sin ^4(x) \, dx=\left (\frac {{\cos \left (x\right )}^3}{8}+\frac {\cos \left (x\right )}{16}\right )\,{\sin \left (x\right )}^5+\frac {3\,x}{128}-\frac {\sin \left (2\,x\right )}{64}+\frac {\sin \left (4\,x\right )}{512} \] Input:
int(cos(x)^4*sin(x)^4,x)
Output:
(3*x)/128 - sin(2*x)/64 + sin(4*x)/512 + sin(x)^5*(cos(x)/16 + cos(x)^3/8)
Time = 0.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.74 \[ \int \cos ^4(x) \sin ^4(x) \, dx=-\frac {\cos \left (x \right ) \sin \left (x \right )^{7}}{8}+\frac {3 \cos \left (x \right ) \sin \left (x \right )^{5}}{16}-\frac {\cos \left (x \right ) \sin \left (x \right )^{3}}{64}-\frac {3 \cos \left (x \right ) \sin \left (x \right )}{128}+\frac {3 x}{128} \] Input:
int(cos(x)^4*sin(x)^4,x)
Output:
( - 16*cos(x)*sin(x)**7 + 24*cos(x)*sin(x)**5 - 2*cos(x)*sin(x)**3 - 3*cos (x)*sin(x) + 3*x)/128