Integrand size = 10, antiderivative size = 46 \[ \int 4 \cos ^4(x) \sin ^4(x) \, dx=\frac {3 x}{32}+\frac {3}{32} \cos (x) \sin (x)+\frac {1}{16} \cos ^3(x) \sin (x)-\frac {1}{4} \cos ^5(x) \sin (x)-\frac {1}{2} \cos ^5(x) \sin ^3(x) \] Output:
3/32*x+3/32*cos(x)*sin(x)+1/16*cos(x)^3*sin(x)-1/4*cos(x)^5*sin(x)-1/2*cos (x)^5*sin(x)^3
Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.52 \[ \int 4 \cos ^4(x) \sin ^4(x) \, dx=4 \left (\frac {3 x}{128}-\frac {1}{128} \sin (4 x)+\frac {\sin (8 x)}{1024}\right ) \] Input:
Integrate[4*Cos[x]^4*Sin[x]^4,x]
Output:
4*((3*x)/128 - Sin[4*x]/128 + Sin[8*x]/1024)
Time = 0.35 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.37, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {27, 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.
\(\displaystyle \int 4 \sin ^4(x) \cos ^4(x) \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 4 \int \cos ^4(x) \sin ^4(x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 4 \int \cos (x)^4 \sin (x)^4dx\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle 4 \left (\frac {3}{8} \int \cos ^4(x) \sin ^2(x)dx-\frac {1}{8} \sin ^3(x) \cos ^5(x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 4 \left (\frac {3}{8} \int \cos (x)^4 \sin (x)^2dx-\frac {1}{8} \sin ^3(x) \cos ^5(x)\right )\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle 4 \left (\frac {3}{8} \left (\frac {1}{6} \int \cos ^4(x)dx-\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin ^3(x) \cos ^5(x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 4 \left (\frac {3}{8} \left (\frac {1}{6} \int \sin \left (x+\frac {\pi }{2}\right )^4dx-\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin ^3(x) \cos ^5(x)\right )\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle 4 \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {3}{4} \int \cos ^2(x)dx+\frac {1}{4} \sin (x) \cos ^3(x)\right )-\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin ^3(x) \cos ^5(x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 4 \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {3}{4} \int \sin \left (x+\frac {\pi }{2}\right )^2dx+\frac {1}{4} \sin (x) \cos ^3(x)\right )-\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin ^3(x) \cos ^5(x)\right )\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle 4 \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {1}{2} \sin (x) \cos (x)\right )+\frac {1}{4} \sin (x) \cos ^3(x)\right )-\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin ^3(x) \cos ^5(x)\right )\) |
\(\Big \downarrow \) 24 |
\(\displaystyle 4 \left (\frac {3}{8} \left (\frac {1}{6} \left (\frac {1}{4} \sin (x) \cos ^3(x)+\frac {3}{4} \left (\frac {x}{2}+\frac {1}{2} \sin (x) \cos (x)\right )\right )-\frac {1}{6} \sin (x) \cos ^5(x)\right )-\frac {1}{8} \sin ^3(x) \cos ^5(x)\right )\) |
Input:
Int[4*Cos[x]^4*Sin[x]^4,x]
Output:
4*(-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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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 = 2.19 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.37
method | result | size |
risch | \(\frac {3 x}{32}+\frac {\sin \left (8 x \right )}{256}-\frac {\sin \left (4 x \right )}{32}\) | \(17\) |
parallelrisch | \(\frac {3 x}{32}+\frac {\sin \left (8 x \right )}{256}-\frac {\sin \left (4 x \right )}{32}\) | \(17\) |
default | \(-\frac {\cos \left (x \right )^{5} \sin \left (x \right )^{3}}{2}-\frac {\cos \left (x \right )^{5} \sin \left (x \right )}{4}+\frac {\left (\cos \left (x \right )^{3}+\frac {3 \cos \left (x \right )}{2}\right ) \sin \left (x \right )}{16}+\frac {3 x}{32}\) | \(36\) |
orering | \(4 x \cos \left (x \right )^{4} \sin \left (x \right )^{4}+\frac {11 \cos \left (x \right )^{3} \sin \left (x \right )^{5}}{32}-\frac {11 \cos \left (x \right )^{5} \sin \left (x \right )^{3}}{32}+\frac {5 x \left (48 \sin \left (x \right )^{6} \cos \left (x \right )^{2}-160 \cos \left (x \right )^{4} \sin \left (x \right )^{4}+48 \cos \left (x \right )^{6} \sin \left (x \right )^{2}\right )}{64}+\frac {3 \sin \left (x \right )^{7} \cos \left (x \right )}{32}-\frac {3 \sin \left (x \right ) \cos \left (x \right )^{7}}{32}+\frac {x \left (-3456 \sin \left (x \right )^{6} \cos \left (x \right )^{2}+9280 \cos \left (x \right )^{4} \sin \left (x \right )^{4}+96 \sin \left (x \right )^{8}-3456 \cos \left (x \right )^{6} \sin \left (x \right )^{2}+96 \cos \left (x \right )^{8}\right )}{1024}\) | \(129\) |
norman | \(\frac {\frac {3 x}{32}-\frac {23 \tan \left (\frac {x}{2}\right )^{3}}{16}+\frac {333 \tan \left (\frac {x}{2}\right )^{5}}{16}-\frac {671 \tan \left (\frac {x}{2}\right )^{7}}{16}+\frac {671 \tan \left (\frac {x}{2}\right )^{9}}{16}-\frac {333 \tan \left (\frac {x}{2}\right )^{11}}{16}+\frac {23 \tan \left (\frac {x}{2}\right )^{13}}{16}+\frac {3 \tan \left (\frac {x}{2}\right )^{15}}{16}+\frac {21 x \tan \left (\frac {x}{2}\right )^{4}}{8}+\frac {21 x \tan \left (\frac {x}{2}\right )^{6}}{4}+\frac {105 x \tan \left (\frac {x}{2}\right )^{8}}{16}+\frac {21 x \tan \left (\frac {x}{2}\right )^{10}}{4}+\frac {21 x \tan \left (\frac {x}{2}\right )^{12}}{8}+\frac {3 x \tan \left (\frac {x}{2}\right )^{14}}{4}+\frac {3 x \tan \left (\frac {x}{2}\right )^{16}}{32}+\frac {3 \tan \left (\frac {x}{2}\right )^{2} x}{4}-\frac {3 \tan \left (\frac {x}{2}\right )}{16}}{\left (1+\tan \left (\frac {x}{2}\right )^{2}\right )^{8}}\) | \(150\) |
Input:
int(4*cos(x)^4*sin(x)^4,x,method=_RETURNVERBOSE)
Output:
3/32*x+1/256*sin(8*x)-1/32*sin(4*x)
Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.67 \[ \int 4 \cos ^4(x) \sin ^4(x) \, dx=\frac {1}{32} \, {\left (16 \, \cos \left (x\right )^{7} - 24 \, \cos \left (x\right )^{5} + 2 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {3}{32} \, x \] Input:
integrate(4*cos(x)^4*sin(x)^4,x, algorithm="fricas")
Output:
1/32*(16*cos(x)^7 - 24*cos(x)^5 + 2*cos(x)^3 + 3*cos(x))*sin(x) + 3/32*x
Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.67 \[ \int 4 \cos ^4(x) \sin ^4(x) \, dx=\frac {3 x}{32} - \frac {\sin ^{3}{\left (2 x \right )} \cos {\left (2 x \right )}}{32} - \frac {3 \sin {\left (2 x \right )} \cos {\left (2 x \right )}}{64} \] Input:
integrate(4*cos(x)**4*sin(x)**4,x)
Output:
3*x/32 - sin(2*x)**3*cos(2*x)/32 - 3*sin(2*x)*cos(2*x)/64
Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.35 \[ \int 4 \cos ^4(x) \sin ^4(x) \, dx=\frac {3}{32} \, x + \frac {1}{256} \, \sin \left (8 \, x\right ) - \frac {1}{32} \, \sin \left (4 \, x\right ) \] Input:
integrate(4*cos(x)^4*sin(x)^4,x, algorithm="maxima")
Output:
3/32*x + 1/256*sin(8*x) - 1/32*sin(4*x)
Time = 0.13 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.35 \[ \int 4 \cos ^4(x) \sin ^4(x) \, dx=\frac {3}{32} \, x + \frac {1}{256} \, \sin \left (8 \, x\right ) - \frac {1}{32} \, \sin \left (4 \, x\right ) \] Input:
integrate(4*cos(x)^4*sin(x)^4,x, algorithm="giac")
Output:
3/32*x + 1/256*sin(8*x) - 1/32*sin(4*x)
Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.72 \[ \int 4 \cos ^4(x) \sin ^4(x) \, dx=\frac {3\,x}{32}-\frac {\sin \left (2\,x\right )}{16}+\frac {\sin \left (4\,x\right )}{128}+4\,{\sin \left (x\right )}^5\,\left (\frac {{\cos \left (x\right )}^3}{8}+\frac {\cos \left (x\right )}{16}\right ) \] Input:
int(4*cos(x)^4*sin(x)^4,x)
Output:
(3*x)/32 - sin(2*x)/16 + sin(4*x)/128 + 4*sin(x)^5*(cos(x)/16 + cos(x)^3/8 )
Time = 0.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.74 \[ \int 4 \cos ^4(x) \sin ^4(x) \, dx=-\frac {\cos \left (x \right ) \sin \left (x \right )^{7}}{2}+\frac {3 \cos \left (x \right ) \sin \left (x \right )^{5}}{4}-\frac {\cos \left (x \right ) \sin \left (x \right )^{3}}{16}-\frac {3 \cos \left (x \right ) \sin \left (x \right )}{32}+\frac {3 x}{32} \] Input:
int(4*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)/32