Integrand size = 9, antiderivative size = 38 \[ \int \cos ^4(x) \cos (4 x) \, dx=\frac {x}{16}+\frac {1}{8} \sin (2 x)+\frac {3}{32} \sin (4 x)+\frac {1}{24} \sin (6 x)+\frac {1}{128} \sin (8 x) \] Output:
1/16*x+1/8*sin(2*x)+3/32*sin(4*x)+1/24*sin(6*x)+1/128*sin(8*x)
Time = 0.01 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00 \[ \int \cos ^4(x) \cos (4 x) \, dx=\frac {x}{16}+\frac {1}{8} \sin (2 x)+\frac {3}{32} \sin (4 x)+\frac {1}{24} \sin (6 x)+\frac {1}{128} \sin (8 x) \] Input:
Integrate[Cos[x]^4*Cos[4*x],x]
Output:
x/16 + Sin[2*x]/8 + (3*Sin[4*x])/32 + Sin[6*x]/24 + Sin[8*x]/128
Time = 0.20 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 4854, 2009}
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.
\(\displaystyle \int \cos ^4(x) \cos (4 x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cos (x)^4 \cos (4 x)dx\) |
\(\Big \downarrow \) 4854 |
\(\displaystyle \int \left (\frac {1}{4} \cos (2 x)+\frac {3}{8} \cos (4 x)+\frac {1}{4} \cos (6 x)+\frac {1}{16} \cos (8 x)+\frac {1}{16}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {x}{16}+\frac {1}{8} \sin (2 x)+\frac {3}{32} \sin (4 x)+\frac {1}{24} \sin (6 x)+\frac {1}{128} \sin (8 x)\) |
Input:
Int[Cos[x]^4*Cos[4*x],x]
Output:
x/16 + Sin[2*x]/8 + (3*Sin[4*x])/32 + Sin[6*x]/24 + Sin[8*x]/128
Int[(F_)[(a_.) + (b_.)*(x_)]^(p_.)*(G_)[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol ] :> Int[ExpandTrigReduce[ActivateTrig[F[a + b*x]^p*G[c + d*x]^q], x], x] / ; FreeQ[{a, b, c, d}, x] && (EqQ[F, sin] || EqQ[F, cos]) && (EqQ[G, sin] || EqQ[G, cos]) && IGtQ[p, 0] && IGtQ[q, 0]
Time = 2.03 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76
method | result | size |
default | \(\frac {x}{16}+\frac {\sin \left (2 x \right )}{8}+\frac {3 \sin \left (4 x \right )}{32}+\frac {\sin \left (6 x \right )}{24}+\frac {\sin \left (8 x \right )}{128}\) | \(29\) |
risch | \(\frac {x}{16}+\frac {\sin \left (2 x \right )}{8}+\frac {3 \sin \left (4 x \right )}{32}+\frac {\sin \left (6 x \right )}{24}+\frac {\sin \left (8 x \right )}{128}\) | \(29\) |
parallelrisch | \(\frac {x}{16}+\frac {\sin \left (2 x \right )}{8}+\frac {3 \sin \left (4 x \right )}{32}+\frac {\sin \left (6 x \right )}{24}+\frac {\sin \left (8 x \right )}{128}\) | \(29\) |
orering | \(x \cos \left (x \right )^{4} \cos \left (4 x \right )-\frac {13 \cos \left (x \right )^{3} \cos \left (4 x \right ) \sin \left (x \right )}{96}+\frac {103 \cos \left (x \right )^{4} \sin \left (4 x \right )}{384}+\frac {205 x \left (12 \cos \left (x \right )^{2} \sin \left (x \right )^{2} \cos \left (4 x \right )+32 \cos \left (x \right )^{3} \sin \left (x \right ) \sin \left (4 x \right )-20 \cos \left (x \right )^{4} \cos \left (4 x \right )\right )}{576}-\frac {19 \cos \left (x \right ) \sin \left (x \right )^{3} \cos \left (4 x \right )}{96}+\frac {9 \cos \left (x \right )^{2} \sin \left (x \right )^{2} \sin \left (4 x \right )}{64}+\frac {91 x \left (24 \sin \left (x \right )^{4} \cos \left (4 x \right )-1344 \cos \left (x \right )^{2} \sin \left (x \right )^{2} \cos \left (4 x \right )+384 \cos \left (x \right ) \sin \left (x \right )^{3} \sin \left (4 x \right )-1664 \cos \left (x \right )^{3} \sin \left (x \right ) \sin \left (4 x \right )+680 \cos \left (x \right )^{4} \cos \left (4 x \right )\right )}{3072}-\frac {25 \sin \left (x \right )^{4} \sin \left (4 x \right )}{384}+\frac {5 x \left (-6240 \sin \left (x \right )^{4} \cos \left (4 x \right )+95232 \cos \left (x \right )^{2} \sin \left (x \right )^{2} \cos \left (4 x \right )-42240 \cos \left (x \right ) \sin \left (x \right )^{3} \sin \left (4 x \right )+88832 \cos \left (x \right )^{3} \sin \left (x \right ) \sin \left (4 x \right )-29600 \cos \left (x \right )^{4} \cos \left (4 x \right )\right )}{6144}+\frac {x \left (653184 \sin \left (x \right )^{4} \cos \left (4 x \right )-6242304 \cos \left (x \right )^{2} \sin \left (x \right )^{2} \cos \left (4 x \right )+3354624 \cos \left (x \right ) \sin \left (x \right )^{3} \sin \left (4 x \right )-5033984 \cos \left (x \right )^{3} \sin \left (x \right ) \sin \left (4 x \right )+1493120 \cos \left (x \right )^{4} \cos \left (4 x \right )\right )}{147456}\) | \(296\) |
Input:
int(cos(x)^4*cos(4*x),x,method=_RETURNVERBOSE)
Output:
1/16*x+1/8*sin(2*x)+3/32*sin(4*x)+1/24*sin(6*x)+1/128*sin(8*x)
Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.82 \[ \int \cos ^4(x) \cos (4 x) \, dx=\frac {1}{48} \, {\left (48 \, \cos \left (x\right )^{7} - 8 \, \cos \left (x\right )^{5} + 2 \, \cos \left (x\right )^{3} + 3 \, \cos \left (x\right )\right )} \sin \left (x\right ) + \frac {1}{16} \, x \] Input:
integrate(cos(x)^4*cos(4*x),x, algorithm="fricas")
Output:
1/48*(48*cos(x)^7 - 8*cos(x)^5 + 2*cos(x)^3 + 3*cos(x))*sin(x) + 1/16*x
Leaf count of result is larger than twice the leaf count of optimal. 139 vs. \(2 (31) = 62\).
Time = 1.24 (sec) , antiderivative size = 139, normalized size of antiderivative = 3.66 \[ \int \cos ^4(x) \cos (4 x) \, dx=\frac {x \sin ^{4}{\left (x \right )} \cos {\left (4 x \right )}}{16} - \frac {x \sin ^{3}{\left (x \right )} \sin {\left (4 x \right )} \cos {\left (x \right )}}{4} - \frac {3 x \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )} \cos {\left (4 x \right )}}{8} + \frac {x \sin {\left (x \right )} \sin {\left (4 x \right )} \cos ^{3}{\left (x \right )}}{4} + \frac {x \cos ^{4}{\left (x \right )} \cos {\left (4 x \right )}}{16} - \frac {\sin ^{4}{\left (x \right )} \sin {\left (4 x \right )}}{24} - \frac {5 \sin ^{3}{\left (x \right )} \cos {\left (x \right )} \cos {\left (4 x \right )}}{48} - \frac {11 \sin {\left (x \right )} \cos ^{3}{\left (x \right )} \cos {\left (4 x \right )}}{48} + \frac {7 \sin {\left (4 x \right )} \cos ^{4}{\left (x \right )}}{24} \] Input:
integrate(cos(x)**4*cos(4*x),x)
Output:
x*sin(x)**4*cos(4*x)/16 - x*sin(x)**3*sin(4*x)*cos(x)/4 - 3*x*sin(x)**2*co s(x)**2*cos(4*x)/8 + x*sin(x)*sin(4*x)*cos(x)**3/4 + x*cos(x)**4*cos(4*x)/ 16 - sin(x)**4*sin(4*x)/24 - 5*sin(x)**3*cos(x)*cos(4*x)/48 - 11*sin(x)*co s(x)**3*cos(4*x)/48 + 7*sin(4*x)*cos(x)**4/24
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79 \[ \int \cos ^4(x) \cos (4 x) \, dx=-\frac {1}{6} \, \sin \left (2 \, x\right )^{3} + \frac {1}{16} \, x + \frac {1}{128} \, \sin \left (8 \, x\right ) + \frac {3}{32} \, \sin \left (4 \, x\right ) + \frac {1}{4} \, \sin \left (2 \, x\right ) \] Input:
integrate(cos(x)^4*cos(4*x),x, algorithm="maxima")
Output:
-1/6*sin(2*x)^3 + 1/16*x + 1/128*sin(8*x) + 3/32*sin(4*x) + 1/4*sin(2*x)
Time = 0.11 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.74 \[ \int \cos ^4(x) \cos (4 x) \, dx=\frac {1}{16} \, x + \frac {1}{128} \, \sin \left (8 \, x\right ) + \frac {1}{24} \, \sin \left (6 \, x\right ) + \frac {3}{32} \, \sin \left (4 \, x\right ) + \frac {1}{8} \, \sin \left (2 \, x\right ) \] Input:
integrate(cos(x)^4*cos(4*x),x, algorithm="giac")
Output:
1/16*x + 1/128*sin(8*x) + 1/24*sin(6*x) + 3/32*sin(4*x) + 1/8*sin(2*x)
Time = 0.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.95 \[ \int \cos ^4(x) \cos (4 x) \, dx=\frac {x}{16}+\frac {\frac {{\mathrm {tan}\left (x\right )}^7}{16}+\frac {11\,{\mathrm {tan}\left (x\right )}^5}{48}+\frac {5\,{\mathrm {tan}\left (x\right )}^3}{48}+\frac {15\,\mathrm {tan}\left (x\right )}{16}}{{\left ({\mathrm {tan}\left (x\right )}^2+1\right )}^4} \] Input:
int(cos(4*x)*cos(x)^4,x)
Output:
x/16 + ((15*tan(x))/16 + (5*tan(x)^3)/48 + (11*tan(x)^5)/48 + tan(x)^7/16) /(tan(x)^2 + 1)^4
Time = 0.16 (sec) , antiderivative size = 102, normalized size of antiderivative = 2.68 \[ \int \cos ^4(x) \cos (4 x) \, dx=-\frac {5 \cos \left (4 x \right ) \cos \left (x \right ) \sin \left (x \right )^{3}}{24}-\frac {\cos \left (4 x \right ) \cos \left (x \right ) \sin \left (x \right )}{16}+\frac {\cos \left (4 x \right ) \sin \left (x \right )^{4} x}{2}-\frac {\cos \left (4 x \right ) \sin \left (x \right )^{2} x}{2}+\frac {\cos \left (4 x \right ) x}{16}-\frac {\cos \left (x \right ) \sin \left (4 x \right ) \sin \left (x \right )^{3} x}{2}+\frac {\cos \left (x \right ) \sin \left (4 x \right ) \sin \left (x \right ) x}{4}-\frac {\sin \left (4 x \right ) \sin \left (x \right )^{4}}{12}-\frac {\sin \left (4 x \right ) \sin \left (x \right )^{2}}{4}+\frac {\sin \left (4 x \right )}{4} \] Input:
int(cos(x)^4*cos(4*x),x)
Output:
( - 10*cos(4*x)*cos(x)*sin(x)**3 - 3*cos(4*x)*cos(x)*sin(x) + 24*cos(4*x)* sin(x)**4*x - 24*cos(4*x)*sin(x)**2*x + 3*cos(4*x)*x - 24*cos(x)*sin(4*x)* sin(x)**3*x + 12*cos(x)*sin(4*x)*sin(x)*x - 4*sin(4*x)*sin(x)**4 - 12*sin( 4*x)*sin(x)**2 + 12*sin(4*x))/48