Integrand size = 29, antiderivative size = 132 \[ \int \cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{16} (6 A+5 C) x+\frac {B \sin (c+d x)}{d}+\frac {(6 A+5 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {(6 A+5 C) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {C \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 B \sin ^3(c+d x)}{3 d}+\frac {B \sin ^5(c+d x)}{5 d} \] Output:
1/16*(6*A+5*C)*x+B*sin(d*x+c)/d+1/16*(6*A+5*C)*cos(d*x+c)*sin(d*x+c)/d+1/2 4*(6*A+5*C)*cos(d*x+c)^3*sin(d*x+c)/d+1/6*C*cos(d*x+c)^5*sin(d*x+c)/d-2/3* B*sin(d*x+c)^3/d+1/5*B*sin(d*x+c)^5/d
Time = 0.30 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.77 \[ \int \cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {960 B \sin (c+d x)-640 B \sin ^3(c+d x)+192 B \sin ^5(c+d x)+5 (72 A c+60 c C+72 A d x+60 C d x+(48 A+45 C) \sin (2 (c+d x))+(6 A+9 C) \sin (4 (c+d x))+C \sin (6 (c+d x)))}{960 d} \] Input:
Integrate[Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
Output:
(960*B*Sin[c + d*x] - 640*B*Sin[c + d*x]^3 + 192*B*Sin[c + d*x]^5 + 5*(72* A*c + 60*c*C + 72*A*d*x + 60*C*d*x + (48*A + 45*C)*Sin[2*(c + d*x)] + (6*A + 9*C)*Sin[4*(c + d*x)] + C*Sin[6*(c + d*x)]))/(960*d)
Time = 0.56 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.95, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {3042, 3502, 3042, 3227, 3042, 3113, 2009, 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.
| \(\displaystyle \int \cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {1}{6} \int \cos ^4(c+d x) (6 A+5 C+6 B \cos (c+d x))dx+\frac {C \sin (c+d x) \cos ^5(c+d x)}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4 \left (6 A+5 C+6 B \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {C \sin (c+d x) \cos ^5(c+d x)}{6 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{6} \left ((6 A+5 C) \int \cos ^4(c+d x)dx+6 B \int \cos ^5(c+d x)dx\right )+\frac {C \sin (c+d x) \cos ^5(c+d x)}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left ((6 A+5 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+6 B \int \sin \left (c+d x+\frac {\pi }{2}\right )^5dx\right )+\frac {C \sin (c+d x) \cos ^5(c+d x)}{6 d}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {1}{6} \left ((6 A+5 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {6 B \int \left (\sin ^4(c+d x)-2 \sin ^2(c+d x)+1\right )d(-\sin (c+d x))}{d}\right )+\frac {C \sin (c+d x) \cos ^5(c+d x)}{6 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{6} \left ((6 A+5 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {6 B \left (-\frac {1}{5} \sin ^5(c+d x)+\frac {2}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {C \sin (c+d x) \cos ^5(c+d x)}{6 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{6} \left ((6 A+5 C) \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {6 B \left (-\frac {1}{5} \sin ^5(c+d x)+\frac {2}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {C \sin (c+d x) \cos ^5(c+d x)}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left ((6 A+5 C) \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {6 B \left (-\frac {1}{5} \sin ^5(c+d x)+\frac {2}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {C \sin (c+d x) \cos ^5(c+d x)}{6 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{6} \left ((6 A+5 C) \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {6 B \left (-\frac {1}{5} \sin ^5(c+d x)+\frac {2}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {C \sin (c+d x) \cos ^5(c+d x)}{6 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{6} \left ((6 A+5 C) \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )-\frac {6 B \left (-\frac {1}{5} \sin ^5(c+d x)+\frac {2}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {C \sin (c+d x) \cos ^5(c+d x)}{6 d}\) |
Input:
Int[Cos[c + d*x]^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2),x]
Output:
(C*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) + ((-6*B*(-Sin[c + d*x] + (2*Sin[c + d*x]^3)/3 - Sin[c + d*x]^5/5))/d + (6*A + 5*C)*((Cos[c + d*x]^3*Sin[c + d *x])/(4*d) + (3*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4))/6
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
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]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -1]
Time = 9.36 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.72
| method | result | size |
| parallelrisch | \(\frac {\left (240 A +225 C \right ) \sin \left (2 d x +2 c \right )+\left (30 A +45 C \right ) \sin \left (4 d x +4 c \right )+100 B \sin \left (3 d x +3 c \right )+12 B \sin \left (5 d x +5 c \right )+5 \sin \left (6 d x +6 c \right ) C +600 B \sin \left (d x +c \right )+360 x \left (A +\frac {5 C}{6}\right ) d}{960 d}\) | \(95\) |
| derivativedivides | \(\frac {C \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(115\) |
| default | \(\frac {C \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(115\) |
| parts | \(\frac {A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {B \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {C \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(120\) |
| risch | \(\frac {3 x A}{8}+\frac {5 C x}{16}+\frac {5 B \sin \left (d x +c \right )}{8 d}+\frac {C \sin \left (6 d x +6 c \right )}{192 d}+\frac {B \sin \left (5 d x +5 c \right )}{80 d}+\frac {\sin \left (4 d x +4 c \right ) A}{32 d}+\frac {3 \sin \left (4 d x +4 c \right ) C}{64 d}+\frac {5 B \sin \left (3 d x +3 c \right )}{48 d}+\frac {\sin \left (2 d x +2 c \right ) A}{4 d}+\frac {15 \sin \left (2 d x +2 c \right ) C}{64 d}\) | \(127\) |
| norman | \(\frac {\left (\frac {3 A}{8}+\frac {5 C}{16}\right ) x +\left (\frac {3 A}{8}+\frac {5 C}{16}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {9 A}{4}+\frac {15 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {9 A}{4}+\frac {15 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {15 A}{2}+\frac {25 C}{4}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {45 A}{8}+\frac {75 C}{16}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {45 A}{8}+\frac {75 C}{16}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-\frac {\left (10 A -208 B +75 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{20 d}-\frac {\left (10 A -16 B +11 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {\left (10 A +16 B +11 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {\left (10 A +208 B +75 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{20 d}-\frac {\left (42 A -112 B -5 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}+\frac {\left (42 A +112 B -5 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}\) | \(301\) |
| orering | \(\text {Expression too large to display}\) | \(8957\) |
Input:
int(cos(d*x+c)^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x,method=_RETURNVERBOSE)
Output:
1/960*((240*A+225*C)*sin(2*d*x+2*c)+(30*A+45*C)*sin(4*d*x+4*c)+100*B*sin(3 *d*x+3*c)+12*B*sin(5*d*x+5*c)+5*sin(6*d*x+6*c)*C+600*B*sin(d*x+c)+360*x*(A +5/6*C)*d)/d
Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.70 \[ \int \cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (6 \, A + 5 \, C\right )} d x + {\left (40 \, C \cos \left (d x + c\right )^{5} + 48 \, B \cos \left (d x + c\right )^{4} + 10 \, {\left (6 \, A + 5 \, C\right )} \cos \left (d x + c\right )^{3} + 64 \, B \cos \left (d x + c\right )^{2} + 15 \, {\left (6 \, A + 5 \, C\right )} \cos \left (d x + c\right ) + 128 \, B\right )} \sin \left (d x + c\right )}{240 \, d} \] Input:
integrate(cos(d*x+c)^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="frica s")
Output:
1/240*(15*(6*A + 5*C)*d*x + (40*C*cos(d*x + c)^5 + 48*B*cos(d*x + c)^4 + 1 0*(6*A + 5*C)*cos(d*x + c)^3 + 64*B*cos(d*x + c)^2 + 15*(6*A + 5*C)*cos(d* x + c) + 128*B)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (121) = 242\).
Time = 0.34 (sec) , antiderivative size = 321, normalized size of antiderivative = 2.43 \[ \int \cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\begin {cases} \frac {3 A x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 A x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 A x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 A \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 A \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} + \frac {8 B \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 B \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {B \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 C x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 C x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 C x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 C x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 C \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 C \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 C \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (A + B \cos {\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**4*(A+B*cos(d*x+c)+C*cos(d*x+c)**2),x)
Output:
Piecewise((3*A*x*sin(c + d*x)**4/8 + 3*A*x*sin(c + d*x)**2*cos(c + d*x)**2 /4 + 3*A*x*cos(c + d*x)**4/8 + 3*A*sin(c + d*x)**3*cos(c + d*x)/(8*d) + 5* A*sin(c + d*x)*cos(c + d*x)**3/(8*d) + 8*B*sin(c + d*x)**5/(15*d) + 4*B*si n(c + d*x)**3*cos(c + d*x)**2/(3*d) + B*sin(c + d*x)*cos(c + d*x)**4/d + 5 *C*x*sin(c + d*x)**6/16 + 15*C*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 15*C *x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 5*C*x*cos(c + d*x)**6/16 + 5*C*sin (c + d*x)**5*cos(c + d*x)/(16*d) + 5*C*sin(c + d*x)**3*cos(c + d*x)**3/(6* d) + 11*C*sin(c + d*x)*cos(c + d*x)**5/(16*d), Ne(d, 0)), (x*(A + B*cos(c) + C*cos(c)**2)*cos(c)**4, True))
Time = 0.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.87 \[ \int \cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A + 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C}{960 \, d} \] Input:
integrate(cos(d*x+c)^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="maxim a")
Output:
1/960*(30*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A + 64*( 3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B - 5*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*C)/d
Time = 0.13 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.83 \[ \int \cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{16} \, {\left (6 \, A + 5 \, C\right )} x + \frac {C \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {B \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {{\left (2 \, A + 3 \, C\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {5 \, B \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (16 \, A + 15 \, C\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {5 \, B \sin \left (d x + c\right )}{8 \, d} \] Input:
integrate(cos(d*x+c)^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, algorithm="giac" )
Output:
1/16*(6*A + 5*C)*x + 1/192*C*sin(6*d*x + 6*c)/d + 1/80*B*sin(5*d*x + 5*c)/ d + 1/64*(2*A + 3*C)*sin(4*d*x + 4*c)/d + 5/48*B*sin(3*d*x + 3*c)/d + 1/64 *(16*A + 15*C)*sin(2*d*x + 2*c)/d + 5/8*B*sin(d*x + c)/d
Time = 0.33 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.95 \[ \int \cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {3\,A\,x}{8}+\frac {5\,C\,x}{16}+\frac {A\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {A\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {5\,B\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {B\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {15\,C\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {3\,C\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {C\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {5\,B\,\sin \left (c+d\,x\right )}{8\,d} \] Input:
int(cos(c + d*x)^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2),x)
Output:
(3*A*x)/8 + (5*C*x)/16 + (A*sin(2*c + 2*d*x))/(4*d) + (A*sin(4*c + 4*d*x)) /(32*d) + (5*B*sin(3*c + 3*d*x))/(48*d) + (B*sin(5*c + 5*d*x))/(80*d) + (1 5*C*sin(2*c + 2*d*x))/(64*d) + (3*C*sin(4*c + 4*d*x))/(64*d) + (C*sin(6*c + 6*d*x))/(192*d) + (5*B*sin(c + d*x))/(8*d)
Time = 0.16 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.97 \[ \int \cos ^4(c+d x) \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} c -60 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a -130 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} c +150 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +165 \cos \left (d x +c \right ) \sin \left (d x +c \right ) c +48 \sin \left (d x +c \right )^{5} b -160 \sin \left (d x +c \right )^{3} b +240 \sin \left (d x +c \right ) b +90 a d x +75 c d x}{240 d} \] Input:
int(cos(d*x+c)^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)
Output:
(40*cos(c + d*x)*sin(c + d*x)**5*c - 60*cos(c + d*x)*sin(c + d*x)**3*a - 1 30*cos(c + d*x)*sin(c + d*x)**3*c + 150*cos(c + d*x)*sin(c + d*x)*a + 165* cos(c + d*x)*sin(c + d*x)*c + 48*sin(c + d*x)**5*b - 160*sin(c + d*x)**3*b + 240*sin(c + d*x)*b + 90*a*d*x + 75*c*d*x)/(240*d)