\(\int \cos (c+d x) (a+b \cos (c+d x))^4 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \, dx\) [964]

Optimal result

Integrand size = 39, antiderivative size = 445 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{16} \left (8 a^4 B+36 a^2 b^2 B+5 b^4 B+8 a^3 b (4 A+3 C)+4 a b^3 (6 A+5 C)\right ) x+\frac {\left (280 a^3 b B+224 a b^3 B+35 a^4 (3 A+2 C)+84 a^2 b^2 (5 A+4 C)+8 b^4 (7 A+6 C)\right ) \sin (c+d x)}{105 d}+\frac {\left (8 a^4 B+36 a^2 b^2 B+5 b^4 B+8 a^3 b (4 A+3 C)+4 a b^3 (6 A+5 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {\left (91 a^3 b B+112 a b^3 B+4 a^4 C+4 b^4 (7 A+6 C)+3 a^2 b^2 (63 A+50 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{105 d}+\frac {b \left (336 a^2 b B+175 b^3 B+24 a^3 C+4 a b^2 (126 A+103 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{840 d}+\frac {\left (14 A b^2+21 a b B+4 a^2 C+12 b^2 C\right ) \cos ^2(c+d x) (a+b \cos (c+d x))^2 \sin (c+d x)}{70 d}+\frac {(7 b B+4 a C) \cos ^2(c+d x) (a+b \cos (c+d x))^3 \sin (c+d x)}{42 d}+\frac {C \cos ^2(c+d x) (a+b \cos (c+d x))^4 \sin (c+d x)}{7 d} \] Output:

1/16*(8*B*a^4+36*B*a^2*b^2+5*B*b^4+8*a^3*b*(4*A+3*C)+4*a*b^3*(6*A+5*C))*x+ 
1/105*(280*B*a^3*b+224*B*a*b^3+35*a^4*(3*A+2*C)+84*a^2*b^2*(5*A+4*C)+8*b^4 
*(7*A+6*C))*sin(d*x+c)/d+1/16*(8*B*a^4+36*B*a^2*b^2+5*B*b^4+8*a^3*b*(4*A+3 
*C)+4*a*b^3*(6*A+5*C))*cos(d*x+c)*sin(d*x+c)/d+1/105*(91*B*a^3*b+112*B*a*b 
^3+4*a^4*C+4*b^4*(7*A+6*C)+3*a^2*b^2*(63*A+50*C))*cos(d*x+c)^2*sin(d*x+c)/ 
d+1/840*b*(336*B*a^2*b+175*B*b^3+24*a^3*C+4*a*b^2*(126*A+103*C))*cos(d*x+c 
)^3*sin(d*x+c)/d+1/70*(14*A*b^2+21*B*a*b+4*C*a^2+12*C*b^2)*cos(d*x+c)^2*(a 
+b*cos(d*x+c))^2*sin(d*x+c)/d+1/42*(7*B*b+4*C*a)*cos(d*x+c)^2*(a+b*cos(d*x 
+c))^3*sin(d*x+c)/d+1/7*C*cos(d*x+c)^2*(a+b*cos(d*x+c))^4*sin(d*x+c)/d
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 3.31 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.19 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {13440 a^3 A b c+10080 a A b^3 c+3360 a^4 B c+15120 a^2 b^2 B c+2100 b^4 B c+10080 a^3 b c C+8400 a b^3 c C+13440 a^3 A b d x+10080 a A b^3 d x+3360 a^4 B d x+15120 a^2 b^2 B d x+2100 b^4 B d x+10080 a^3 b C d x+8400 a b^3 C d x+105 \left (192 a^3 b B+160 a b^3 B+16 a^4 (4 A+3 C)+48 a^2 b^2 (6 A+5 C)+5 b^4 (8 A+7 C)\right ) \sin (c+d x)+105 \left (16 a^4 B+96 a^2 b^2 B+15 b^4 B+64 a^3 b (A+C)+4 a b^3 (16 A+15 C)\right ) \sin (2 (c+d x))+3360 a^2 A b^2 \sin (3 (c+d x))+700 A b^4 \sin (3 (c+d x))+2240 a^3 b B \sin (3 (c+d x))+2800 a b^3 B \sin (3 (c+d x))+560 a^4 C \sin (3 (c+d x))+4200 a^2 b^2 C \sin (3 (c+d x))+735 b^4 C \sin (3 (c+d x))+840 a A b^3 \sin (4 (c+d x))+1260 a^2 b^2 B \sin (4 (c+d x))+315 b^4 B \sin (4 (c+d x))+840 a^3 b C \sin (4 (c+d x))+1260 a b^3 C \sin (4 (c+d x))+84 A b^4 \sin (5 (c+d x))+336 a b^3 B \sin (5 (c+d x))+504 a^2 b^2 C \sin (5 (c+d x))+147 b^4 C \sin (5 (c+d x))+35 b^4 B \sin (6 (c+d x))+140 a b^3 C \sin (6 (c+d x))+15 b^4 C \sin (7 (c+d x))}{6720 d} \] Input:

Integrate[Cos[c + d*x]*(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[ 
c + d*x]^2),x]
 

Output:

(13440*a^3*A*b*c + 10080*a*A*b^3*c + 3360*a^4*B*c + 15120*a^2*b^2*B*c + 21 
00*b^4*B*c + 10080*a^3*b*c*C + 8400*a*b^3*c*C + 13440*a^3*A*b*d*x + 10080* 
a*A*b^3*d*x + 3360*a^4*B*d*x + 15120*a^2*b^2*B*d*x + 2100*b^4*B*d*x + 1008 
0*a^3*b*C*d*x + 8400*a*b^3*C*d*x + 105*(192*a^3*b*B + 160*a*b^3*B + 16*a^4 
*(4*A + 3*C) + 48*a^2*b^2*(6*A + 5*C) + 5*b^4*(8*A + 7*C))*Sin[c + d*x] + 
105*(16*a^4*B + 96*a^2*b^2*B + 15*b^4*B + 64*a^3*b*(A + C) + 4*a*b^3*(16*A 
 + 15*C))*Sin[2*(c + d*x)] + 3360*a^2*A*b^2*Sin[3*(c + d*x)] + 700*A*b^4*S 
in[3*(c + d*x)] + 2240*a^3*b*B*Sin[3*(c + d*x)] + 2800*a*b^3*B*Sin[3*(c + 
d*x)] + 560*a^4*C*Sin[3*(c + d*x)] + 4200*a^2*b^2*C*Sin[3*(c + d*x)] + 735 
*b^4*C*Sin[3*(c + d*x)] + 840*a*A*b^3*Sin[4*(c + d*x)] + 1260*a^2*b^2*B*Si 
n[4*(c + d*x)] + 315*b^4*B*Sin[4*(c + d*x)] + 840*a^3*b*C*Sin[4*(c + d*x)] 
 + 1260*a*b^3*C*Sin[4*(c + d*x)] + 84*A*b^4*Sin[5*(c + d*x)] + 336*a*b^3*B 
*Sin[5*(c + d*x)] + 504*a^2*b^2*C*Sin[5*(c + d*x)] + 147*b^4*C*Sin[5*(c + 
d*x)] + 35*b^4*B*Sin[6*(c + d*x)] + 140*a*b^3*C*Sin[6*(c + d*x)] + 15*b^4* 
C*Sin[7*(c + d*x)])/(6720*d)
 

Rubi [A] (verified)

Time = 1.94 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3528, 3042, 3528, 3042, 3528, 3042, 3512, 3042, 3502, 27, 3042, 3213}

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.

\(\displaystyle \int \cos (c+d x) (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {1}{7} \int \cos (c+d x) (a+b \cos (c+d x))^3 \left ((7 b B+4 a C) \cos ^2(c+d x)+(7 A b+6 C b+7 a B) \cos (c+d x)+a (7 A+2 C)\right )dx+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left ((7 b B+4 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(7 A b+6 C b+7 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+a (7 A+2 C)\right )dx+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {1}{7} \left (\frac {1}{6} \int \cos (c+d x) (a+b \cos (c+d x))^2 \left (3 \left (4 C a^2+21 b B a+14 A b^2+12 b^2 C\right ) \cos ^2(c+d x)+\left (42 B a^2+84 A b a+68 b C a+35 b^2 B\right ) \cos (c+d x)+2 a (21 a A+7 b B+10 a C)\right )dx+\frac {(4 a C+7 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (3 \left (4 C a^2+21 b B a+14 A b^2+12 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (42 B a^2+84 A b a+68 b C a+35 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a (21 a A+7 b B+10 a C)\right )dx+\frac {(4 a C+7 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \int \cos (c+d x) (a+b \cos (c+d x)) \left (\left (24 C a^3+336 b B a^2+4 b^2 (126 A+103 C) a+175 b^3 B\right ) \cos ^2(c+d x)+\left (210 B a^3+(630 A b+488 C b) a^2+497 b^2 B a+24 b^3 (7 A+6 C)\right ) \cos (c+d x)+2 a \left ((105 A+62 C) a^2+98 b B a+6 b^2 (7 A+6 C)\right )\right )dx+\frac {3 \sin (c+d x) \cos ^2(c+d x) \left (4 a^2 C+21 a b B+14 A b^2+12 b^2 C\right ) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {(4 a C+7 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (\left (24 C a^3+336 b B a^2+4 b^2 (126 A+103 C) a+175 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (210 B a^3+(630 A b+488 C b) a^2+497 b^2 B a+24 b^3 (7 A+6 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 a \left ((105 A+62 C) a^2+98 b B a+6 b^2 (7 A+6 C)\right )\right )dx+\frac {3 \sin (c+d x) \cos ^2(c+d x) \left (4 a^2 C+21 a b B+14 A b^2+12 b^2 C\right ) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {(4 a C+7 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}\)

\(\Big \downarrow \) 3512

\(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int \cos (c+d x) \left (8 \left ((105 A+62 C) a^2+98 b B a+6 b^2 (7 A+6 C)\right ) a^2+24 \left (4 C a^4+91 b B a^3+3 b^2 (63 A+50 C) a^2+112 b^3 B a+4 b^4 (7 A+6 C)\right ) \cos ^2(c+d x)+105 \left (8 B a^4+8 b (4 A+3 C) a^3+36 b^2 B a^2+4 b^3 (6 A+5 C) a+5 b^4 B\right ) \cos (c+d x)\right )dx+\frac {b \sin (c+d x) \cos ^3(c+d x) \left (24 a^3 C+336 a^2 b B+4 a b^2 (126 A+103 C)+175 b^3 B\right )}{4 d}\right )+\frac {3 \sin (c+d x) \cos ^2(c+d x) \left (4 a^2 C+21 a b B+14 A b^2+12 b^2 C\right ) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {(4 a C+7 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (8 \left ((105 A+62 C) a^2+98 b B a+6 b^2 (7 A+6 C)\right ) a^2+24 \left (4 C a^4+91 b B a^3+3 b^2 (63 A+50 C) a^2+112 b^3 B a+4 b^4 (7 A+6 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+105 \left (8 B a^4+8 b (4 A+3 C) a^3+36 b^2 B a^2+4 b^3 (6 A+5 C) a+5 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {b \sin (c+d x) \cos ^3(c+d x) \left (24 a^3 C+336 a^2 b B+4 a b^2 (126 A+103 C)+175 b^3 B\right )}{4 d}\right )+\frac {3 \sin (c+d x) \cos ^2(c+d x) \left (4 a^2 C+21 a b B+14 A b^2+12 b^2 C\right ) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {(4 a C+7 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int 3 \cos (c+d x) \left (8 \left (35 (3 A+2 C) a^4+280 b B a^3+84 b^2 (5 A+4 C) a^2+224 b^3 B a+8 b^4 (7 A+6 C)\right )+105 \left (8 B a^4+8 b (4 A+3 C) a^3+36 b^2 B a^2+4 b^3 (6 A+5 C) a+5 b^4 B\right ) \cos (c+d x)\right )dx+\frac {8 \sin (c+d x) \cos ^2(c+d x) \left (4 a^4 C+91 a^3 b B+3 a^2 b^2 (63 A+50 C)+112 a b^3 B+4 b^4 (7 A+6 C)\right )}{d}\right )+\frac {b \sin (c+d x) \cos ^3(c+d x) \left (24 a^3 C+336 a^2 b B+4 a b^2 (126 A+103 C)+175 b^3 B\right )}{4 d}\right )+\frac {3 \sin (c+d x) \cos ^2(c+d x) \left (4 a^2 C+21 a b B+14 A b^2+12 b^2 C\right ) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {(4 a C+7 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \cos (c+d x) \left (8 \left (35 (3 A+2 C) a^4+280 b B a^3+84 b^2 (5 A+4 C) a^2+224 b^3 B a+8 b^4 (7 A+6 C)\right )+105 \left (8 B a^4+8 b (4 A+3 C) a^3+36 b^2 B a^2+4 b^3 (6 A+5 C) a+5 b^4 B\right ) \cos (c+d x)\right )dx+\frac {8 \sin (c+d x) \cos ^2(c+d x) \left (4 a^4 C+91 a^3 b B+3 a^2 b^2 (63 A+50 C)+112 a b^3 B+4 b^4 (7 A+6 C)\right )}{d}\right )+\frac {b \sin (c+d x) \cos ^3(c+d x) \left (24 a^3 C+336 a^2 b B+4 a b^2 (126 A+103 C)+175 b^3 B\right )}{4 d}\right )+\frac {3 \sin (c+d x) \cos ^2(c+d x) \left (4 a^2 C+21 a b B+14 A b^2+12 b^2 C\right ) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {(4 a C+7 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (8 \left (35 (3 A+2 C) a^4+280 b B a^3+84 b^2 (5 A+4 C) a^2+224 b^3 B a+8 b^4 (7 A+6 C)\right )+105 \left (8 B a^4+8 b (4 A+3 C) a^3+36 b^2 B a^2+4 b^3 (6 A+5 C) a+5 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {8 \sin (c+d x) \cos ^2(c+d x) \left (4 a^4 C+91 a^3 b B+3 a^2 b^2 (63 A+50 C)+112 a b^3 B+4 b^4 (7 A+6 C)\right )}{d}\right )+\frac {b \sin (c+d x) \cos ^3(c+d x) \left (24 a^3 C+336 a^2 b B+4 a b^2 (126 A+103 C)+175 b^3 B\right )}{4 d}\right )+\frac {3 \sin (c+d x) \cos ^2(c+d x) \left (4 a^2 C+21 a b B+14 A b^2+12 b^2 C\right ) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {(4 a C+7 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}\)

\(\Big \downarrow \) 3213

\(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {3 \sin (c+d x) \cos ^2(c+d x) \left (4 a^2 C+21 a b B+14 A b^2+12 b^2 C\right ) (a+b \cos (c+d x))^2}{5 d}+\frac {1}{5} \left (\frac {b \sin (c+d x) \cos ^3(c+d x) \left (24 a^3 C+336 a^2 b B+4 a b^2 (126 A+103 C)+175 b^3 B\right )}{4 d}+\frac {1}{4} \left (\frac {8 \sin (c+d x) \left (35 a^4 (3 A+2 C)+280 a^3 b B+84 a^2 b^2 (5 A+4 C)+224 a b^3 B+8 b^4 (7 A+6 C)\right )}{d}+\frac {8 \sin (c+d x) \cos ^2(c+d x) \left (4 a^4 C+91 a^3 b B+3 a^2 b^2 (63 A+50 C)+112 a b^3 B+4 b^4 (7 A+6 C)\right )}{d}+\frac {105 \sin (c+d x) \cos (c+d x) \left (8 a^4 B+8 a^3 b (4 A+3 C)+36 a^2 b^2 B+4 a b^3 (6 A+5 C)+5 b^4 B\right )}{2 d}+\frac {105}{2} x \left (8 a^4 B+8 a^3 b (4 A+3 C)+36 a^2 b^2 B+4 a b^3 (6 A+5 C)+5 b^4 B\right )\right )\right )\right )+\frac {(4 a C+7 b B) \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^3}{6 d}\right )+\frac {C \sin (c+d x) \cos ^2(c+d x) (a+b \cos (c+d x))^4}{7 d}\)

Input:

Int[Cos[c + d*x]*(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d* 
x]^2),x]
 

Output:

(C*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^4*Sin[c + d*x])/(7*d) + (((7*b*B + 
4*a*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(6*d) + ((3*(14 
*A*b^2 + 21*a*b*B + 4*a^2*C + 12*b^2*C)*Cos[c + d*x]^2*(a + b*Cos[c + d*x] 
)^2*Sin[c + d*x])/(5*d) + ((b*(336*a^2*b*B + 175*b^3*B + 24*a^3*C + 4*a*b^ 
2*(126*A + 103*C))*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + ((105*(8*a^4*B + 3 
6*a^2*b^2*B + 5*b^4*B + 8*a^3*b*(4*A + 3*C) + 4*a*b^3*(6*A + 5*C))*x)/2 + 
(8*(280*a^3*b*B + 224*a*b^3*B + 35*a^4*(3*A + 2*C) + 84*a^2*b^2*(5*A + 4*C 
) + 8*b^4*(7*A + 6*C))*Sin[c + d*x])/d + (105*(8*a^4*B + 36*a^2*b^2*B + 5* 
b^4*B + 8*a^3*b*(4*A + 3*C) + 4*a*b^3*(6*A + 5*C))*Cos[c + d*x]*Sin[c + d* 
x])/(2*d) + (8*(91*a^3*b*B + 112*a*b^3*B + 4*a^4*C + 4*b^4*(7*A + 6*C) + 3 
*a^2*b^2*(63*A + 50*C))*Cos[c + d*x]^2*Sin[c + d*x])/d)/4)/5)/6)/7
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3213

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.) 
*(x_)]), x_Symbol] :> Simp[(2*a*c + b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Co 
s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free 
Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
 

rule 3502

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]
 

rule 3512

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f 
_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si 
n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3))   Int[(a + b*Si 
n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + 
A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 
, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 
0] && NeQ[a^2 - b^2, 0] &&  !LtQ[m, -1]
 

rule 3528

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) 
+ (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ 
.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x 
])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + 
n + 2))   Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* 
d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a 
*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + 
 n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} 
, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ 
m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 505, normalized size of antiderivative = 1.13

\[\frac {A \,a^{4} \sin \left (d x +c \right )+B \,a^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{4} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 A \,a^{3} b \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 B \,a^{3} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 a^{3} b C \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 )+2 A \,a^{2} b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+6 B \,a^{2} b^{2} \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 )+\frac {6 C \,a^{2} b^{2} \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}+4 a A \,b^{3} \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 )+\frac {4 B a \,b^{3} \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}+4 C a \,b^{3} \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 {A \,b^{4} \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}+B \,b^{4} \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 {C \,b^{4} \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\]

Input:

int(cos(d*x+c)*(a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)
 

Output:

1/d*(A*a^4*sin(d*x+c)+B*a^4*(1/2*cos(d*x+c)*sin(d*x+c)+1/2*d*x+1/2*c)+1/3* 
a^4*C*(2+cos(d*x+c)^2)*sin(d*x+c)+4*A*a^3*b*(1/2*cos(d*x+c)*sin(d*x+c)+1/2 
*d*x+1/2*c)+4/3*B*a^3*b*(2+cos(d*x+c)^2)*sin(d*x+c)+4*a^3*b*C*(1/4*(cos(d* 
x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+2*A*a^2*b^2*(2+cos(d*x+c) 
^2)*sin(d*x+c)+6*B*a^2*b^2*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3 
/8*d*x+3/8*c)+6/5*C*a^2*b^2*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c) 
+4*a*A*b^3*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)+4/ 
5*B*a*b^3*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+4*C*a*b^3*(1/6*(c 
os(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c)+ 
1/5*A*b^4*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)+B*b^4*(1/6*(cos(d 
*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c)+1/7* 
C*b^4*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c))
 

Fricas [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.80 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (8 \, B a^{4} + 8 \, {\left (4 \, A + 3 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 4 \, {\left (6 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} d x + {\left (240 \, C b^{4} \cos \left (d x + c\right )^{6} + 280 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{5} + 560 \, {\left (3 \, A + 2 \, C\right )} a^{4} + 4480 \, B a^{3} b + 1344 \, {\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 3584 \, B a b^{3} + 128 \, {\left (7 \, A + 6 \, C\right )} b^{4} + 48 \, {\left (42 \, C a^{2} b^{2} + 28 \, B a b^{3} + {\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 70 \, {\left (24 \, C a^{3} b + 36 \, B a^{2} b^{2} + 4 \, {\left (6 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (35 \, C a^{4} + 140 \, B a^{3} b + 42 \, {\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 112 \, B a b^{3} + 4 \, {\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 105 \, {\left (8 \, B a^{4} + 8 \, {\left (4 \, A + 3 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 4 \, {\left (6 \, A + 5 \, C\right )} a b^{3} + 5 \, B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, d} \] Input:

integrate(cos(d*x+c)*(a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, 
 algorithm="fricas")
 

Output:

1/1680*(105*(8*B*a^4 + 8*(4*A + 3*C)*a^3*b + 36*B*a^2*b^2 + 4*(6*A + 5*C)* 
a*b^3 + 5*B*b^4)*d*x + (240*C*b^4*cos(d*x + c)^6 + 280*(4*C*a*b^3 + B*b^4) 
*cos(d*x + c)^5 + 560*(3*A + 2*C)*a^4 + 4480*B*a^3*b + 1344*(5*A + 4*C)*a^ 
2*b^2 + 3584*B*a*b^3 + 128*(7*A + 6*C)*b^4 + 48*(42*C*a^2*b^2 + 28*B*a*b^3 
 + (7*A + 6*C)*b^4)*cos(d*x + c)^4 + 70*(24*C*a^3*b + 36*B*a^2*b^2 + 4*(6* 
A + 5*C)*a*b^3 + 5*B*b^4)*cos(d*x + c)^3 + 16*(35*C*a^4 + 140*B*a^3*b + 42 
*(5*A + 4*C)*a^2*b^2 + 112*B*a*b^3 + 4*(7*A + 6*C)*b^4)*cos(d*x + c)^2 + 1 
05*(8*B*a^4 + 8*(4*A + 3*C)*a^3*b + 36*B*a^2*b^2 + 4*(6*A + 5*C)*a*b^3 + 5 
*B*b^4)*cos(d*x + c))*sin(d*x + c))/d
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1334 vs. \(2 (461) = 922\).

Time = 0.61 (sec) , antiderivative size = 1334, normalized size of antiderivative = 3.00 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)*(a+b*cos(d*x+c))**4*(A+B*cos(d*x+c)+C*cos(d*x+c)**2), 
x)
 

Output:

Piecewise((A*a**4*sin(c + d*x)/d + 2*A*a**3*b*x*sin(c + d*x)**2 + 2*A*a**3 
*b*x*cos(c + d*x)**2 + 2*A*a**3*b*sin(c + d*x)*cos(c + d*x)/d + 4*A*a**2*b 
**2*sin(c + d*x)**3/d + 6*A*a**2*b**2*sin(c + d*x)*cos(c + d*x)**2/d + 3*A 
*a*b**3*x*sin(c + d*x)**4/2 + 3*A*a*b**3*x*sin(c + d*x)**2*cos(c + d*x)**2 
 + 3*A*a*b**3*x*cos(c + d*x)**4/2 + 3*A*a*b**3*sin(c + d*x)**3*cos(c + d*x 
)/(2*d) + 5*A*a*b**3*sin(c + d*x)*cos(c + d*x)**3/(2*d) + 8*A*b**4*sin(c + 
 d*x)**5/(15*d) + 4*A*b**4*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + A*b**4* 
sin(c + d*x)*cos(c + d*x)**4/d + B*a**4*x*sin(c + d*x)**2/2 + B*a**4*x*cos 
(c + d*x)**2/2 + B*a**4*sin(c + d*x)*cos(c + d*x)/(2*d) + 8*B*a**3*b*sin(c 
 + d*x)**3/(3*d) + 4*B*a**3*b*sin(c + d*x)*cos(c + d*x)**2/d + 9*B*a**2*b* 
*2*x*sin(c + d*x)**4/4 + 9*B*a**2*b**2*x*sin(c + d*x)**2*cos(c + d*x)**2/2 
 + 9*B*a**2*b**2*x*cos(c + d*x)**4/4 + 9*B*a**2*b**2*sin(c + d*x)**3*cos(c 
 + d*x)/(4*d) + 15*B*a**2*b**2*sin(c + d*x)*cos(c + d*x)**3/(4*d) + 32*B*a 
*b**3*sin(c + d*x)**5/(15*d) + 16*B*a*b**3*sin(c + d*x)**3*cos(c + d*x)**2 
/(3*d) + 4*B*a*b**3*sin(c + d*x)*cos(c + d*x)**4/d + 5*B*b**4*x*sin(c + d* 
x)**6/16 + 15*B*b**4*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 15*B*b**4*x*si 
n(c + d*x)**2*cos(c + d*x)**4/16 + 5*B*b**4*x*cos(c + d*x)**6/16 + 5*B*b** 
4*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 5*B*b**4*sin(c + d*x)**3*cos(c + d 
*x)**3/(6*d) + 11*B*b**4*sin(c + d*x)*cos(c + d*x)**5/(16*d) + 2*C*a**4*si 
n(c + d*x)**3/(3*d) + C*a**4*sin(c + d*x)*cos(c + d*x)**2/d + 3*C*a**3*...
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.12 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 2240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 6720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} b - 8960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} b + 840 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} b - 13440 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b^{2} + 1260 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b^{2} + 2688 \, {\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 )} C a^{2} b^{2} + 840 \, {\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 a b^{3} + 1792 \, {\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 a b^{3} - 140 \, {\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 a b^{3} + 448 \, {\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 )} A b^{4} - 35 \, {\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 )} B b^{4} - 192 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} C b^{4} + 6720 \, A a^{4} \sin \left (d x + c\right )}{6720 \, d} \] Input:

integrate(cos(d*x+c)*(a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, 
 algorithm="maxima")
 

Output:

1/6720*(1680*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^4 - 2240*(sin(d*x + c)^3 
 - 3*sin(d*x + c))*C*a^4 + 6720*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^3*b - 
 8960*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^3*b + 840*(12*d*x + 12*c + sin 
(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^3*b - 13440*(sin(d*x + c)^3 - 3*si 
n(d*x + c))*A*a^2*b^2 + 1260*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d 
*x + 2*c))*B*a^2*b^2 + 2688*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin 
(d*x + c))*C*a^2*b^2 + 840*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x 
 + 2*c))*A*a*b^3 + 1792*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x 
 + c))*B*a*b^3 - 140*(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*a*b^3 + 448*(3*sin(d*x + c)^5 - 10*sin(d*x 
+ c)^3 + 15*sin(d*x + c))*A*b^4 - 35*(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))*B*b^4 - 192*(5*sin(d*x + c)^7 
 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*C*b^4 + 6720*A 
*a^4*sin(d*x + c))/d
                                                                                    
                                                                                    
 

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 390, normalized size of antiderivative = 0.88 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {C b^{4} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {1}{16} \, {\left (8 \, B a^{4} + 32 \, A a^{3} b + 24 \, C a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3} + 20 \, C a b^{3} + 5 \, B b^{4}\right )} x + \frac {{\left (4 \, C a b^{3} + B b^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (24 \, C a^{2} b^{2} + 16 \, B a b^{3} + 4 \, A b^{4} + 7 \, C b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (8 \, C a^{3} b + 12 \, B a^{2} b^{2} + 8 \, A a b^{3} + 12 \, C a b^{3} + 3 \, B b^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (16 \, C a^{4} + 64 \, B a^{3} b + 96 \, A a^{2} b^{2} + 120 \, C a^{2} b^{2} + 80 \, B a b^{3} + 20 \, A b^{4} + 21 \, C b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (16 \, B a^{4} + 64 \, A a^{3} b + 64 \, C a^{3} b + 96 \, B a^{2} b^{2} + 64 \, A a b^{3} + 60 \, C a b^{3} + 15 \, B b^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (64 \, A a^{4} + 48 \, C a^{4} + 192 \, B a^{3} b + 288 \, A a^{2} b^{2} + 240 \, C a^{2} b^{2} + 160 \, B a b^{3} + 40 \, A b^{4} + 35 \, C b^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \] Input:

integrate(cos(d*x+c)*(a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x, 
 algorithm="giac")
 

Output:

1/448*C*b^4*sin(7*d*x + 7*c)/d + 1/16*(8*B*a^4 + 32*A*a^3*b + 24*C*a^3*b + 
 36*B*a^2*b^2 + 24*A*a*b^3 + 20*C*a*b^3 + 5*B*b^4)*x + 1/192*(4*C*a*b^3 + 
B*b^4)*sin(6*d*x + 6*c)/d + 1/320*(24*C*a^2*b^2 + 16*B*a*b^3 + 4*A*b^4 + 7 
*C*b^4)*sin(5*d*x + 5*c)/d + 1/64*(8*C*a^3*b + 12*B*a^2*b^2 + 8*A*a*b^3 + 
12*C*a*b^3 + 3*B*b^4)*sin(4*d*x + 4*c)/d + 1/192*(16*C*a^4 + 64*B*a^3*b + 
96*A*a^2*b^2 + 120*C*a^2*b^2 + 80*B*a*b^3 + 20*A*b^4 + 21*C*b^4)*sin(3*d*x 
 + 3*c)/d + 1/64*(16*B*a^4 + 64*A*a^3*b + 64*C*a^3*b + 96*B*a^2*b^2 + 64*A 
*a*b^3 + 60*C*a*b^3 + 15*B*b^4)*sin(2*d*x + 2*c)/d + 1/64*(64*A*a^4 + 48*C 
*a^4 + 192*B*a^3*b + 288*A*a^2*b^2 + 240*C*a^2*b^2 + 160*B*a*b^3 + 40*A*b^ 
4 + 35*C*b^4)*sin(d*x + c)/d
 

Mupad [B] (verification not implemented)

Time = 3.34 (sec) , antiderivative size = 675, normalized size of antiderivative = 1.52 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {B\,a^4\,x}{2}+\frac {5\,B\,b^4\,x}{16}+\frac {3\,A\,a\,b^3\,x}{2}+2\,A\,a^3\,b\,x+\frac {5\,C\,a\,b^3\,x}{4}+\frac {3\,C\,a^3\,b\,x}{2}+\frac {A\,a^4\,\sin \left (c+d\,x\right )}{d}+\frac {5\,A\,b^4\,\sin \left (c+d\,x\right )}{8\,d}+\frac {3\,C\,a^4\,\sin \left (c+d\,x\right )}{4\,d}+\frac {35\,C\,b^4\,\sin \left (c+d\,x\right )}{64\,d}+\frac {9\,B\,a^2\,b^2\,x}{4}+\frac {B\,a^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {5\,A\,b^4\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {A\,b^4\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {15\,B\,b^4\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {C\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {3\,B\,b^4\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {B\,b^4\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {7\,C\,b^4\,\sin \left (3\,c+3\,d\,x\right )}{64\,d}+\frac {7\,C\,b^4\,\sin \left (5\,c+5\,d\,x\right )}{320\,d}+\frac {C\,b^4\,\sin \left (7\,c+7\,d\,x\right )}{448\,d}+\frac {A\,a\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {A\,a^3\,b\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {A\,a\,b^3\,\sin \left (4\,c+4\,d\,x\right )}{8\,d}+\frac {9\,A\,a^2\,b^2\,\sin \left (c+d\,x\right )}{2\,d}+\frac {5\,B\,a\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {B\,a^3\,b\,\sin \left (3\,c+3\,d\,x\right )}{3\,d}+\frac {B\,a\,b^3\,\sin \left (5\,c+5\,d\,x\right )}{20\,d}+\frac {15\,C\,a\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{16\,d}+\frac {C\,a^3\,b\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {3\,C\,a\,b^3\,\sin \left (4\,c+4\,d\,x\right )}{16\,d}+\frac {C\,a^3\,b\,\sin \left (4\,c+4\,d\,x\right )}{8\,d}+\frac {C\,a\,b^3\,\sin \left (6\,c+6\,d\,x\right )}{48\,d}+\frac {15\,C\,a^2\,b^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {A\,a^2\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{2\,d}+\frac {3\,B\,a^2\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {3\,B\,a^2\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{16\,d}+\frac {5\,C\,a^2\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{8\,d}+\frac {3\,C\,a^2\,b^2\,\sin \left (5\,c+5\,d\,x\right )}{40\,d}+\frac {5\,B\,a\,b^3\,\sin \left (c+d\,x\right )}{2\,d}+\frac {3\,B\,a^3\,b\,\sin \left (c+d\,x\right )}{d} \] Input:

int(cos(c + d*x)*(a + b*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos(c + d* 
x)^2),x)
 

Output:

(B*a^4*x)/2 + (5*B*b^4*x)/16 + (3*A*a*b^3*x)/2 + 2*A*a^3*b*x + (5*C*a*b^3* 
x)/4 + (3*C*a^3*b*x)/2 + (A*a^4*sin(c + d*x))/d + (5*A*b^4*sin(c + d*x))/( 
8*d) + (3*C*a^4*sin(c + d*x))/(4*d) + (35*C*b^4*sin(c + d*x))/(64*d) + (9* 
B*a^2*b^2*x)/4 + (B*a^4*sin(2*c + 2*d*x))/(4*d) + (5*A*b^4*sin(3*c + 3*d*x 
))/(48*d) + (A*b^4*sin(5*c + 5*d*x))/(80*d) + (15*B*b^4*sin(2*c + 2*d*x))/ 
(64*d) + (C*a^4*sin(3*c + 3*d*x))/(12*d) + (3*B*b^4*sin(4*c + 4*d*x))/(64* 
d) + (B*b^4*sin(6*c + 6*d*x))/(192*d) + (7*C*b^4*sin(3*c + 3*d*x))/(64*d) 
+ (7*C*b^4*sin(5*c + 5*d*x))/(320*d) + (C*b^4*sin(7*c + 7*d*x))/(448*d) + 
(A*a*b^3*sin(2*c + 2*d*x))/d + (A*a^3*b*sin(2*c + 2*d*x))/d + (A*a*b^3*sin 
(4*c + 4*d*x))/(8*d) + (9*A*a^2*b^2*sin(c + d*x))/(2*d) + (5*B*a*b^3*sin(3 
*c + 3*d*x))/(12*d) + (B*a^3*b*sin(3*c + 3*d*x))/(3*d) + (B*a*b^3*sin(5*c 
+ 5*d*x))/(20*d) + (15*C*a*b^3*sin(2*c + 2*d*x))/(16*d) + (C*a^3*b*sin(2*c 
 + 2*d*x))/d + (3*C*a*b^3*sin(4*c + 4*d*x))/(16*d) + (C*a^3*b*sin(4*c + 4* 
d*x))/(8*d) + (C*a*b^3*sin(6*c + 6*d*x))/(48*d) + (15*C*a^2*b^2*sin(c + d* 
x))/(4*d) + (A*a^2*b^2*sin(3*c + 3*d*x))/(2*d) + (3*B*a^2*b^2*sin(2*c + 2* 
d*x))/(2*d) + (3*B*a^2*b^2*sin(4*c + 4*d*x))/(16*d) + (5*C*a^2*b^2*sin(3*c 
 + 3*d*x))/(8*d) + (3*C*a^2*b^2*sin(5*c + 5*d*x))/(40*d) + (5*B*a*b^3*sin( 
c + d*x))/(2*d) + (3*B*a^3*b*sin(c + d*x))/d
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.07 \[ \int \cos (c+d x) (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {-4200 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{2} b^{3}+4200 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{4} b +10500 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b^{3}+2016 \sin \left (d x +c \right )^{5} a^{2} b^{2} c -6720 \sin \left (d x +c \right )^{3} a^{2} b^{2} c +10080 \sin \left (d x +c \right ) a^{2} b^{2} c +4200 a^{4} b d x +6300 a^{2} b^{3} d x +280 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b^{5}-910 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{5}+1155 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{5}+525 b^{5} d x +1680 \sin \left (d x +c \right ) a^{5}+1120 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a \,b^{3} c -1680 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{3} b c -3640 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a \,b^{3} c +4200 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3} b c +4620 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a \,b^{3} c +2520 a^{3} b c d x +2100 a \,b^{3} c d x -240 \sin \left (d x +c \right )^{7} b^{4} c +1680 \sin \left (d x +c \right )^{5} a \,b^{4}+1008 \sin \left (d x +c \right )^{5} b^{4} c -560 \sin \left (d x +c \right )^{3} a^{4} c -5600 \sin \left (d x +c \right )^{3} a^{3} b^{2}-5600 \sin \left (d x +c \right )^{3} a \,b^{4}-1680 \sin \left (d x +c \right )^{3} b^{4} c +1680 \sin \left (d x +c \right ) a^{4} c +16800 \sin \left (d x +c \right ) a^{3} b^{2}+8400 \sin \left (d x +c \right ) a \,b^{4}+1680 \sin \left (d x +c \right ) b^{4} c}{1680 d} \] Input:

int(cos(d*x+c)*(a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2),x)
 

Output:

(1120*cos(c + d*x)*sin(c + d*x)**5*a*b**3*c + 280*cos(c + d*x)*sin(c + d*x 
)**5*b**5 - 1680*cos(c + d*x)*sin(c + d*x)**3*a**3*b*c - 4200*cos(c + d*x) 
*sin(c + d*x)**3*a**2*b**3 - 3640*cos(c + d*x)*sin(c + d*x)**3*a*b**3*c - 
910*cos(c + d*x)*sin(c + d*x)**3*b**5 + 4200*cos(c + d*x)*sin(c + d*x)*a** 
4*b + 4200*cos(c + d*x)*sin(c + d*x)*a**3*b*c + 10500*cos(c + d*x)*sin(c + 
 d*x)*a**2*b**3 + 4620*cos(c + d*x)*sin(c + d*x)*a*b**3*c + 1155*cos(c + d 
*x)*sin(c + d*x)*b**5 - 240*sin(c + d*x)**7*b**4*c + 2016*sin(c + d*x)**5* 
a**2*b**2*c + 1680*sin(c + d*x)**5*a*b**4 + 1008*sin(c + d*x)**5*b**4*c - 
560*sin(c + d*x)**3*a**4*c - 5600*sin(c + d*x)**3*a**3*b**2 - 6720*sin(c + 
 d*x)**3*a**2*b**2*c - 5600*sin(c + d*x)**3*a*b**4 - 1680*sin(c + d*x)**3* 
b**4*c + 1680*sin(c + d*x)*a**5 + 1680*sin(c + d*x)*a**4*c + 16800*sin(c + 
 d*x)*a**3*b**2 + 10080*sin(c + d*x)*a**2*b**2*c + 8400*sin(c + d*x)*a*b** 
4 + 1680*sin(c + d*x)*b**4*c + 4200*a**4*b*d*x + 2520*a**3*b*c*d*x + 6300* 
a**2*b**3*d*x + 2100*a*b**3*c*d*x + 525*b**5*d*x)/(1680*d)