Integrand size = 41, antiderivative size = 372 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{16} \left (24 a^3 b B+32 a b^3 B+8 b^4 (A+2 C)+12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)\right ) x+\frac {\left (8 a^4 B+60 a^2 b^2 B+15 b^4 B+20 a b^3 (2 A+3 C)+8 a^3 b (4 A+5 C)\right ) \sin (c+d x)}{15 d}+\frac {\left (24 A b^4+360 a^3 b B+336 a b^3 B+15 a^4 (5 A+6 C)+10 a^2 b^2 (49 A+66 C)\right ) \cos (c+d x) \sin (c+d x)}{240 d}+\frac {a \left (4 A b^3+16 a^3 B+36 a b^2 B+a^2 b (39 A+50 C)\right ) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac {\left (12 A b^2+48 a b B+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{120 d}+\frac {(2 A b+3 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{6 d} \]
1/16*(24*B*a^3*b+32*B*a*b^3+8*b^4*(A+2*C)+12*a^2*b^2*(3*A+4*C)+a^4*(5*A+6* C))*x+1/15*(8*B*a^4+60*B*a^2*b^2+15*B*b^4+20*a*b^3*(2*A+3*C)+8*a^3*b*(4*A+ 5*C))*sin(d*x+c)/d+1/240*(24*A*b^4+360*B*a^3*b+336*B*a*b^3+15*a^4*(5*A+6*C )+10*a^2*b^2*(49*A+66*C))*cos(d*x+c)*sin(d*x+c)/d+1/60*a*(4*A*b^3+16*B*a^3 +36*B*a*b^2+a^2*b*(39*A+50*C))*cos(d*x+c)^2*sin(d*x+c)/d+1/120*(12*A*b^2+4 8*B*a*b+5*a^2*(5*A+6*C))*cos(d*x+c)^3*(a+b*sec(d*x+c))^2*sin(d*x+c)/d+1/15 *(2*A*b+3*B*a)*cos(d*x+c)^4*(a+b*sec(d*x+c))^3*sin(d*x+c)/d+1/6*A*cos(d*x+ c)^5*(a+b*sec(d*x+c))^4*sin(d*x+c)/d
Time = 3.86 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.16 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {300 a^4 A c+2160 a^2 A b^2 c+480 A b^4 c+1440 a^3 b B c+1920 a b^3 B c+360 a^4 c C+2880 a^2 b^2 c C+960 b^4 c C+300 a^4 A d x+2160 a^2 A b^2 d x+480 A b^4 d x+1440 a^3 b B d x+1920 a b^3 B d x+360 a^4 C d x+2880 a^2 b^2 C d x+960 b^4 C d x+120 \left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) \sin (c+d x)+15 \left (16 A b^4+64 a^3 b B+64 a b^3 B+96 a^2 b^2 (A+C)+a^4 (15 A+16 C)\right ) \sin (2 (c+d x))+400 a^3 A b \sin (3 (c+d x))+320 a A b^3 \sin (3 (c+d x))+100 a^4 B \sin (3 (c+d x))+480 a^2 b^2 B \sin (3 (c+d x))+320 a^3 b C \sin (3 (c+d x))+45 a^4 A \sin (4 (c+d x))+180 a^2 A b^2 \sin (4 (c+d x))+120 a^3 b B \sin (4 (c+d x))+30 a^4 C \sin (4 (c+d x))+48 a^3 A b \sin (5 (c+d x))+12 a^4 B \sin (5 (c+d x))+5 a^4 A \sin (6 (c+d x))}{960 d} \]
(300*a^4*A*c + 2160*a^2*A*b^2*c + 480*A*b^4*c + 1440*a^3*b*B*c + 1920*a*b^ 3*B*c + 360*a^4*c*C + 2880*a^2*b^2*c*C + 960*b^4*c*C + 300*a^4*A*d*x + 216 0*a^2*A*b^2*d*x + 480*A*b^4*d*x + 1440*a^3*b*B*d*x + 1920*a*b^3*B*d*x + 36 0*a^4*C*d*x + 2880*a^2*b^2*C*d*x + 960*b^4*C*d*x + 120*(5*a^4*B + 36*a^2*b ^2*B + 8*b^4*B + 8*a*b^3*(3*A + 4*C) + 4*a^3*b*(5*A + 6*C))*Sin[c + d*x] + 15*(16*A*b^4 + 64*a^3*b*B + 64*a*b^3*B + 96*a^2*b^2*(A + C) + a^4*(15*A + 16*C))*Sin[2*(c + d*x)] + 400*a^3*A*b*Sin[3*(c + d*x)] + 320*a*A*b^3*Sin[ 3*(c + d*x)] + 100*a^4*B*Sin[3*(c + d*x)] + 480*a^2*b^2*B*Sin[3*(c + d*x)] + 320*a^3*b*C*Sin[3*(c + d*x)] + 45*a^4*A*Sin[4*(c + d*x)] + 180*a^2*A*b^ 2*Sin[4*(c + d*x)] + 120*a^3*b*B*Sin[4*(c + d*x)] + 30*a^4*C*Sin[4*(c + d* x)] + 48*a^3*A*b*Sin[5*(c + d*x)] + 12*a^4*B*Sin[5*(c + d*x)] + 5*a^4*A*Si n[6*(c + d*x)])/(960*d)
Time = 2.39 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.366, Rules used = {3042, 4582, 3042, 4582, 3042, 4582, 3042, 4562, 27, 3042, 4535, 3042, 3117, 4533, 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.
| \(\displaystyle \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{6} \int \cos ^5(c+d x) (a+b \sec (c+d x))^3 \left (b (A+6 C) \sec ^2(c+d x)+(5 a A+6 b B+6 a C) \sec (c+d x)+2 (2 A b+3 a B)\right )dx+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (b (A+6 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(5 a A+6 b B+6 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+2 (2 A b+3 a B)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \cos ^4(c+d x) (a+b \sec (c+d x))^2 \left (5 (5 A+6 C) a^2+48 b B a+12 A b^2+3 b (3 A b+10 C b+2 a B) \sec ^2(c+d x)+2 \left (12 B a^2+b (23 A+30 C) a+15 b^2 B\right ) \sec (c+d x)\right )dx+\frac {2 (3 a B+2 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (5 (5 A+6 C) a^2+48 b B a+12 A b^2+3 b (3 A b+10 C b+2 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 \left (12 B a^2+b (23 A+30 C) a+15 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {2 (3 a B+2 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (b \left (5 (5 A+6 C) a^2+72 b B a+24 b^2 (2 A+5 C)\right ) \sec ^2(c+d x)+\left (15 (5 A+6 C) a^3+264 b B a^2+8 b^2 (32 A+45 C) a+120 b^3 B\right ) \sec (c+d x)+6 \left (16 B a^3+b (39 A+50 C) a^2+36 b^2 B a+4 A b^3\right )\right )dx+\frac {\sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (b \left (5 (5 A+6 C) a^2+72 b B a+24 b^2 (2 A+5 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (15 (5 A+6 C) a^3+264 b B a^2+8 b^2 (32 A+45 C) a+120 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+6 \left (16 B a^3+b (39 A+50 C) a^2+36 b^2 B a+4 A b^3\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {\sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 4562 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {2 a \sin (c+d x) \cos ^2(c+d x) \left (16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right )}{d}-\frac {1}{3} \int -3 \cos ^2(c+d x) \left (15 (5 A+6 C) a^4+360 b B a^3+10 b^2 (49 A+66 C) a^2+336 b^3 B a+24 A b^4+b^2 \left (5 (5 A+6 C) a^2+72 b B a+24 b^2 (2 A+5 C)\right ) \sec ^2(c+d x)+8 \left (8 B a^4+8 b (4 A+5 C) a^3+60 b^2 B a^2+20 b^3 (2 A+3 C) a+15 b^4 B\right ) \sec (c+d x)\right )dx\right )+\frac {\sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \cos ^2(c+d x) \left (15 (5 A+6 C) a^4+360 b B a^3+10 b^2 (49 A+66 C) a^2+336 b^3 B a+24 A b^4+b^2 \left (5 (5 A+6 C) a^2+72 b B a+24 b^2 (2 A+5 C)\right ) \sec ^2(c+d x)+8 \left (8 B a^4+8 b (4 A+5 C) a^3+60 b^2 B a^2+20 b^3 (2 A+3 C) a+15 b^4 B\right ) \sec (c+d x)\right )dx+\frac {2 a \sin (c+d x) \cos ^2(c+d x) \left (16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right )}{d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \frac {15 (5 A+6 C) a^4+360 b B a^3+10 b^2 (49 A+66 C) a^2+336 b^3 B a+24 A b^4+b^2 \left (5 (5 A+6 C) a^2+72 b B a+24 b^2 (2 A+5 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+8 \left (8 B a^4+8 b (4 A+5 C) a^3+60 b^2 B a^2+20 b^3 (2 A+3 C) a+15 b^4 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {2 a \sin (c+d x) \cos ^2(c+d x) \left (16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right )}{d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (8 \left (8 a^4 B+8 a^3 b (4 A+5 C)+60 a^2 b^2 B+20 a b^3 (2 A+3 C)+15 b^4 B\right ) \int \cos (c+d x)dx+\int \cos ^2(c+d x) \left (15 (5 A+6 C) a^4+360 b B a^3+10 b^2 (49 A+66 C) a^2+336 b^3 B a+24 A b^4+b^2 \left (5 (5 A+6 C) a^2+72 b B a+24 b^2 (2 A+5 C)\right ) \sec ^2(c+d x)\right )dx+\frac {2 a \sin (c+d x) \cos ^2(c+d x) \left (16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right )}{d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (8 \left (8 a^4 B+8 a^3 b (4 A+5 C)+60 a^2 b^2 B+20 a b^3 (2 A+3 C)+15 b^4 B\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+\int \frac {15 (5 A+6 C) a^4+360 b B a^3+10 b^2 (49 A+66 C) a^2+336 b^3 B a+24 A b^4+b^2 \left (5 (5 A+6 C) a^2+72 b B a+24 b^2 (2 A+5 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {2 a \sin (c+d x) \cos ^2(c+d x) \left (16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right )}{d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \frac {15 (5 A+6 C) a^4+360 b B a^3+10 b^2 (49 A+66 C) a^2+336 b^3 B a+24 A b^4+b^2 \left (5 (5 A+6 C) a^2+72 b B a+24 b^2 (2 A+5 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {2 a \sin (c+d x) \cos ^2(c+d x) \left (16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right )}{d}+\frac {8 \sin (c+d x) \left (8 a^4 B+8 a^3 b (4 A+5 C)+60 a^2 b^2 B+20 a b^3 (2 A+3 C)+15 b^4 B\right )}{d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 4533 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\frac {15}{2} \left (a^4 (5 A+6 C)+24 a^3 b B+12 a^2 b^2 (3 A+4 C)+32 a b^3 B+8 b^4 (A+2 C)\right ) \int 1dx+\frac {2 a \sin (c+d x) \cos ^2(c+d x) \left (16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right )}{d}+\frac {8 \sin (c+d x) \left (8 a^4 B+8 a^3 b (4 A+5 C)+60 a^2 b^2 B+20 a b^3 (2 A+3 C)+15 b^4 B\right )}{d}+\frac {\sin (c+d x) \cos (c+d x) \left (15 a^4 (5 A+6 C)+360 a^3 b B+10 a^2 b^2 (49 A+66 C)+336 a b^3 B+24 A b^4\right )}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{4 d}\right )+\frac {2 (3 a B+2 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^4}{6 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {\sin (c+d x) \cos ^3(c+d x) \left (5 a^2 (5 A+6 C)+48 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{4 d}+\frac {1}{4} \left (\frac {2 a \sin (c+d x) \cos ^2(c+d x) \left (16 a^3 B+a^2 b (39 A+50 C)+36 a b^2 B+4 A b^3\right )}{d}+\frac {8 \sin (c+d x) \left (8 a^4 B+8 a^3 b (4 A+5 C)+60 a^2 b^2 B+20 a b^3 (2 A+3 C)+15 b^4 B\right )}{d}+\frac {\sin (c+d x) \cos (c+d x) \left (15 a^4 (5 A+6 C)+360 a^3 b B+10 a^2 b^2 (49 A+66 C)+336 a b^3 B+24 A b^4\right )}{2 d}+\frac {15}{2} x \left (a^4 (5 A+6 C)+24 a^3 b B+12 a^2 b^2 (3 A+4 C)+32 a b^3 B+8 b^4 (A+2 C)\right )\right )\right )+\frac {2 (3 a B+2 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^3}{5 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^4}{6 d}\) |
(A*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(6*d) + ((2*(2*A*b + 3*a*B)*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(5*d) + (((12 *A*b^2 + 48*a*b*B + 5*a^2*(5*A + 6*C))*Cos[c + d*x]^3*(a + b*Sec[c + d*x]) ^2*Sin[c + d*x])/(4*d) + ((15*(24*a^3*b*B + 32*a*b^3*B + 8*b^4*(A + 2*C) + 12*a^2*b^2*(3*A + 4*C) + a^4*(5*A + 6*C))*x)/2 + (8*(8*a^4*B + 60*a^2*b^2 *B + 15*b^4*B + 20*a*b^3*(2*A + 3*C) + 8*a^3*b*(4*A + 5*C))*Sin[c + d*x])/ d + ((24*A*b^4 + 360*a^3*b*B + 336*a*b^3*B + 15*a^4*(5*A + 6*C) + 10*a^2*b ^2*(49*A + 66*C))*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (2*a*(4*A*b^3 + 16*a^ 3*B + 36*a*b^2*B + a^2*b*(39*A + 50*C))*Cos[c + d*x]^2*Sin[c + d*x])/d)/4) /5)/6
3.9.95.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[A*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + Simp[(C*m + A*(m + 1))/(b^2*m) Int[(b*Csc[e + f*x])^(m + 2), x], x] /; Fr eeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* (B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b Int[(b*Cs c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) , x] /; FreeQ[{b, e, f, A, B, C, m}, x]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Si mp[1/(d*n) Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (n*(a*C + B* b) + A*a*(n + 1))*Csc[e + f*x] + b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[ {a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d* Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Cs c[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a , b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]
Time = 1.02 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.75
| method | result | size |
| parallelrisch | \(\frac {\left (\left (225 A +240 C \right ) a^{4}+960 B \,a^{3} b +1440 a^{2} b^{2} \left (A +C \right )+960 B a \,b^{3}+240 A \,b^{4}\right ) \sin \left (2 d x +2 c \right )+400 a \left (\frac {B \,a^{3}}{4}+b \left (A +\frac {4 C}{5}\right ) a^{2}+\frac {6 B a \,b^{2}}{5}+\frac {4 A \,b^{3}}{5}\right ) \sin \left (3 d x +3 c \right )+45 a^{2} \left (a^{2} \left (A +\frac {2 C}{3}\right )+\frac {8 B a b}{3}+4 A \,b^{2}\right ) \sin \left (4 d x +4 c \right )+\left (48 A \,a^{3} b +12 B \,a^{4}\right ) \sin \left (5 d x +5 c \right )+5 a^{4} A \sin \left (6 d x +6 c \right )+\left (600 B \,a^{4}+2400 \left (A +\frac {6 C}{5}\right ) a^{3} b +4320 B \,a^{2} b^{2}+2880 a \left (A +\frac {4 C}{3}\right ) b^{3}+960 B \,b^{4}\right ) \sin \left (d x +c \right )+300 d x \left (\left (A +\frac {6 C}{5}\right ) a^{4}+\frac {24 B \,a^{3} b}{5}+\frac {36 b^{2} \left (A +\frac {4 C}{3}\right ) a^{2}}{5}+\frac {32 B a \,b^{3}}{5}+\frac {8 b^{4} \left (A +2 C \right )}{5}\right )}{960 d}\) | \(278\) |
| derivativedivides | \(\frac {a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 A \,a^{3} b \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {B \,a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+6 A \,a^{2} b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 B \,a^{3} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{4} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a A \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+2 B \,a^{2} b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {4 a^{3} b C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+A \,b^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 B a \,b^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 C \,a^{2} b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,b^{4} \sin \left (d x +c \right )+4 C a \,b^{3} \sin \left (d x +c \right )+C \,b^{4} \left (d x +c \right )}{d}\) | \(431\) |
| default | \(\frac {a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 A \,a^{3} b \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {B \,a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+6 A \,a^{2} b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 B \,a^{3} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{4} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a A \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+2 B \,a^{2} b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {4 a^{3} b C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+A \,b^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 B a \,b^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 C \,a^{2} b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,b^{4} \sin \left (d x +c \right )+4 C a \,b^{3} \sin \left (d x +c \right )+C \,b^{4} \left (d x +c \right )}{d}\) | \(431\) |
| risch | \(\frac {3 a^{4} A \sin \left (4 d x +4 c \right )}{64 d}+\frac {5 \sin \left (3 d x +3 c \right ) A \,a^{3} b}{12 d}+\frac {x A \,b^{4}}{2}+\frac {5 \sin \left (d x +c \right ) A \,a^{3} b}{2 d}+\frac {3 \sin \left (d x +c \right ) a A \,b^{3}}{d}+\frac {9 \sin \left (d x +c \right ) B \,a^{2} b^{2}}{2 d}+\frac {3 \sin \left (d x +c \right ) a^{3} b C}{d}+\frac {4 \sin \left (d x +c \right ) C a \,b^{3}}{d}+\frac {\sin \left (5 d x +5 c \right ) A \,a^{3} b}{20 d}+\frac {3 \sin \left (4 d x +4 c \right ) A \,a^{2} b^{2}}{16 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{3} b}{8 d}+\frac {\sin \left (3 d x +3 c \right ) a A \,b^{3}}{3 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{2} b^{2}}{2 d}+\frac {\sin \left (3 d x +3 c \right ) a^{3} b C}{3 d}+\frac {3 \sin \left (2 d x +2 c \right ) A \,a^{2} b^{2}}{2 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{3} b}{d}+\frac {\sin \left (2 d x +2 c \right ) B a \,b^{3}}{d}+\frac {3 \sin \left (2 d x +2 c \right ) C \,a^{2} b^{2}}{2 d}+\frac {a^{4} A \sin \left (6 d x +6 c \right )}{192 d}+\frac {\sin \left (4 d x +4 c \right ) a^{4} C}{32 d}+\frac {15 \sin \left (2 d x +2 c \right ) a^{4} A}{64 d}+\frac {\sin \left (2 d x +2 c \right ) a^{4} C}{4 d}+\frac {5 a^{4} A x}{16}+x C \,b^{4}+\frac {3 a^{4} x C}{8}+\frac {5 \sin \left (d x +c \right ) B \,a^{4}}{8 d}+\frac {\sin \left (d x +c \right ) B \,b^{4}}{d}+\frac {\sin \left (5 d x +5 c \right ) B \,a^{4}}{80 d}+\frac {5 \sin \left (3 d x +3 c \right ) B \,a^{4}}{48 d}+2 x B a \,b^{3}+3 x C \,a^{2} b^{2}+\frac {\sin \left (2 d x +2 c \right ) A \,b^{4}}{4 d}+\frac {3 B \,a^{3} b x}{2}+\frac {9 A \,a^{2} b^{2} x}{4}\) | \(535\) |
1/960*(((225*A+240*C)*a^4+960*B*a^3*b+1440*a^2*b^2*(A+C)+960*B*a*b^3+240*A *b^4)*sin(2*d*x+2*c)+400*a*(1/4*B*a^3+b*(A+4/5*C)*a^2+6/5*B*a*b^2+4/5*A*b^ 3)*sin(3*d*x+3*c)+45*a^2*(a^2*(A+2/3*C)+8/3*B*a*b+4*A*b^2)*sin(4*d*x+4*c)+ (48*A*a^3*b+12*B*a^4)*sin(5*d*x+5*c)+5*a^4*A*sin(6*d*x+6*c)+(600*B*a^4+240 0*(A+6/5*C)*a^3*b+4320*B*a^2*b^2+2880*a*(A+4/3*C)*b^3+960*B*b^4)*sin(d*x+c )+300*d*x*((A+6/5*C)*a^4+24/5*B*a^3*b+36/5*b^2*(A+4/3*C)*a^2+32/5*B*a*b^3+ 8/5*b^4*(A+2*C)))/d
Time = 0.28 (sec) , antiderivative size = 292, normalized size of antiderivative = 0.78 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 24 \, B a^{3} b + 12 \, {\left (3 \, A + 4 \, C\right )} a^{2} b^{2} + 32 \, B a b^{3} + 8 \, {\left (A + 2 \, C\right )} b^{4}\right )} d x + {\left (40 \, A a^{4} \cos \left (d x + c\right )^{5} + 128 \, B a^{4} + 128 \, {\left (4 \, A + 5 \, C\right )} a^{3} b + 960 \, B a^{2} b^{2} + 320 \, {\left (2 \, A + 3 \, C\right )} a b^{3} + 240 \, B b^{4} + 48 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{4} + 10 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 24 \, B a^{3} b + 36 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} + 32 \, {\left (2 \, B a^{4} + 2 \, {\left (4 \, A + 5 \, C\right )} a^{3} b + 15 \, B a^{2} b^{2} + 10 \, A a b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{4} + 24 \, B a^{3} b + 12 \, {\left (3 \, A + 4 \, C\right )} a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]
integrate(cos(d*x+c)^6*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="fricas")
1/240*(15*((5*A + 6*C)*a^4 + 24*B*a^3*b + 12*(3*A + 4*C)*a^2*b^2 + 32*B*a* b^3 + 8*(A + 2*C)*b^4)*d*x + (40*A*a^4*cos(d*x + c)^5 + 128*B*a^4 + 128*(4 *A + 5*C)*a^3*b + 960*B*a^2*b^2 + 320*(2*A + 3*C)*a*b^3 + 240*B*b^4 + 48*( B*a^4 + 4*A*a^3*b)*cos(d*x + c)^4 + 10*((5*A + 6*C)*a^4 + 24*B*a^3*b + 36* A*a^2*b^2)*cos(d*x + c)^3 + 32*(2*B*a^4 + 2*(4*A + 5*C)*a^3*b + 15*B*a^2*b ^2 + 10*A*a*b^3)*cos(d*x + c)^2 + 15*((5*A + 6*C)*a^4 + 24*B*a^3*b + 12*(3 *A + 4*C)*a^2*b^2 + 32*B*a*b^3 + 8*A*b^4)*cos(d*x + c))*sin(d*x + c))/d
Timed out. \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Time = 0.23 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.12 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 256 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{3} b - 120 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} b + 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} b - 180 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b^{2} + 1920 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} b^{2} - 1440 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} b^{2} + 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a b^{3} - 960 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b^{3} - 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{4} - 960 \, {\left (d x + c\right )} C b^{4} - 3840 \, C a b^{3} \sin \left (d x + c\right ) - 960 \, B b^{4} \sin \left (d x + c\right )}{960 \, d} \]
integrate(cos(d*x+c)^6*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="maxima")
-1/960*(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))*A*a^4 - 64*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*si n(d*x + c))*B*a^4 - 30*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2 *c))*C*a^4 - 256*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))* A*a^3*b - 120*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^ 3*b + 1280*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^3*b - 180*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^2*b^2 + 1920*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^2*b^2 - 1440*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^2* b^2 + 1280*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a*b^3 - 960*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a*b^3 - 240*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*b^4 - 9 60*(d*x + c)*C*b^4 - 3840*C*a*b^3*sin(d*x + c) - 960*B*b^4*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 1578 vs. \(2 (358) = 716\).
Time = 0.38 (sec) , antiderivative size = 1578, normalized size of antiderivative = 4.24 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
integrate(cos(d*x+c)^6*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="giac")
1/240*(15*(5*A*a^4 + 6*C*a^4 + 24*B*a^3*b + 36*A*a^2*b^2 + 48*C*a^2*b^2 + 32*B*a*b^3 + 8*A*b^4 + 16*C*b^4)*(d*x + c) - 2*(165*A*a^4*tan(1/2*d*x + 1/ 2*c)^11 - 240*B*a^4*tan(1/2*d*x + 1/2*c)^11 + 150*C*a^4*tan(1/2*d*x + 1/2* c)^11 - 960*A*a^3*b*tan(1/2*d*x + 1/2*c)^11 + 600*B*a^3*b*tan(1/2*d*x + 1/ 2*c)^11 - 960*C*a^3*b*tan(1/2*d*x + 1/2*c)^11 + 900*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 - 1440*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 + 720*C*a^2*b^2*tan(1 /2*d*x + 1/2*c)^11 - 960*A*a*b^3*tan(1/2*d*x + 1/2*c)^11 + 480*B*a*b^3*tan (1/2*d*x + 1/2*c)^11 - 960*C*a*b^3*tan(1/2*d*x + 1/2*c)^11 + 120*A*b^4*tan (1/2*d*x + 1/2*c)^11 - 240*B*b^4*tan(1/2*d*x + 1/2*c)^11 - 25*A*a^4*tan(1/ 2*d*x + 1/2*c)^9 - 560*B*a^4*tan(1/2*d*x + 1/2*c)^9 + 210*C*a^4*tan(1/2*d* x + 1/2*c)^9 - 2240*A*a^3*b*tan(1/2*d*x + 1/2*c)^9 + 840*B*a^3*b*tan(1/2*d *x + 1/2*c)^9 - 3520*C*a^3*b*tan(1/2*d*x + 1/2*c)^9 + 1260*A*a^2*b^2*tan(1 /2*d*x + 1/2*c)^9 - 5280*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 2160*C*a^2*b^2 *tan(1/2*d*x + 1/2*c)^9 - 3520*A*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 1440*B*a*b ^3*tan(1/2*d*x + 1/2*c)^9 - 4800*C*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 360*A*b^ 4*tan(1/2*d*x + 1/2*c)^9 - 1200*B*b^4*tan(1/2*d*x + 1/2*c)^9 + 450*A*a^4*t an(1/2*d*x + 1/2*c)^7 - 1248*B*a^4*tan(1/2*d*x + 1/2*c)^7 + 60*C*a^4*tan(1 /2*d*x + 1/2*c)^7 - 4992*A*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 240*B*a^3*b*tan( 1/2*d*x + 1/2*c)^7 - 5760*C*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 360*A*a^2*b^2*t an(1/2*d*x + 1/2*c)^7 - 8640*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 1440*C*...
Time = 20.37 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.44 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {5\,A\,a^4\,x}{16}+\frac {A\,b^4\,x}{2}+\frac {3\,C\,a^4\,x}{8}+C\,b^4\,x+2\,B\,a\,b^3\,x+\frac {3\,B\,a^3\,b\,x}{2}+\frac {5\,B\,a^4\,\sin \left (c+d\,x\right )}{8\,d}+\frac {B\,b^4\,\sin \left (c+d\,x\right )}{d}+\frac {9\,A\,a^2\,b^2\,x}{4}+3\,C\,a^2\,b^2\,x+\frac {15\,A\,a^4\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {3\,A\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {A\,a^4\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {A\,b^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {5\,B\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {B\,a^4\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {C\,a^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {A\,a\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{3\,d}+\frac {5\,A\,a^3\,b\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {A\,a^3\,b\,\sin \left (5\,c+5\,d\,x\right )}{20\,d}+\frac {B\,a\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {B\,a^3\,b\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {B\,a^3\,b\,\sin \left (4\,c+4\,d\,x\right )}{8\,d}+\frac {9\,B\,a^2\,b^2\,\sin \left (c+d\,x\right )}{2\,d}+\frac {C\,a^3\,b\,\sin \left (3\,c+3\,d\,x\right )}{3\,d}+\frac {3\,A\,a^2\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {3\,A\,a^2\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{16\,d}+\frac {B\,a^2\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{2\,d}+\frac {3\,C\,a^2\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {3\,A\,a\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {5\,A\,a^3\,b\,\sin \left (c+d\,x\right )}{2\,d}+\frac {4\,C\,a\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {3\,C\,a^3\,b\,\sin \left (c+d\,x\right )}{d} \]
(5*A*a^4*x)/16 + (A*b^4*x)/2 + (3*C*a^4*x)/8 + C*b^4*x + 2*B*a*b^3*x + (3* B*a^3*b*x)/2 + (5*B*a^4*sin(c + d*x))/(8*d) + (B*b^4*sin(c + d*x))/d + (9* A*a^2*b^2*x)/4 + 3*C*a^2*b^2*x + (15*A*a^4*sin(2*c + 2*d*x))/(64*d) + (3*A *a^4*sin(4*c + 4*d*x))/(64*d) + (A*a^4*sin(6*c + 6*d*x))/(192*d) + (A*b^4* sin(2*c + 2*d*x))/(4*d) + (5*B*a^4*sin(3*c + 3*d*x))/(48*d) + (B*a^4*sin(5 *c + 5*d*x))/(80*d) + (C*a^4*sin(2*c + 2*d*x))/(4*d) + (C*a^4*sin(4*c + 4* d*x))/(32*d) + (A*a*b^3*sin(3*c + 3*d*x))/(3*d) + (5*A*a^3*b*sin(3*c + 3*d *x))/(12*d) + (A*a^3*b*sin(5*c + 5*d*x))/(20*d) + (B*a*b^3*sin(2*c + 2*d*x ))/d + (B*a^3*b*sin(2*c + 2*d*x))/d + (B*a^3*b*sin(4*c + 4*d*x))/(8*d) + ( 9*B*a^2*b^2*sin(c + d*x))/(2*d) + (C*a^3*b*sin(3*c + 3*d*x))/(3*d) + (3*A* a^2*b^2*sin(2*c + 2*d*x))/(2*d) + (3*A*a^2*b^2*sin(4*c + 4*d*x))/(16*d) + (B*a^2*b^2*sin(3*c + 3*d*x))/(2*d) + (3*C*a^2*b^2*sin(2*c + 2*d*x))/(2*d) + (3*A*a*b^3*sin(c + d*x))/d + (5*A*a^3*b*sin(c + d*x))/(2*d) + (4*C*a*b^3 *sin(c + d*x))/d + (3*C*a^3*b*sin(c + d*x))/d