Integrand size = 41, antiderivative size = 438 \[ \int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{16} \left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) x+\frac {\left (336 a^3 b B+371 a b^3 B+12 a^4 (6 A+7 C)+b^4 (74 A+105 C)+3 a^2 b^2 (162 A+203 C)\right ) \sin (c+d x)}{105 d}+\frac {\left (5 a^4 B+36 a^2 b^2 B+8 b^4 B+8 a b^3 (3 A+4 C)+4 a^3 b (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (24 A b^3+175 a^3 B+336 a b^2 B+a^2 (412 A b+504 b C)\right ) \cos ^3(c+d x) \sin (c+d x)}{840 d}+\frac {\left (4 A b^2+21 a b B+2 a^2 (6 A+7 C)\right ) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{70 d}+\frac {(4 A b+7 a B) \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{42 d}+\frac {A \cos ^6(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{7 d}-\frac {\left (4 A b^4+112 a^3 b B+91 a b^3 B+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right ) \sin ^3(c+d x)}{105 d} \]
1/16*(5*B*a^4+36*B*a^2*b^2+8*B*b^4+8*a*b^3*(3*A+4*C)+4*a^3*b*(5*A+6*C))*x+ 1/105*(336*B*a^3*b+371*B*a*b^3+12*a^4*(6*A+7*C)+b^4*(74*A+105*C)+3*a^2*b^2 *(162*A+203*C))*sin(d*x+c)/d+1/16*(5*B*a^4+36*B*a^2*b^2+8*B*b^4+8*a*b^3*(3 *A+4*C)+4*a^3*b*(5*A+6*C))*cos(d*x+c)*sin(d*x+c)/d+1/840*a*(24*A*b^3+175*B *a^3+336*B*a*b^2+a^2*(412*A*b+504*C*b))*cos(d*x+c)^3*sin(d*x+c)/d+1/70*(4* A*b^2+21*B*a*b+2*a^2*(6*A+7*C))*cos(d*x+c)^4*(a+b*sec(d*x+c))^2*sin(d*x+c) /d+1/42*(4*A*b+7*B*a)*cos(d*x+c)^5*(a+b*sec(d*x+c))^3*sin(d*x+c)/d+1/7*A*c os(d*x+c)^6*(a+b*sec(d*x+c))^4*sin(d*x+c)/d-1/105*(4*A*b^4+112*B*a^3*b+91* B*a*b^3+4*a^4*(6*A+7*C)+3*a^2*b^2*(50*A+63*C))*sin(d*x+c)^3/d
Time = 3.58 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.21 \[ \int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {8400 a^3 A b c+10080 a A b^3 c+2100 a^4 B c+15120 a^2 b^2 B c+3360 b^4 B c+10080 a^3 b c C+13440 a b^3 c C+8400 a^3 A b d x+10080 a A b^3 d x+2100 a^4 B d x+15120 a^2 b^2 B d x+3360 b^4 B d x+10080 a^3 b C d x+13440 a b^3 C d x+105 \left (160 a^3 b B+192 a b^3 B+16 b^4 (3 A+4 C)+48 a^2 b^2 (5 A+6 C)+5 a^4 (7 A+8 C)\right ) \sin (c+d x)+105 \left (15 a^4 B+96 a^2 b^2 B+16 b^4 B+64 a b^3 (A+C)+a^3 (60 A b+64 b C)\right ) \sin (2 (c+d x))+735 a^4 A \sin (3 (c+d x))+4200 a^2 A b^2 \sin (3 (c+d x))+560 A b^4 \sin (3 (c+d x))+2800 a^3 b B \sin (3 (c+d x))+2240 a b^3 B \sin (3 (c+d x))+700 a^4 C \sin (3 (c+d x))+3360 a^2 b^2 C \sin (3 (c+d x))+1260 a^3 A b \sin (4 (c+d x))+840 a A b^3 \sin (4 (c+d x))+315 a^4 B \sin (4 (c+d x))+1260 a^2 b^2 B \sin (4 (c+d x))+840 a^3 b C \sin (4 (c+d x))+147 a^4 A \sin (5 (c+d x))+504 a^2 A b^2 \sin (5 (c+d x))+336 a^3 b B \sin (5 (c+d x))+84 a^4 C \sin (5 (c+d x))+140 a^3 A b \sin (6 (c+d x))+35 a^4 B \sin (6 (c+d x))+15 a^4 A \sin (7 (c+d x))}{6720 d} \]
(8400*a^3*A*b*c + 10080*a*A*b^3*c + 2100*a^4*B*c + 15120*a^2*b^2*B*c + 336 0*b^4*B*c + 10080*a^3*b*c*C + 13440*a*b^3*c*C + 8400*a^3*A*b*d*x + 10080*a *A*b^3*d*x + 2100*a^4*B*d*x + 15120*a^2*b^2*B*d*x + 3360*b^4*B*d*x + 10080 *a^3*b*C*d*x + 13440*a*b^3*C*d*x + 105*(160*a^3*b*B + 192*a*b^3*B + 16*b^4 *(3*A + 4*C) + 48*a^2*b^2*(5*A + 6*C) + 5*a^4*(7*A + 8*C))*Sin[c + d*x] + 105*(15*a^4*B + 96*a^2*b^2*B + 16*b^4*B + 64*a*b^3*(A + C) + a^3*(60*A*b + 64*b*C))*Sin[2*(c + d*x)] + 735*a^4*A*Sin[3*(c + d*x)] + 4200*a^2*A*b^2*S in[3*(c + d*x)] + 560*A*b^4*Sin[3*(c + d*x)] + 2800*a^3*b*B*Sin[3*(c + d*x )] + 2240*a*b^3*B*Sin[3*(c + d*x)] + 700*a^4*C*Sin[3*(c + d*x)] + 3360*a^2 *b^2*C*Sin[3*(c + d*x)] + 1260*a^3*A*b*Sin[4*(c + d*x)] + 840*a*A*b^3*Sin[ 4*(c + d*x)] + 315*a^4*B*Sin[4*(c + d*x)] + 1260*a^2*b^2*B*Sin[4*(c + d*x) ] + 840*a^3*b*C*Sin[4*(c + d*x)] + 147*a^4*A*Sin[5*(c + d*x)] + 504*a^2*A* b^2*Sin[5*(c + d*x)] + 336*a^3*b*B*Sin[5*(c + d*x)] + 84*a^4*C*Sin[5*(c + d*x)] + 140*a^3*A*b*Sin[6*(c + d*x)] + 35*a^4*B*Sin[6*(c + d*x)] + 15*a^4* A*Sin[7*(c + d*x)])/(6720*d)
Time = 2.77 (sec) , antiderivative size = 409, normalized size of antiderivative = 0.93, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.439, Rules used = {3042, 4582, 3042, 4582, 3042, 4582, 3042, 4562, 25, 3042, 4535, 3042, 3115, 24, 4532, 3042, 3492, 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.
| \(\displaystyle \int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^7}dx\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{7} \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (b (2 A+7 C) \sec ^2(c+d x)+(6 a A+7 b B+7 a C) \sec (c+d x)+4 A b+7 a B\right )dx+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (b (2 A+7 C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+(6 a A+7 b B+7 a C) \csc \left (c+d x+\frac {\pi }{2}\right )+4 A b+7 a B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{6} \int \cos ^5(c+d x) (a+b \sec (c+d x))^2 \left (2 b (10 A b+21 C b+7 a B) \sec ^2(c+d x)+\left (35 B a^2+68 A b a+84 b C a+42 b^2 B\right ) \sec (c+d x)+3 \left (2 (6 A+7 C) a^2+21 b B a+4 A b^2\right )\right )dx+\frac {(7 a B+4 A b) \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{6} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (2 b (10 A b+21 C b+7 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (35 B a^2+68 A b a+84 b C a+42 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (2 (6 A+7 C) a^2+21 b B a+4 A b^2\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {(7 a B+4 A b) \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 4582 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (175 B a^3+(412 A b+504 C b) a^2+336 b^2 B a+24 A b^3+2 b \left (6 (6 A+7 C) a^2+98 b B a+b^2 (62 A+105 C)\right ) \sec ^2(c+d x)+\left (24 (6 A+7 C) a^3+497 b B a^2+2 b^2 (244 A+315 C) a+210 b^3 B\right ) \sec (c+d x)\right )dx+\frac {3 \sin (c+d x) \cos ^4(c+d x) \left (2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {(7 a B+4 A b) \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (175 B a^3+(412 A b+504 C b) a^2+336 b^2 B a+24 A b^3+2 b \left (6 (6 A+7 C) a^2+98 b B a+b^2 (62 A+105 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (24 (6 A+7 C) a^3+497 b B a^2+2 b^2 (244 A+315 C) a+210 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {3 \sin (c+d x) \cos ^4(c+d x) \left (2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {(7 a B+4 A b) \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 4562 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {a \sin (c+d x) \cos ^3(c+d x) \left (175 a^3 B+a^2 (412 A b+504 b C)+336 a b^2 B+24 A b^3\right )}{4 d}-\frac {1}{4} \int -\cos ^3(c+d x) \left (8 b^2 \left (6 (6 A+7 C) a^2+98 b B a+b^2 (62 A+105 C)\right ) \sec ^2(c+d x)+105 \left (5 B a^4+4 b (5 A+6 C) a^3+36 b^2 B a^2+8 b^3 (3 A+4 C) a+8 b^4 B\right ) \sec (c+d x)+24 \left (4 (6 A+7 C) a^4+112 b B a^3+3 b^2 (50 A+63 C) a^2+91 b^3 B a+4 A b^4\right )\right )dx\right )+\frac {3 \sin (c+d x) \cos ^4(c+d x) \left (2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {(7 a B+4 A b) \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int \cos ^3(c+d x) \left (8 b^2 \left (6 (6 A+7 C) a^2+98 b B a+b^2 (62 A+105 C)\right ) \sec ^2(c+d x)+105 \left (5 B a^4+4 b (5 A+6 C) a^3+36 b^2 B a^2+8 b^3 (3 A+4 C) a+8 b^4 B\right ) \sec (c+d x)+24 \left (4 (6 A+7 C) a^4+112 b B a^3+3 b^2 (50 A+63 C) a^2+91 b^3 B a+4 A b^4\right )\right )dx+\frac {a \sin (c+d x) \cos ^3(c+d x) \left (175 a^3 B+a^2 (412 A b+504 b C)+336 a b^2 B+24 A b^3\right )}{4 d}\right )+\frac {3 \sin (c+d x) \cos ^4(c+d x) \left (2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {(7 a B+4 A b) \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int \frac {8 b^2 \left (6 (6 A+7 C) a^2+98 b B a+b^2 (62 A+105 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+105 \left (5 B a^4+4 b (5 A+6 C) a^3+36 b^2 B a^2+8 b^3 (3 A+4 C) a+8 b^4 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+24 \left (4 (6 A+7 C) a^4+112 b B a^3+3 b^2 (50 A+63 C) a^2+91 b^3 B a+4 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a \sin (c+d x) \cos ^3(c+d x) \left (175 a^3 B+a^2 (412 A b+504 b C)+336 a b^2 B+24 A b^3\right )}{4 d}\right )+\frac {3 \sin (c+d x) \cos ^4(c+d x) \left (2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {(7 a B+4 A b) \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (105 \left (5 a^4 B+4 a^3 b (5 A+6 C)+36 a^2 b^2 B+8 a b^3 (3 A+4 C)+8 b^4 B\right ) \int \cos ^2(c+d x)dx+\int \cos ^3(c+d x) \left (8 b^2 \left (6 (6 A+7 C) a^2+98 b B a+b^2 (62 A+105 C)\right ) \sec ^2(c+d x)+24 \left (4 (6 A+7 C) a^4+112 b B a^3+3 b^2 (50 A+63 C) a^2+91 b^3 B a+4 A b^4\right )\right )dx\right )+\frac {a \sin (c+d x) \cos ^3(c+d x) \left (175 a^3 B+a^2 (412 A b+504 b C)+336 a b^2 B+24 A b^3\right )}{4 d}\right )+\frac {3 \sin (c+d x) \cos ^4(c+d x) \left (2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {(7 a B+4 A b) \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (105 \left (5 a^4 B+4 a^3 b (5 A+6 C)+36 a^2 b^2 B+8 a b^3 (3 A+4 C)+8 b^4 B\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\int \frac {8 b^2 \left (6 (6 A+7 C) a^2+98 b B a+b^2 (62 A+105 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+24 \left (4 (6 A+7 C) a^4+112 b B a^3+3 b^2 (50 A+63 C) a^2+91 b^3 B a+4 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx\right )+\frac {a \sin (c+d x) \cos ^3(c+d x) \left (175 a^3 B+a^2 (412 A b+504 b C)+336 a b^2 B+24 A b^3\right )}{4 d}\right )+\frac {3 \sin (c+d x) \cos ^4(c+d x) \left (2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {(7 a B+4 A b) \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \frac {8 b^2 \left (6 (6 A+7 C) a^2+98 b B a+b^2 (62 A+105 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+24 \left (4 (6 A+7 C) a^4+112 b B a^3+3 b^2 (50 A+63 C) a^2+91 b^3 B a+4 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+105 \left (5 a^4 B+4 a^3 b (5 A+6 C)+36 a^2 b^2 B+8 a b^3 (3 A+4 C)+8 b^4 B\right ) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )\right )+\frac {a \sin (c+d x) \cos ^3(c+d x) \left (175 a^3 B+a^2 (412 A b+504 b C)+336 a b^2 B+24 A b^3\right )}{4 d}\right )+\frac {3 \sin (c+d x) \cos ^4(c+d x) \left (2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {(7 a B+4 A b) \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \frac {8 b^2 \left (6 (6 A+7 C) a^2+98 b B a+b^2 (62 A+105 C)\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+24 \left (4 (6 A+7 C) a^4+112 b B a^3+3 b^2 (50 A+63 C) a^2+91 b^3 B a+4 A b^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+105 \left (5 a^4 B+4 a^3 b (5 A+6 C)+36 a^2 b^2 B+8 a b^3 (3 A+4 C)+8 b^4 B\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a \sin (c+d x) \cos ^3(c+d x) \left (175 a^3 B+a^2 (412 A b+504 b C)+336 a b^2 B+24 A b^3\right )}{4 d}\right )+\frac {3 \sin (c+d x) \cos ^4(c+d x) \left (2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {(7 a B+4 A b) \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 4532 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \cos (c+d x) \left (8 \left (6 (6 A+7 C) a^2+98 b B a+b^2 (62 A+105 C)\right ) b^2+24 \left (4 (6 A+7 C) a^4+112 b B a^3+3 b^2 (50 A+63 C) a^2+91 b^3 B a+4 A b^4\right ) \cos ^2(c+d x)\right )dx+105 \left (5 a^4 B+4 a^3 b (5 A+6 C)+36 a^2 b^2 B+8 a b^3 (3 A+4 C)+8 b^4 B\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a \sin (c+d x) \cos ^3(c+d x) \left (175 a^3 B+a^2 (412 A b+504 b C)+336 a b^2 B+24 A b^3\right )}{4 d}\right )+\frac {3 \sin (c+d x) \cos ^4(c+d x) \left (2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {(7 a B+4 A b) \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (8 \left (6 (6 A+7 C) a^2+98 b B a+b^2 (62 A+105 C)\right ) b^2+24 \left (4 (6 A+7 C) a^4+112 b B a^3+3 b^2 (50 A+63 C) a^2+91 b^3 B a+4 A b^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+105 \left (5 a^4 B+4 a^3 b (5 A+6 C)+36 a^2 b^2 B+8 a b^3 (3 A+4 C)+8 b^4 B\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a \sin (c+d x) \cos ^3(c+d x) \left (175 a^3 B+a^2 (412 A b+504 b C)+336 a b^2 B+24 A b^3\right )}{4 d}\right )+\frac {3 \sin (c+d x) \cos ^4(c+d x) \left (2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {(7 a B+4 A b) \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3492 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (105 \left (5 a^4 B+4 a^3 b (5 A+6 C)+36 a^2 b^2 B+8 a b^3 (3 A+4 C)+8 b^4 B\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {\int \left (8 \left (12 (6 A+7 C) a^4+336 b B a^3+3 b^2 (162 A+203 C) a^2+371 b^3 B a+b^4 (74 A+105 C)\right )-24 \left (4 (6 A+7 C) a^4+112 b B a^3+3 b^2 (50 A+63 C) a^2+91 b^3 B a+4 A b^4\right ) \sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}\right )+\frac {a \sin (c+d x) \cos ^3(c+d x) \left (175 a^3 B+a^2 (412 A b+504 b C)+336 a b^2 B+24 A b^3\right )}{4 d}\right )+\frac {3 \sin (c+d x) \cos ^4(c+d x) \left (2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right ) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {(7 a B+4 A b) \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{6} \left (\frac {3 \sin (c+d x) \cos ^4(c+d x) \left (2 a^2 (6 A+7 C)+21 a b B+4 A b^2\right ) (a+b \sec (c+d x))^2}{5 d}+\frac {1}{5} \left (\frac {a \sin (c+d x) \cos ^3(c+d x) \left (175 a^3 B+a^2 (412 A b+504 b C)+336 a b^2 B+24 A b^3\right )}{4 d}+\frac {1}{4} \left (105 \left (5 a^4 B+4 a^3 b (5 A+6 C)+36 a^2 b^2 B+8 a b^3 (3 A+4 C)+8 b^4 B\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {8 \sin ^3(c+d x) \left (4 a^4 (6 A+7 C)+112 a^3 b B+3 a^2 b^2 (50 A+63 C)+91 a b^3 B+4 A b^4\right )-8 \sin (c+d x) \left (12 a^4 (6 A+7 C)+336 a^3 b B+3 a^2 b^2 (162 A+203 C)+371 a b^3 B+b^4 (74 A+105 C)\right )}{d}\right )\right )\right )+\frac {(7 a B+4 A b) \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\right )+\frac {A \sin (c+d x) \cos ^6(c+d x) (a+b \sec (c+d x))^4}{7 d}\) |
(A*Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*Sin[c + d*x])/(7*d) + (((4*A*b + 7*a*B)*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + ((3*(4* A*b^2 + 21*a*b*B + 2*a^2*(6*A + 7*C))*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^ 2*Sin[c + d*x])/(5*d) + ((a*(24*A*b^3 + 175*a^3*B + 336*a*b^2*B + a^2*(412 *A*b + 504*b*C))*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (105*(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))*(x/2 + (Co s[c + d*x]*Sin[c + d*x])/(2*d)) - (-8*(336*a^3*b*B + 371*a*b^3*B + 12*a^4* (6*A + 7*C) + b^4*(74*A + 105*C) + 3*a^2*b^2*(162*A + 203*C))*Sin[c + d*x] + 8*(4*A*b^4 + 112*a^3*b*B + 91*a*b^3*B + 4*a^4*(6*A + 7*C) + 3*a^2*b^2*( 50*A + 63*C))*Sin[c + d*x]^3)/d)/4)/5)/6)/7
3.9.96.3.1 Defintions of rubi rules used
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[sin[(e_.) + (f_.)*(x_)]^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[-f^(-1) Subst[Int[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2 ), x], x, Cos[e + f*x]], x] /; FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2, 0]
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Int[(C + A*Sin[e + f*x]^2)/Sin[e + f*x]^(m + 2), x] /; FreeQ[ {e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && ILtQ[(m + 1)/2, 0]
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 = 1.34 (sec) , antiderivative size = 335, normalized size of antiderivative = 0.76
| method | result | size |
| parallelrisch | \(\frac {\left (1575 B \,a^{4}+6300 \left (A +\frac {16 C}{15}\right ) b \,a^{3}+10080 B \,a^{2} b^{2}+6720 b^{3} \left (A +C \right ) a +1680 B \,b^{4}\right ) \sin \left (2 d x +2 c \right )+\left (\left (735 A +700 C \right ) a^{4}+2800 B \,a^{3} b +4200 b^{2} \left (A +\frac {4 C}{5}\right ) a^{2}+2240 B a \,b^{3}+560 A \,b^{4}\right ) \sin \left (3 d x +3 c \right )+1260 \left (\frac {B \,a^{3}}{4}+b \left (A +\frac {2 C}{3}\right ) a^{2}+B a \,b^{2}+\frac {2 A \,b^{3}}{3}\right ) a \sin \left (4 d x +4 c \right )+\left (\left (147 A +84 C \right ) a^{4}+336 B \,a^{3} b +504 A \,a^{2} b^{2}\right ) \sin \left (5 d x +5 c \right )+\left (140 A \,a^{3} b +35 B \,a^{4}\right ) \sin \left (6 d x +6 c \right )+15 a^{4} A \sin \left (7 d x +7 c \right )+\left (\left (3675 A +4200 C \right ) a^{4}+16800 B \,a^{3} b +25200 b^{2} \left (A +\frac {6 C}{5}\right ) a^{2}+20160 B a \,b^{3}+5040 b^{4} \left (A +\frac {4 C}{3}\right )\right ) \sin \left (d x +c \right )+8400 d \left (\frac {B \,a^{4}}{4}+\left (A +\frac {6 C}{5}\right ) a^{3} b +\frac {9 B \,a^{2} b^{2}}{5}+\frac {6 a \left (A +\frac {4 C}{3}\right ) b^{3}}{5}+\frac {2 B \,b^{4}}{5}\right ) x}{6720 d}\) | \(335\) |
| derivativedivides | \(\frac {\frac {a^{4} A \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}+4 A \,a^{3} b \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 )+B \,a^{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 {6 A \,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}+\frac {4 B \,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 {a^{4} C \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 )+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 )+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 )+\frac {A \,b^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+\frac {4 B a \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+2 C \,a^{2} b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+B \,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 C 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 )+C \sin \left (d x +c \right ) b^{4}}{d}\) | \(505\) |
| default | \(\frac {\frac {a^{4} A \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}+4 A \,a^{3} b \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 )+B \,a^{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 {6 A \,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}+\frac {4 B \,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 {a^{4} C \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 )+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 )+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 )+\frac {A \,b^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+\frac {4 B a \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+2 C \,a^{2} b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+B \,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 C 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 )+C \sin \left (d x +c \right ) b^{4}}{d}\) | \(505\) |
| risch | \(\frac {7 a^{4} A \sin \left (3 d x +3 c \right )}{64 d}+\frac {15 \sin \left (2 d x +2 c \right ) B \,a^{4}}{64 d}+\frac {\sin \left (2 d x +2 c \right ) B \,b^{4}}{4 d}+\frac {3 \sin \left (4 d x +4 c \right ) B \,a^{2} b^{2}}{16 d}+\frac {\sin \left (4 d x +4 c \right ) a^{3} b C}{8 d}+\frac {\sin \left (3 d x +3 c \right ) B a \,b^{3}}{3 d}+\frac {\sin \left (3 d x +3 c \right ) C \,a^{2} b^{2}}{2 d}+\frac {\sin \left (2 d x +2 c \right ) C a \,b^{3}}{d}+\frac {15 \sin \left (d x +c \right ) A \,a^{2} b^{2}}{4 d}+\frac {5 \sin \left (d x +c \right ) B \,a^{3} b}{2 d}+\frac {3 \sin \left (d x +c \right ) B a \,b^{3}}{d}+\frac {9 \sin \left (d x +c \right ) C \,a^{2} b^{2}}{2 d}+\frac {\sin \left (6 d x +6 c \right ) A \,a^{3} b}{48 d}+\frac {3 \sin \left (5 d x +5 c \right ) A \,a^{2} b^{2}}{40 d}+\frac {\sin \left (5 d x +5 c \right ) B \,a^{3} b}{20 d}+\frac {\sin \left (4 d x +4 c \right ) a A \,b^{3}}{8 d}+\frac {5 B \,a^{4} x}{16}+\frac {5 a^{3} A b x}{4}+\frac {35 \sin \left (d x +c \right ) a^{4} A}{64 d}+\frac {5 \sin \left (d x +c \right ) a^{4} C}{8 d}+\frac {5 \sin \left (3 d x +3 c \right ) a^{4} C}{48 d}+\frac {15 \sin \left (2 d x +2 c \right ) A \,a^{3} b}{16 d}+\frac {\sin \left (2 d x +2 c \right ) a A \,b^{3}}{d}+\frac {3 \sin \left (2 d x +2 c \right ) B \,a^{2} b^{2}}{2 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} b C}{d}+\frac {3 \sin \left (4 d x +4 c \right ) A \,a^{3} b}{16 d}+\frac {5 \sin \left (3 d x +3 c \right ) A \,a^{2} b^{2}}{8 d}+\frac {5 \sin \left (3 d x +3 c \right ) B \,a^{3} b}{12 d}+\frac {\sin \left (5 d x +5 c \right ) a^{4} C}{80 d}+\frac {a^{4} A \sin \left (7 d x +7 c \right )}{448 d}+\frac {x B \,b^{4}}{2}+\frac {7 a^{4} A \sin \left (5 d x +5 c \right )}{320 d}+\frac {\sin \left (3 d x +3 c \right ) A \,b^{4}}{12 d}+2 x C a \,b^{3}+\frac {3 \sin \left (4 d x +4 c \right ) B \,a^{4}}{64 d}+\frac {3 \sin \left (d x +c \right ) A \,b^{4}}{4 d}+\frac {\sin \left (d x +c \right ) C \,b^{4}}{d}+\frac {\sin \left (6 d x +6 c \right ) B \,a^{4}}{192 d}+\frac {3 A a \,b^{3} x}{2}+\frac {9 B \,a^{2} b^{2} x}{4}+\frac {3 C \,a^{3} b x}{2}\) | \(676\) |
1/6720*((1575*B*a^4+6300*(A+16/15*C)*b*a^3+10080*B*a^2*b^2+6720*b^3*(A+C)* a+1680*B*b^4)*sin(2*d*x+2*c)+((735*A+700*C)*a^4+2800*B*a^3*b+4200*b^2*(A+4 /5*C)*a^2+2240*B*a*b^3+560*A*b^4)*sin(3*d*x+3*c)+1260*(1/4*B*a^3+b*(A+2/3* C)*a^2+B*a*b^2+2/3*A*b^3)*a*sin(4*d*x+4*c)+((147*A+84*C)*a^4+336*B*a^3*b+5 04*A*a^2*b^2)*sin(5*d*x+5*c)+(140*A*a^3*b+35*B*a^4)*sin(6*d*x+6*c)+15*a^4* A*sin(7*d*x+7*c)+((3675*A+4200*C)*a^4+16800*B*a^3*b+25200*b^2*(A+6/5*C)*a^ 2+20160*B*a*b^3+5040*b^4*(A+4/3*C))*sin(d*x+c)+8400*d*(1/4*B*a^4+(A+6/5*C) *a^3*b+9/5*B*a^2*b^2+6/5*a*(A+4/3*C)*b^3+2/5*B*b^4)*x)/d
Time = 0.32 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.81 \[ \int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (5 \, B a^{4} + 4 \, {\left (5 \, A + 6 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 8 \, {\left (3 \, A + 4 \, C\right )} a b^{3} + 8 \, B b^{4}\right )} d x + {\left (240 \, A a^{4} \cos \left (d x + c\right )^{6} + 280 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{5} + 128 \, {\left (6 \, A + 7 \, C\right )} a^{4} + 3584 \, B a^{3} b + 1344 \, {\left (4 \, A + 5 \, C\right )} a^{2} b^{2} + 4480 \, B a b^{3} + 560 \, {\left (2 \, A + 3 \, C\right )} b^{4} + 48 \, {\left ({\left (6 \, A + 7 \, C\right )} a^{4} + 28 \, B a^{3} b + 42 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} + 70 \, {\left (5 \, B a^{4} + 4 \, {\left (5 \, A + 6 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 24 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (4 \, {\left (6 \, A + 7 \, C\right )} a^{4} + 112 \, B a^{3} b + 42 \, {\left (4 \, A + 5 \, C\right )} a^{2} b^{2} + 140 \, B a b^{3} + 35 \, A b^{4}\right )} \cos \left (d x + c\right )^{2} + 105 \, {\left (5 \, B a^{4} + 4 \, {\left (5 \, A + 6 \, C\right )} a^{3} b + 36 \, B a^{2} b^{2} + 8 \, {\left (3 \, A + 4 \, C\right )} a b^{3} + 8 \, B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, d} \]
integrate(cos(d*x+c)^7*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="fricas")
1/1680*(105*(5*B*a^4 + 4*(5*A + 6*C)*a^3*b + 36*B*a^2*b^2 + 8*(3*A + 4*C)* a*b^3 + 8*B*b^4)*d*x + (240*A*a^4*cos(d*x + c)^6 + 280*(B*a^4 + 4*A*a^3*b) *cos(d*x + c)^5 + 128*(6*A + 7*C)*a^4 + 3584*B*a^3*b + 1344*(4*A + 5*C)*a^ 2*b^2 + 4480*B*a*b^3 + 560*(2*A + 3*C)*b^4 + 48*((6*A + 7*C)*a^4 + 28*B*a^ 3*b + 42*A*a^2*b^2)*cos(d*x + c)^4 + 70*(5*B*a^4 + 4*(5*A + 6*C)*a^3*b + 3 6*B*a^2*b^2 + 24*A*a*b^3)*cos(d*x + c)^3 + 16*(4*(6*A + 7*C)*a^4 + 112*B*a ^3*b + 42*(4*A + 5*C)*a^2*b^2 + 140*B*a*b^3 + 35*A*b^4)*cos(d*x + c)^2 + 1 05*(5*B*a^4 + 4*(5*A + 6*C)*a^3*b + 36*B*a^2*b^2 + 8*(3*A + 4*C)*a*b^3 + 8 *B*b^4)*cos(d*x + c))*sin(d*x + c))/d
Timed out. \[ \int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Time = 0.24 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.14 \[ \int \cos ^7(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 {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 )} A a^{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 a^{4} - 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 )} C a^{4} + 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 )} A a^{3} b - 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^{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 - 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 )} 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} + 13440 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \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} + 8960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a b^{3} - 6720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{3} + 2240 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{4} - 1680 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b^{4} - 6720 \, C b^{4} \sin \left (d x + c\right )}{6720 \, d} \]
integrate(cos(d*x+c)^7*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="maxima")
-1/6720*(192*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 3 5*sin(d*x + c))*A*a^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*a^4 - 448*(3*sin(d*x + c)^5 - 10*sin( d*x + c)^3 + 15*sin(d*x + c))*C*a^4 + 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))*A*a^3*b - 1792*(3*sin(d* x + c)^5 - 10*sin(d*x + c)^3 + 15*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 - 2688*(3*sin(d*x + c) ^5 - 10*sin(d*x + c)^3 + 15*sin(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 + 13440*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^2*b^2 - 840*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*s in(2*d*x + 2*c))*A*a*b^3 + 8960*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a*b^3 - 6720*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a*b^3 + 2240*(sin(d*x + c)^3 - 3 *sin(d*x + c))*A*b^4 - 1680*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*b^4 - 6720* C*b^4*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 1815 vs. \(2 (423) = 846\).
Time = 0.40 (sec) , antiderivative size = 1815, normalized size of antiderivative = 4.14 \[ \int \cos ^7(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
integrate(cos(d*x+c)^7*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="giac")
1/1680*(105*(5*B*a^4 + 20*A*a^3*b + 24*C*a^3*b + 36*B*a^2*b^2 + 24*A*a*b^3 + 32*C*a*b^3 + 8*B*b^4)*(d*x + c) + 2*(1680*A*a^4*tan(1/2*d*x + 1/2*c)^13 - 1155*B*a^4*tan(1/2*d*x + 1/2*c)^13 + 1680*C*a^4*tan(1/2*d*x + 1/2*c)^13 - 4620*A*a^3*b*tan(1/2*d*x + 1/2*c)^13 + 6720*B*a^3*b*tan(1/2*d*x + 1/2*c )^13 - 4200*C*a^3*b*tan(1/2*d*x + 1/2*c)^13 + 10080*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^13 - 6300*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^13 + 10080*C*a^2*b^2*tan (1/2*d*x + 1/2*c)^13 - 4200*A*a*b^3*tan(1/2*d*x + 1/2*c)^13 + 6720*B*a*b^3 *tan(1/2*d*x + 1/2*c)^13 - 3360*C*a*b^3*tan(1/2*d*x + 1/2*c)^13 + 1680*A*b ^4*tan(1/2*d*x + 1/2*c)^13 - 840*B*b^4*tan(1/2*d*x + 1/2*c)^13 + 1680*C*b^ 4*tan(1/2*d*x + 1/2*c)^13 + 3360*A*a^4*tan(1/2*d*x + 1/2*c)^11 - 980*B*a^4 *tan(1/2*d*x + 1/2*c)^11 + 5600*C*a^4*tan(1/2*d*x + 1/2*c)^11 - 3920*A*a^3 *b*tan(1/2*d*x + 1/2*c)^11 + 22400*B*a^3*b*tan(1/2*d*x + 1/2*c)^11 - 10080 *C*a^3*b*tan(1/2*d*x + 1/2*c)^11 + 33600*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 - 15120*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 + 47040*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 - 10080*A*a*b^3*tan(1/2*d*x + 1/2*c)^11 + 31360*B*a*b^3*tan(1/2 *d*x + 1/2*c)^11 - 13440*C*a*b^3*tan(1/2*d*x + 1/2*c)^11 + 7840*A*b^4*tan( 1/2*d*x + 1/2*c)^11 - 3360*B*b^4*tan(1/2*d*x + 1/2*c)^11 + 10080*C*b^4*tan (1/2*d*x + 1/2*c)^11 + 14448*A*a^4*tan(1/2*d*x + 1/2*c)^9 - 2975*B*a^4*tan (1/2*d*x + 1/2*c)^9 + 12656*C*a^4*tan(1/2*d*x + 1/2*c)^9 - 11900*A*a^3*b*t an(1/2*d*x + 1/2*c)^9 + 50624*B*a^3*b*tan(1/2*d*x + 1/2*c)^9 - 7560*C*a...
Time = 21.87 (sec) , antiderivative size = 675, normalized size of antiderivative = 1.54 \[ \int \cos ^7(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\,B\,a^4\,x}{16}+\frac {B\,b^4\,x}{2}+\frac {3\,A\,a\,b^3\,x}{2}+\frac {5\,A\,a^3\,b\,x}{4}+2\,C\,a\,b^3\,x+\frac {3\,C\,a^3\,b\,x}{2}+\frac {35\,A\,a^4\,\sin \left (c+d\,x\right )}{64\,d}+\frac {3\,A\,b^4\,\sin \left (c+d\,x\right )}{4\,d}+\frac {5\,C\,a^4\,\sin \left (c+d\,x\right )}{8\,d}+\frac {C\,b^4\,\sin \left (c+d\,x\right )}{d}+\frac {9\,B\,a^2\,b^2\,x}{4}+\frac {7\,A\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{64\,d}+\frac {7\,A\,a^4\,\sin \left (5\,c+5\,d\,x\right )}{320\,d}+\frac {A\,a^4\,\sin \left (7\,c+7\,d\,x\right )}{448\,d}+\frac {15\,B\,a^4\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {A\,b^4\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {3\,B\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {B\,a^4\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {B\,b^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {5\,C\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {C\,a^4\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {A\,a\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {15\,A\,a^3\,b\,\sin \left (2\,c+2\,d\,x\right )}{16\,d}+\frac {A\,a\,b^3\,\sin \left (4\,c+4\,d\,x\right )}{8\,d}+\frac {3\,A\,a^3\,b\,\sin \left (4\,c+4\,d\,x\right )}{16\,d}+\frac {A\,a^3\,b\,\sin \left (6\,c+6\,d\,x\right )}{48\,d}+\frac {15\,A\,a^2\,b^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {B\,a\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{3\,d}+\frac {5\,B\,a^3\,b\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {B\,a^3\,b\,\sin \left (5\,c+5\,d\,x\right )}{20\,d}+\frac {C\,a\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {C\,a^3\,b\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {C\,a^3\,b\,\sin \left (4\,c+4\,d\,x\right )}{8\,d}+\frac {9\,C\,a^2\,b^2\,\sin \left (c+d\,x\right )}{2\,d}+\frac {5\,A\,a^2\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{8\,d}+\frac {3\,A\,a^2\,b^2\,\sin \left (5\,c+5\,d\,x\right )}{40\,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 {C\,a^2\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{2\,d}+\frac {3\,B\,a\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {5\,B\,a^3\,b\,\sin \left (c+d\,x\right )}{2\,d} \]
(5*B*a^4*x)/16 + (B*b^4*x)/2 + (3*A*a*b^3*x)/2 + (5*A*a^3*b*x)/4 + 2*C*a*b ^3*x + (3*C*a^3*b*x)/2 + (35*A*a^4*sin(c + d*x))/(64*d) + (3*A*b^4*sin(c + d*x))/(4*d) + (5*C*a^4*sin(c + d*x))/(8*d) + (C*b^4*sin(c + d*x))/d + (9* B*a^2*b^2*x)/4 + (7*A*a^4*sin(3*c + 3*d*x))/(64*d) + (7*A*a^4*sin(5*c + 5* d*x))/(320*d) + (A*a^4*sin(7*c + 7*d*x))/(448*d) + (15*B*a^4*sin(2*c + 2*d *x))/(64*d) + (A*b^4*sin(3*c + 3*d*x))/(12*d) + (3*B*a^4*sin(4*c + 4*d*x)) /(64*d) + (B*a^4*sin(6*c + 6*d*x))/(192*d) + (B*b^4*sin(2*c + 2*d*x))/(4*d ) + (5*C*a^4*sin(3*c + 3*d*x))/(48*d) + (C*a^4*sin(5*c + 5*d*x))/(80*d) + (A*a*b^3*sin(2*c + 2*d*x))/d + (15*A*a^3*b*sin(2*c + 2*d*x))/(16*d) + (A*a *b^3*sin(4*c + 4*d*x))/(8*d) + (3*A*a^3*b*sin(4*c + 4*d*x))/(16*d) + (A*a^ 3*b*sin(6*c + 6*d*x))/(48*d) + (15*A*a^2*b^2*sin(c + d*x))/(4*d) + (B*a*b^ 3*sin(3*c + 3*d*x))/(3*d) + (5*B*a^3*b*sin(3*c + 3*d*x))/(12*d) + (B*a^3*b *sin(5*c + 5*d*x))/(20*d) + (C*a*b^3*sin(2*c + 2*d*x))/d + (C*a^3*b*sin(2* c + 2*d*x))/d + (C*a^3*b*sin(4*c + 4*d*x))/(8*d) + (9*C*a^2*b^2*sin(c + d* x))/(2*d) + (5*A*a^2*b^2*sin(3*c + 3*d*x))/(8*d) + (3*A*a^2*b^2*sin(5*c + 5*d*x))/(40*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) + (C*a^2*b^2*sin(3*c + 3*d*x))/(2*d) + (3*B*a*b^3*sin( c + d*x))/d + (5*B*a^3*b*sin(c + d*x))/(2*d)