Integrand size = 41, antiderivative size = 314 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=b^4 C x+\frac {\left (3 a^4 B+24 a^2 b^2 B+8 b^4 B+16 a b^3 (A+2 C)+4 a^3 b (3 A+4 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (12 A b^4+80 a^3 b B+95 a b^3 B+4 a^4 (4 A+5 C)+2 a^2 b^2 (56 A+85 C)\right ) \tan (c+d x)}{30 d}+\frac {a \left (24 A b^3+45 a^3 B+130 a b^2 B+4 a^2 b (29 A+40 C)\right ) \sec (c+d x) \tan (c+d x)}{120 d}+\frac {\left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)\right ) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{60 d}+\frac {(4 A b+5 a B) (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{20 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^4(c+d x) \tan (c+d x)}{5 d} \]
b^4*C*x+1/8*(3*B*a^4+24*B*a^2*b^2+8*B*b^4+16*a*b^3*(A+2*C)+4*a^3*b*(3*A+4* C))*arctanh(sin(d*x+c))/d+1/30*(12*A*b^4+80*B*a^3*b+95*B*a*b^3+4*a^4*(4*A+ 5*C)+2*a^2*b^2*(56*A+85*C))*tan(d*x+c)/d+1/120*a*(24*A*b^3+45*B*a^3+130*B* a*b^2+4*a^2*b*(29*A+40*C))*sec(d*x+c)*tan(d*x+c)/d+1/60*(12*A*b^2+35*B*a*b +4*a^2*(4*A+5*C))*(a+b*cos(d*x+c))^2*sec(d*x+c)^2*tan(d*x+c)/d+1/20*(4*A*b +5*B*a)*(a+b*cos(d*x+c))^3*sec(d*x+c)^3*tan(d*x+c)/d+1/5*A*(a+b*cos(d*x+c) )^4*sec(d*x+c)^4*tan(d*x+c)/d
Time = 2.14 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.73 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {120 b^4 C d x+15 \left (3 a^4 B+24 a^2 b^2 B+8 b^4 B+16 a b^3 (A+2 C)+4 a^3 b (3 A+4 C)\right ) \text {arctanh}(\sin (c+d x))+15 \left (8 \left (A b^4+4 a^3 b B+4 a b^3 B+a^4 (A+C)+6 a^2 b^2 (A+C)\right )+a \left (16 A b^3+3 a^3 B+24 a b^2 B+4 a^2 b (3 A+4 C)\right ) \sec (c+d x)+2 a^3 (4 A b+a B) \sec ^3(c+d x)\right ) \tan (c+d x)+40 a^2 \left (6 A b^2+4 a b B+a^2 (2 A+C)\right ) \tan ^3(c+d x)+24 a^4 A \tan ^5(c+d x)}{120 d} \]
(120*b^4*C*d*x + 15*(3*a^4*B + 24*a^2*b^2*B + 8*b^4*B + 16*a*b^3*(A + 2*C) + 4*a^3*b*(3*A + 4*C))*ArcTanh[Sin[c + d*x]] + 15*(8*(A*b^4 + 4*a^3*b*B + 4*a*b^3*B + a^4*(A + C) + 6*a^2*b^2*(A + C)) + a*(16*A*b^3 + 3*a^3*B + 24 *a*b^2*B + 4*a^2*b*(3*A + 4*C))*Sec[c + d*x] + 2*a^3*(4*A*b + a*B)*Sec[c + d*x]^3)*Tan[c + d*x] + 40*a^2*(6*A*b^2 + 4*a*b*B + a^2*(2*A + C))*Tan[c + d*x]^3 + 24*a^4*A*Tan[c + d*x]^5)/(120*d)
Time = 2.24 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.06, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.390, Rules used = {3042, 3526, 3042, 3526, 3042, 3526, 3042, 3510, 25, 3042, 3500, 27, 3042, 3214, 3042, 4257}
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.
| \(\displaystyle \int \sec ^6(c+d x) (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{5} \int (a+b \cos (c+d x))^3 \left (5 b C \cos ^2(c+d x)+(4 a A+5 b B+5 a C) \cos (c+d x)+4 A b+5 a B\right ) \sec ^5(c+d x)dx+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (5 b C \sin \left (c+d x+\frac {\pi }{2}\right )^2+(4 a A+5 b B+5 a C) \sin \left (c+d x+\frac {\pi }{2}\right )+4 A b+5 a B\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int (a+b \cos (c+d x))^2 \left (4 (4 A+5 C) a^2+35 b B a+12 A b^2+20 b^2 C \cos ^2(c+d x)+\left (15 B a^2+28 A b a+40 b C a+20 b^2 B\right ) \cos (c+d x)\right ) \sec ^4(c+d x)dx+\frac {(5 a B+4 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (4 (4 A+5 C) a^2+35 b B a+12 A b^2+20 b^2 C \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (15 B a^2+28 A b a+40 b C a+20 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {(5 a B+4 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int (a+b \cos (c+d x)) \left (45 B a^3+4 b (29 A+40 C) a^2+130 b^2 B a+24 A b^3+60 b^3 C \cos ^2(c+d x)+\left (8 (4 A+5 C) a^3+115 b B a^2+36 b^2 (3 A+5 C) a+60 b^3 B\right ) \cos (c+d x)\right ) \sec ^3(c+d x)dx+\frac {\tan (c+d x) \sec ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (45 B a^3+4 b (29 A+40 C) a^2+130 b^2 B a+24 A b^3+60 b^3 C \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (8 (4 A+5 C) a^3+115 b B a^2+36 b^2 (3 A+5 C) a+60 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {\tan (c+d x) \sec ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {a \tan (c+d x) \sec (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{2 d}-\frac {1}{2} \int -\left (\left (120 C \cos ^2(c+d x) b^4+4 \left (4 (4 A+5 C) a^4+80 b B a^3+2 b^2 (56 A+85 C) a^2+95 b^3 B a+12 A b^4\right )+15 \left (3 B a^4+4 b (3 A+4 C) a^3+24 b^2 B a^2+16 b^3 (A+2 C) a+8 b^4 B\right ) \cos (c+d x)\right ) \sec ^2(c+d x)\right )dx\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \left (120 C \cos ^2(c+d x) b^4+4 \left (4 (4 A+5 C) a^4+80 b B a^3+2 b^2 (56 A+85 C) a^2+95 b^3 B a+12 A b^4\right )+15 \left (3 B a^4+4 b (3 A+4 C) a^3+24 b^2 B a^2+16 b^3 (A+2 C) a+8 b^4 B\right ) \cos (c+d x)\right ) \sec ^2(c+d x)dx+\frac {a \tan (c+d x) \sec (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{2 d}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {120 C \sin \left (c+d x+\frac {\pi }{2}\right )^2 b^4+4 \left (4 (4 A+5 C) a^4+80 b B a^3+2 b^2 (56 A+85 C) a^2+95 b^3 B a+12 A b^4\right )+15 \left (3 B a^4+4 b (3 A+4 C) a^3+24 b^2 B a^2+16 b^3 (A+2 C) a+8 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a \tan (c+d x) \sec (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{2 d}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int 15 \left (3 B a^4+4 b (3 A+4 C) a^3+24 b^2 B a^2+16 b^3 (A+2 C) a+8 b^4 B+8 b^4 C \cos (c+d x)\right ) \sec (c+d x)dx+\frac {4 \tan (c+d x) \left (4 a^4 (4 A+5 C)+80 a^3 b B+2 a^2 b^2 (56 A+85 C)+95 a b^3 B+12 A b^4\right )}{d}\right )+\frac {a \tan (c+d x) \sec (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{2 d}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 \int \left (3 B a^4+4 b (3 A+4 C) a^3+24 b^2 B a^2+16 b^3 (A+2 C) a+8 b^4 B+8 b^4 C \cos (c+d x)\right ) \sec (c+d x)dx+\frac {4 \tan (c+d x) \left (4 a^4 (4 A+5 C)+80 a^3 b B+2 a^2 b^2 (56 A+85 C)+95 a b^3 B+12 A b^4\right )}{d}\right )+\frac {a \tan (c+d x) \sec (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{2 d}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 \int \frac {3 B a^4+4 b (3 A+4 C) a^3+24 b^2 B a^2+16 b^3 (A+2 C) a+8 b^4 B+8 b^4 C \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {4 \tan (c+d x) \left (4 a^4 (4 A+5 C)+80 a^3 b B+2 a^2 b^2 (56 A+85 C)+95 a b^3 B+12 A b^4\right )}{d}\right )+\frac {a \tan (c+d x) \sec (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{2 d}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 \left (\left (3 a^4 B+4 a^3 b (3 A+4 C)+24 a^2 b^2 B+16 a b^3 (A+2 C)+8 b^4 B\right ) \int \sec (c+d x)dx+8 b^4 C x\right )+\frac {4 \tan (c+d x) \left (4 a^4 (4 A+5 C)+80 a^3 b B+2 a^2 b^2 (56 A+85 C)+95 a b^3 B+12 A b^4\right )}{d}\right )+\frac {a \tan (c+d x) \sec (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{2 d}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (15 \left (\left (3 a^4 B+4 a^3 b (3 A+4 C)+24 a^2 b^2 B+16 a b^3 (A+2 C)+8 b^4 B\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+8 b^4 C x\right )+\frac {4 \tan (c+d x) \left (4 a^4 (4 A+5 C)+80 a^3 b B+2 a^2 b^2 (56 A+85 C)+95 a b^3 B+12 A b^4\right )}{d}\right )+\frac {a \tan (c+d x) \sec (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{2 d}\right )+\frac {\tan (c+d x) \sec ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {(5 a B+4 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {\tan (c+d x) \sec ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \cos (c+d x))^2}{3 d}+\frac {1}{3} \left (\frac {a \tan (c+d x) \sec (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{2 d}+\frac {1}{2} \left (15 \left (\frac {\left (3 a^4 B+4 a^3 b (3 A+4 C)+24 a^2 b^2 B+16 a b^3 (A+2 C)+8 b^4 B\right ) \text {arctanh}(\sin (c+d x))}{d}+8 b^4 C x\right )+\frac {4 \tan (c+d x) \left (4 a^4 (4 A+5 C)+80 a^3 b B+2 a^2 b^2 (56 A+85 C)+95 a b^3 B+12 A b^4\right )}{d}\right )\right )\right )+\frac {(5 a B+4 A b) \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}\right )+\frac {A \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^4}{5 d}\) |
(A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + (((4*A*b + 5*a*B)*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + (((12*A *b^2 + 35*a*b*B + 4*a^2*(4*A + 5*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^2 *Tan[c + d*x])/(3*d) + ((a*(24*A*b^3 + 45*a^3*B + 130*a*b^2*B + 4*a^2*b*(2 9*A + 40*C))*Sec[c + d*x]*Tan[c + d*x])/(2*d) + (15*(8*b^4*C*x + ((3*a^4*B + 24*a^2*b^2*B + 8*b^4*B + 16*a*b^3*(A + 2*C) + 4*a^3*b*(3*A + 4*C))*ArcT anh[Sin[c + d*x]])/d) + (4*(12*A*b^4 + 80*a^3*b*B + 95*a*b^3*B + 4*a^4*(4* A + 5*C) + 2*a^2*b^2*(56*A + 85*C))*Tan[c + d*x])/d)/2)/3)/4)/5
3.10.71.3.1 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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d Int[1/(c + d *Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* (a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x ])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A *b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
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[(-(b*c - a*d))*(A*b^2 - a*b*B + a^2*C)*Cos[ e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b^2*f*(m + 1)*(a^2 - b^2))), x] - S imp[1/(b^2*(m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[b*( m + 1)*((b*B - a*C)*(b*c - a*d) - A*b*(a*c - b*d)) + (b*B*(a^2*d + b^2*d*(m + 1) - a*b*c*(m + 2)) + (b*c - a*d)*(A*b^2*(m + 2) + C*(a^2 + b^2*(m + 1)) ))*Sin[e + f*x] - b*C*d*(m + 1)*(a^2 - b^2)*Sin[e + f*x]^2, x], x], x] /; F reeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
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^2*C - B*c*d + A*d^2))*Cos[e + f*x ]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 - d^2))), x] + Simp[1/(d*(n + 1)*(c^2 - d^2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x ] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f *x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d , 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -1]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.75 (sec) , antiderivative size = 280, normalized size of antiderivative = 0.89
| method | result | size |
| parts | \(-\frac {A \,a^{4} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {\left (B \,b^{4}+4 C a \,b^{3}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (A \,b^{4}+4 B a \,b^{3}+6 C \,a^{2} b^{2}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 a A \,b^{3}+6 B \,a^{2} b^{2}+4 a^{3} b C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b +a^{4} C \right ) \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {C \,b^{4} \left (d x +c \right )}{d}\) | \(280\) |
| derivativedivides | \(\frac {-A \,a^{4} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+B \,a^{4} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-a^{4} C \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 A \,a^{3} b \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 B \,a^{3} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 a^{3} b C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-6 A \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+6 B \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 C \,a^{2} b^{2} \tan \left (d x +c \right )+4 a A \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 B a \,b^{3} \tan \left (d x +c \right )+4 C a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{4} \tan \left (d x +c \right )+B \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \,b^{4} \left (d x +c \right )}{d}\) | \(421\) |
| default | \(\frac {-A \,a^{4} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+B \,a^{4} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-a^{4} C \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 A \,a^{3} b \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-4 B \,a^{3} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 a^{3} b C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-6 A \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+6 B \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 C \,a^{2} b^{2} \tan \left (d x +c \right )+4 a A \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 B a \,b^{3} \tan \left (d x +c \right )+4 C a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{4} \tan \left (d x +c \right )+B \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \,b^{4} \left (d x +c \right )}{d}\) | \(421\) |
| parallelrisch | \(\frac {-900 \left (\frac {\cos \left (5 d x +5 c \right )}{5}+\cos \left (3 d x +3 c \right )+2 \cos \left (d x +c \right )\right ) \left (\frac {B \,a^{4}}{4}+b \left (A +\frac {4 C}{3}\right ) a^{3}+2 B \,a^{2} b^{2}+\frac {4 a \,b^{3} \left (A +2 C \right )}{3}+\frac {2 B \,b^{4}}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+900 \left (\frac {\cos \left (5 d x +5 c \right )}{5}+\cos \left (3 d x +3 c \right )+2 \cos \left (d x +c \right )\right ) \left (\frac {B \,a^{4}}{4}+b \left (A +\frac {4 C}{3}\right ) a^{3}+2 B \,a^{2} b^{2}+\frac {4 a \,b^{3} \left (A +2 C \right )}{3}+\frac {2 B \,b^{4}}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+600 C \,b^{4} d x \cos \left (3 d x +3 c \right )+120 C \,b^{4} d x \cos \left (5 d x +5 c \right )+\left (\left (320 A +400 C \right ) a^{4}+1600 B \,a^{3} b +2400 \left (A +\frac {9 C}{10}\right ) b^{2} a^{2}+1440 B a \,b^{3}+360 A \,b^{4}\right ) \sin \left (3 d x +3 c \right )+\left (\left (64 A +80 C \right ) a^{4}+320 B \,a^{3} b +480 \left (A +\frac {3 C}{2}\right ) b^{2} a^{2}+480 B a \,b^{3}+120 A \,b^{4}\right ) \sin \left (5 d x +5 c \right )+1680 \left (\frac {B \,a^{3}}{4}+\left (A +\frac {4 C}{7}\right ) a^{2} b +\frac {6 B a \,b^{2}}{7}+\frac {4 A \,b^{3}}{7}\right ) a \sin \left (2 d x +2 c \right )+360 \left (\frac {B \,a^{3}}{4}+\left (A +\frac {4 C}{3}\right ) a^{2} b +2 B a \,b^{2}+\frac {4 A \,b^{3}}{3}\right ) a \sin \left (4 d x +4 c \right )+1200 C \,b^{4} d x \cos \left (d x +c \right )+640 \sin \left (d x +c \right ) \left (a^{4} \left (A +\frac {C}{2}\right )+2 B \,a^{3} b +3 \left (A +\frac {3 C}{4}\right ) a^{2} b^{2}+\frac {3 B a \,b^{3}}{2}+\frac {3 A \,b^{4}}{8}\right )}{600 \left (\frac {\cos \left (5 d x +5 c \right )}{5}+\cos \left (3 d x +3 c \right )+2 \cos \left (d x +c \right )\right ) d}\) | \(503\) |
| risch | \(\text {Expression too large to display}\) | \(1096\) |
-A*a^4/d*(-8/15-1/5*sec(d*x+c)^4-4/15*sec(d*x+c)^2)*tan(d*x+c)+(4*A*a^3*b+ B*a^4)/d*(-(-1/4*sec(d*x+c)^3-3/8*sec(d*x+c))*tan(d*x+c)+3/8*ln(sec(d*x+c) +tan(d*x+c)))+(B*b^4+4*C*a*b^3)/d*ln(sec(d*x+c)+tan(d*x+c))+(A*b^4+4*B*a*b ^3+6*C*a^2*b^2)/d*tan(d*x+c)+(4*A*a*b^3+6*B*a^2*b^2+4*C*a^3*b)/d*(1/2*sec( d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))-(6*A*a^2*b^2+4*B*a^3*b+C* a^4)/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+C*b^4/d*(d*x+c)
Time = 0.31 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.08 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {240 \, C b^{4} d x \cos \left (d x + c\right )^{5} + 15 \, {\left (3 \, B a^{4} + 4 \, {\left (3 \, A + 4 \, C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 16 \, {\left (A + 2 \, C\right )} a b^{3} + 8 \, B b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) - 15 \, {\left (3 \, B a^{4} + 4 \, {\left (3 \, A + 4 \, C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 16 \, {\left (A + 2 \, C\right )} a b^{3} + 8 \, B b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (24 \, A a^{4} + 8 \, {\left (2 \, {\left (4 \, A + 5 \, C\right )} a^{4} + 40 \, B a^{3} b + 30 \, {\left (2 \, A + 3 \, C\right )} a^{2} b^{2} + 60 \, B a b^{3} + 15 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} + 15 \, {\left (3 \, B a^{4} + 4 \, {\left (3 \, A + 4 \, C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 16 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left ({\left (4 \, A + 5 \, C\right )} a^{4} + 20 \, B a^{3} b + 30 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 30 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d \cos \left (d x + c\right )^{5}} \]
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6, x, algorithm="fricas")
1/240*(240*C*b^4*d*x*cos(d*x + c)^5 + 15*(3*B*a^4 + 4*(3*A + 4*C)*a^3*b + 24*B*a^2*b^2 + 16*(A + 2*C)*a*b^3 + 8*B*b^4)*cos(d*x + c)^5*log(sin(d*x + c) + 1) - 15*(3*B*a^4 + 4*(3*A + 4*C)*a^3*b + 24*B*a^2*b^2 + 16*(A + 2*C)* a*b^3 + 8*B*b^4)*cos(d*x + c)^5*log(-sin(d*x + c) + 1) + 2*(24*A*a^4 + 8*( 2*(4*A + 5*C)*a^4 + 40*B*a^3*b + 30*(2*A + 3*C)*a^2*b^2 + 60*B*a*b^3 + 15* A*b^4)*cos(d*x + c)^4 + 15*(3*B*a^4 + 4*(3*A + 4*C)*a^3*b + 24*B*a^2*b^2 + 16*A*a*b^3)*cos(d*x + c)^3 + 8*((4*A + 5*C)*a^4 + 20*B*a^3*b + 30*A*a^2*b ^2)*cos(d*x + c)^2 + 30*(B*a^4 + 4*A*a^3*b)*cos(d*x + c))*sin(d*x + c))/(d *cos(d*x + c)^5)
Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\text {Timed out} \]
Time = 0.24 (sec) , antiderivative size = 511, normalized size of antiderivative = 1.63 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {16 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A a^{4} + 80 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} + 320 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} b + 480 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{2} b^{2} + 240 \, {\left (d x + c\right )} C b^{4} - 15 \, B a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 60 \, A a^{3} b {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 240 \, C a^{3} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 360 \, B a^{2} b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 240 \, A a b^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 480 \, C a b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 120 \, B b^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 1440 \, C a^{2} b^{2} \tan \left (d x + c\right ) + 960 \, B a b^{3} \tan \left (d x + c\right ) + 240 \, A b^{4} \tan \left (d x + c\right )}{240 \, d} \]
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6, x, algorithm="maxima")
1/240*(16*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*A*a^4 + 80*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^4 + 320*(tan(d*x + c)^3 + 3*tan( d*x + c))*B*a^3*b + 480*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^2*b^2 + 240* (d*x + c)*C*b^4 - 15*B*a^4*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 60*A*a^3*b*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c ) - 1)) - 240*C*a^3*b*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 360*B*a^2*b^2*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) - 240*A*a*b^ 3*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d *x + c) - 1)) + 480*C*a*b^3*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1) ) + 120*B*b^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 1440*C*a^2 *b^2*tan(d*x + c) + 960*B*a*b^3*tan(d*x + c) + 240*A*b^4*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 1140 vs. \(2 (302) = 604\).
Time = 0.41 (sec) , antiderivative size = 1140, normalized size of antiderivative = 3.63 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\text {Too large to display} \]
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6, x, algorithm="giac")
1/120*(120*(d*x + c)*C*b^4 + 15*(3*B*a^4 + 12*A*a^3*b + 16*C*a^3*b + 24*B* a^2*b^2 + 16*A*a*b^3 + 32*C*a*b^3 + 8*B*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 15*(3*B*a^4 + 12*A*a^3*b + 16*C*a^3*b + 24*B*a^2*b^2 + 16*A*a*b^3 + 32*C*a*b^3 + 8*B*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(120*A*a^4* tan(1/2*d*x + 1/2*c)^9 - 75*B*a^4*tan(1/2*d*x + 1/2*c)^9 + 120*C*a^4*tan(1 /2*d*x + 1/2*c)^9 - 300*A*a^3*b*tan(1/2*d*x + 1/2*c)^9 + 480*B*a^3*b*tan(1 /2*d*x + 1/2*c)^9 - 240*C*a^3*b*tan(1/2*d*x + 1/2*c)^9 + 720*A*a^2*b^2*tan (1/2*d*x + 1/2*c)^9 - 360*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 720*C*a^2*b^2 *tan(1/2*d*x + 1/2*c)^9 - 240*A*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 480*B*a*b^3 *tan(1/2*d*x + 1/2*c)^9 + 120*A*b^4*tan(1/2*d*x + 1/2*c)^9 - 160*A*a^4*tan (1/2*d*x + 1/2*c)^7 + 30*B*a^4*tan(1/2*d*x + 1/2*c)^7 - 320*C*a^4*tan(1/2* d*x + 1/2*c)^7 + 120*A*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 1280*B*a^3*b*tan(1/2 *d*x + 1/2*c)^7 + 480*C*a^3*b*tan(1/2*d*x + 1/2*c)^7 - 1920*A*a^2*b^2*tan( 1/2*d*x + 1/2*c)^7 + 720*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 2880*C*a^2*b^2 *tan(1/2*d*x + 1/2*c)^7 + 480*A*a*b^3*tan(1/2*d*x + 1/2*c)^7 - 1920*B*a*b^ 3*tan(1/2*d*x + 1/2*c)^7 - 480*A*b^4*tan(1/2*d*x + 1/2*c)^7 + 464*A*a^4*ta n(1/2*d*x + 1/2*c)^5 + 400*C*a^4*tan(1/2*d*x + 1/2*c)^5 + 1600*B*a^3*b*tan (1/2*d*x + 1/2*c)^5 + 2400*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 4320*C*a^2*b ^2*tan(1/2*d*x + 1/2*c)^5 + 2880*B*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 720*A*b^ 4*tan(1/2*d*x + 1/2*c)^5 - 160*A*a^4*tan(1/2*d*x + 1/2*c)^3 - 30*B*a^4*...
Time = 6.02 (sec) , antiderivative size = 4068, normalized size of antiderivative = 12.96 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\text {Too large to display} \]
(atan(((tan(c/2 + (d*x)/2)*((9*B^2*a^8)/2 + 32*B^2*b^8 + 32*C^2*b^8 + 128* A^2*a^2*b^6 + 192*A^2*a^4*b^4 + 72*A^2*a^6*b^2 + 192*B^2*a^2*b^6 + 312*B^2 *a^4*b^4 + 72*B^2*a^6*b^2 + 512*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 128*C^2*a^ 6*b^2 + 128*A*B*a*b^7 + 36*A*B*a^7*b + 256*B*C*a*b^7 + 48*B*C*a^7*b + 480* A*B*a^3*b^5 + 336*A*B*a^5*b^3 + 512*A*C*a^2*b^6 + 640*A*C*a^4*b^4 + 192*A* C*a^6*b^2 + 896*B*C*a^3*b^5 + 480*B*C*a^5*b^3) + ((3*B*a^4)/8 + B*b^4 + 3* B*a^2*b^2 + 2*A*a*b^3 + (3*A*a^3*b)/2 + 4*C*a*b^3 + 2*C*a^3*b)*(12*B*a^4 + 32*B*b^4 + 32*C*b^4 + 96*B*a^2*b^2 + 64*A*a*b^3 + 48*A*a^3*b + 128*C*a*b^ 3 + 64*C*a^3*b))*((3*B*a^4)/8 + B*b^4 + 3*B*a^2*b^2 + 2*A*a*b^3 + (3*A*a^3 *b)/2 + 4*C*a*b^3 + 2*C*a^3*b)*1i + (tan(c/2 + (d*x)/2)*((9*B^2*a^8)/2 + 3 2*B^2*b^8 + 32*C^2*b^8 + 128*A^2*a^2*b^6 + 192*A^2*a^4*b^4 + 72*A^2*a^6*b^ 2 + 192*B^2*a^2*b^6 + 312*B^2*a^4*b^4 + 72*B^2*a^6*b^2 + 512*C^2*a^2*b^6 + 512*C^2*a^4*b^4 + 128*C^2*a^6*b^2 + 128*A*B*a*b^7 + 36*A*B*a^7*b + 256*B* C*a*b^7 + 48*B*C*a^7*b + 480*A*B*a^3*b^5 + 336*A*B*a^5*b^3 + 512*A*C*a^2*b ^6 + 640*A*C*a^4*b^4 + 192*A*C*a^6*b^2 + 896*B*C*a^3*b^5 + 480*B*C*a^5*b^3 ) - ((3*B*a^4)/8 + B*b^4 + 3*B*a^2*b^2 + 2*A*a*b^3 + (3*A*a^3*b)/2 + 4*C*a *b^3 + 2*C*a^3*b)*(12*B*a^4 + 32*B*b^4 + 32*C*b^4 + 96*B*a^2*b^2 + 64*A*a* b^3 + 48*A*a^3*b + 128*C*a*b^3 + 64*C*a^3*b))*((3*B*a^4)/8 + B*b^4 + 3*B*a ^2*b^2 + 2*A*a*b^3 + (3*A*a^3*b)/2 + 4*C*a*b^3 + 2*C*a^3*b)*1i)/((tan(c/2 + (d*x)/2)*((9*B^2*a^8)/2 + 32*B^2*b^8 + 32*C^2*b^8 + 128*A^2*a^2*b^6 +...