Integrand size = 33, antiderivative size = 355 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \text {arctanh}(\sin (c+d x))}{4 d}+\frac {\left (35 b^4 (2 A+3 C)+84 a^2 b^2 (4 A+5 C)+8 a^4 (6 A+7 C)\right ) \tan (c+d x)}{105 d}+\frac {a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \sec (c+d x) \tan (c+d x)}{4 d}+\frac {\left (4 A b^4+4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)\right ) \sec ^2(c+d x) \tan (c+d x)}{105 d}+\frac {a b \left (6 A b^2+a^2 (103 A+126 C)\right ) \sec ^3(c+d x) \tan (c+d x)}{210 d}+\frac {\left (2 A b^2+a^2 (6 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^4(c+d x) \tan (c+d x)}{35 d}+\frac {2 A b (a+b \cos (c+d x))^3 \sec ^5(c+d x) \tan (c+d x)}{21 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^6(c+d x) \tan (c+d x)}{7 d} \]
1/4*a*b*(2*b^2*(3*A+4*C)+a^2*(5*A+6*C))*arctanh(sin(d*x+c))/d+1/105*(35*b^ 4*(2*A+3*C)+84*a^2*b^2*(4*A+5*C)+8*a^4*(6*A+7*C))*tan(d*x+c)/d+1/4*a*b*(2* b^2*(3*A+4*C)+a^2*(5*A+6*C))*sec(d*x+c)*tan(d*x+c)/d+1/105*(4*A*b^4+4*a^4* (6*A+7*C)+3*a^2*b^2*(50*A+63*C))*sec(d*x+c)^2*tan(d*x+c)/d+1/210*a*b*(6*A* b^2+a^2*(103*A+126*C))*sec(d*x+c)^3*tan(d*x+c)/d+1/35*(2*A*b^2+a^2*(6*A+7* C))*(a+b*cos(d*x+c))^2*sec(d*x+c)^4*tan(d*x+c)/d+2/21*A*b*(a+b*cos(d*x+c)) ^3*sec(d*x+c)^5*tan(d*x+c)/d+1/7*A*(a+b*cos(d*x+c))^4*sec(d*x+c)^6*tan(d*x +c)/d
Time = 2.22 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.66 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {105 a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \text {arctanh}(\sin (c+d x))+\tan (c+d x) \left (420 \left (a^4+6 a^2 b^2+b^4\right ) (A+C)+105 a b \left (2 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \sec (c+d x)+70 a b \left (6 A b^2+a^2 (5 A+6 C)\right ) \sec ^3(c+d x)+280 a^3 A b \sec ^5(c+d x)+140 \left (A b^4+6 a^2 b^2 (2 A+C)+a^4 (3 A+2 C)\right ) \tan ^2(c+d x)+84 a^2 \left (6 A b^2+a^2 (3 A+C)\right ) \tan ^4(c+d x)+60 a^4 A \tan ^6(c+d x)\right )}{420 d} \]
(105*a*b*(2*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*ArcTanh[Sin[c + d*x]] + Tan [c + d*x]*(420*(a^4 + 6*a^2*b^2 + b^4)*(A + C) + 105*a*b*(2*b^2*(3*A + 4*C ) + a^2*(5*A + 6*C))*Sec[c + d*x] + 70*a*b*(6*A*b^2 + a^2*(5*A + 6*C))*Sec [c + d*x]^3 + 280*a^3*A*b*Sec[c + d*x]^5 + 140*(A*b^4 + 6*a^2*b^2*(2*A + C ) + a^4*(3*A + 2*C))*Tan[c + d*x]^2 + 84*a^2*(6*A*b^2 + a^2*(3*A + C))*Tan [c + d*x]^4 + 60*a^4*A*Tan[c + d*x]^6))/(420*d)
Time = 2.55 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.98, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {3042, 3527, 3042, 3526, 27, 3042, 3526, 3042, 3510, 27, 3042, 3500, 27, 3042, 3227, 3042, 4254, 24, 4255, 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.
| \(\displaystyle \int \sec ^8(c+d x) (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^8}dx\) |
| \(\Big \downarrow \) 3527 |
| \(\displaystyle \frac {1}{7} \int (a+b \cos (c+d x))^3 \left (b (2 A+7 C) \cos ^2(c+d x)+a (6 A+7 C) \cos (c+d x)+4 A b\right ) \sec ^7(c+d x)dx+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (b (2 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a (6 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right )+4 A b\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^7}dx+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{6} \int 2 (a+b \cos (c+d x))^2 \left (b^2 (10 A+21 C) \cos ^2(c+d x)+2 a b (17 A+21 C) \cos (c+d x)+3 \left ((6 A+7 C) a^2+2 A b^2\right )\right ) \sec ^6(c+d x)dx+\frac {2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \int (a+b \cos (c+d x))^2 \left (b^2 (10 A+21 C) \cos ^2(c+d x)+2 a b (17 A+21 C) \cos (c+d x)+3 \left ((6 A+7 C) a^2+2 A b^2\right )\right ) \sec ^6(c+d x)dx+\frac {2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (b^2 (10 A+21 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+2 a b (17 A+21 C) \sin \left (c+d x+\frac {\pi }{2}\right )+3 \left ((6 A+7 C) a^2+2 A b^2\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^6}dx+\frac {2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3526 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \int (a+b \cos (c+d x)) \left (b \left (6 (6 A+7 C) a^2+b^2 (62 A+105 C)\right ) \cos ^2(c+d x)+a \left (12 (6 A+7 C) a^2+b^2 (244 A+315 C)\right ) \cos (c+d x)+2 b \left ((103 A+126 C) a^2+6 A b^2\right )\right ) \sec ^5(c+d x)dx+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (b \left (6 (6 A+7 C) a^2+b^2 (62 A+105 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+a \left (12 (6 A+7 C) a^2+b^2 (244 A+315 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+2 b \left ((103 A+126 C) a^2+6 A b^2\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3510 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 d}-\frac {1}{4} \int -2 \left (2 b^2 \left (6 (6 A+7 C) a^2+b^2 (62 A+105 C)\right ) \cos ^2(c+d x)+105 a b \left ((5 A+6 C) a^2+2 b^2 (3 A+4 C)\right ) \cos (c+d x)+6 \left (4 (6 A+7 C) a^4+3 b^2 (50 A+63 C) a^2+4 A b^4\right )\right ) \sec ^4(c+d x)dx\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \int \left (2 b^2 \left (6 (6 A+7 C) a^2+b^2 (62 A+105 C)\right ) \cos ^2(c+d x)+105 a b \left ((5 A+6 C) a^2+2 b^2 (3 A+4 C)\right ) \cos (c+d x)+6 \left (4 (6 A+7 C) a^4+3 b^2 (50 A+63 C) a^2+4 A b^4\right )\right ) \sec ^4(c+d x)dx+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \int \frac {2 b^2 \left (6 (6 A+7 C) a^2+b^2 (62 A+105 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+105 a b \left ((5 A+6 C) a^2+2 b^2 (3 A+4 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+6 \left (4 (6 A+7 C) a^4+3 b^2 (50 A+63 C) a^2+4 A b^4\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3500 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \left (\frac {1}{3} \int 3 \left (105 a b \left ((5 A+6 C) a^2+2 b^2 (3 A+4 C)\right )+2 \left (8 (6 A+7 C) a^4+84 b^2 (4 A+5 C) a^2+35 b^4 (2 A+3 C)\right ) \cos (c+d x)\right ) \sec ^3(c+d x)dx+\frac {2 \left (4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)+4 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \left (\int \left (105 a b \left ((5 A+6 C) a^2+2 b^2 (3 A+4 C)\right )+2 \left (8 (6 A+7 C) a^4+84 b^2 (4 A+5 C) a^2+35 b^4 (2 A+3 C)\right ) \cos (c+d x)\right ) \sec ^3(c+d x)dx+\frac {2 \left (4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)+4 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \left (\int \frac {105 a b \left ((5 A+6 C) a^2+2 b^2 (3 A+4 C)\right )+2 \left (8 (6 A+7 C) a^4+84 b^2 (4 A+5 C) a^2+35 b^4 (2 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {2 \left (4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)+4 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \left (105 a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \int \sec ^3(c+d x)dx+2 \left (8 a^4 (6 A+7 C)+84 a^2 b^2 (4 A+5 C)+35 b^4 (2 A+3 C)\right ) \int \sec ^2(c+d x)dx+\frac {2 \left (4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)+4 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \left (105 a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+2 \left (8 a^4 (6 A+7 C)+84 a^2 b^2 (4 A+5 C)+35 b^4 (2 A+3 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {2 \left (4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)+4 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 4254 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \left (105 a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {2 \left (8 a^4 (6 A+7 C)+84 a^2 b^2 (4 A+5 C)+35 b^4 (2 A+3 C)\right ) \int 1d(-\tan (c+d x))}{d}+\frac {2 \left (4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)+4 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \left (105 a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+\frac {2 \left (8 a^4 (6 A+7 C)+84 a^2 b^2 (4 A+5 C)+35 b^4 (2 A+3 C)\right ) \tan (c+d x)}{d}+\frac {2 \left (4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)+4 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \left (105 a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \left (\frac {1}{2} \int \sec (c+d x)dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {2 \left (8 a^4 (6 A+7 C)+84 a^2 b^2 (4 A+5 C)+35 b^4 (2 A+3 C)\right ) \tan (c+d x)}{d}+\frac {2 \left (4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)+4 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {1}{5} \left (\frac {1}{2} \left (105 a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {2 \left (8 a^4 (6 A+7 C)+84 a^2 b^2 (4 A+5 C)+35 b^4 (2 A+3 C)\right ) \tan (c+d x)}{d}+\frac {2 \left (4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)+4 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )+\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 d}\right )+\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}\right )+\frac {2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{3} \left (\frac {3 \left (a^2 (6 A+7 C)+2 A b^2\right ) \tan (c+d x) \sec ^4(c+d x) (a+b \cos (c+d x))^2}{5 d}+\frac {1}{5} \left (\frac {a b \left (a^2 (103 A+126 C)+6 A b^2\right ) \tan (c+d x) \sec ^3(c+d x)}{2 d}+\frac {1}{2} \left (105 a b \left (a^2 (5 A+6 C)+2 b^2 (3 A+4 C)\right ) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )+\frac {2 \left (8 a^4 (6 A+7 C)+84 a^2 b^2 (4 A+5 C)+35 b^4 (2 A+3 C)\right ) \tan (c+d x)}{d}+\frac {2 \left (4 a^4 (6 A+7 C)+3 a^2 b^2 (50 A+63 C)+4 A b^4\right ) \tan (c+d x) \sec ^2(c+d x)}{d}\right )\right )\right )+\frac {2 A b \tan (c+d x) \sec ^5(c+d x) (a+b \cos (c+d x))^3}{3 d}\right )+\frac {A \tan (c+d x) \sec ^6(c+d x) (a+b \cos (c+d x))^4}{7 d}\) |
(A*(a + b*Cos[c + d*x])^4*Sec[c + d*x]^6*Tan[c + d*x])/(7*d) + ((2*A*b*(a + b*Cos[c + d*x])^3*Sec[c + d*x]^5*Tan[c + d*x])/(3*d) + ((3*(2*A*b^2 + a^ 2*(6*A + 7*C))*(a + b*Cos[c + d*x])^2*Sec[c + d*x]^4*Tan[c + d*x])/(5*d) + ((a*b*(6*A*b^2 + a^2*(103*A + 126*C))*Sec[c + d*x]^3*Tan[c + d*x])/(2*d) + ((2*(35*b^4*(2*A + 3*C) + 84*a^2*b^2*(4*A + 5*C) + 8*a^4*(6*A + 7*C))*Ta n[c + d*x])/d + (2*(4*A*b^4 + 4*a^4*(6*A + 7*C) + 3*a^2*b^2*(50*A + 63*C)) *Sec[c + d*x]^2*Tan[c + d*x])/d + 105*a*b*(2*b^2*(3*A + 4*C) + a^2*(5*A + 6*C))*(ArcTanh[Sin[c + d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)))/2 )/5)/3)/7
3.6.58.3.1 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[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(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[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(c^2*C + 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*c*m + a*d*(n + 1)) - (A*d*(a*d*(n + 2) - b *c*(n + 1)) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b*( A*d^2*(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, 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_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 14.07 (sec) , antiderivative size = 317, normalized size of antiderivative = 0.89
| method | result | size |
| parts | \(-\frac {a^{4} A \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )}{d}-\frac {\left (A \,b^{4}+6 C \,a^{2} b^{2}\right ) \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (4 A a \,b^{3}+4 C \,a^{3} b \right ) \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (6 A \,a^{2} b^{2}+C \,a^{4}\right ) \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )}{d}+\frac {C \,b^{4} \tan \left (d x +c \right )}{d}+\frac {4 A \,a^{3} b \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )}{d}+\frac {4 C a \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(317\) |
| derivativedivides | \(\frac {-a^{4} A \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )-C \,a^{4} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+4 A \,a^{3} b \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+4 C \,a^{3} b \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 A \,a^{2} b^{2} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )-6 C \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 A a \,b^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+4 C a \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-A \,b^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+C \tan \left (d x +c \right ) b^{4}}{d}\) | \(384\) |
| default | \(\frac {-a^{4} A \left (-\frac {16}{35}-\frac {\left (\sec ^{6}\left (d x +c \right )\right )}{7}-\frac {6 \left (\sec ^{4}\left (d x +c \right )\right )}{35}-\frac {8 \left (\sec ^{2}\left (d x +c \right )\right )}{35}\right ) \tan \left (d x +c \right )-C \,a^{4} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )+4 A \,a^{3} b \left (-\left (-\frac {\left (\sec ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sec ^{3}\left (d x +c \right )\right )}{24}-\frac {5 \sec \left (d x +c \right )}{16}\right ) \tan \left (d x +c \right )+\frac {5 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{16}\right )+4 C \,a^{3} b \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-6 A \,a^{2} b^{2} \left (-\frac {8}{15}-\frac {\left (\sec ^{4}\left (d x +c \right )\right )}{5}-\frac {4 \left (\sec ^{2}\left (d x +c \right )\right )}{15}\right ) \tan \left (d x +c \right )-6 C \,a^{2} b^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 A a \,b^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+4 C a \,b^{3} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-A \,b^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+C \tan \left (d x +c \right ) b^{4}}{d}\) | \(384\) |
| parallelrisch | \(\frac {-3675 b \left (\cos \left (5 d x +5 c \right )+\frac {\cos \left (7 d x +7 c \right )}{7}+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )\right ) a \left (\left (A +\frac {6 C}{5}\right ) a^{2}+\frac {6 \left (A +\frac {4 C}{3}\right ) b^{2}}{5}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+3675 b \left (\cos \left (5 d x +5 c \right )+\frac {\cos \left (7 d x +7 c \right )}{7}+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )\right ) a \left (\left (A +\frac {6 C}{5}\right ) a^{2}+\frac {6 \left (A +\frac {4 C}{3}\right ) b^{2}}{5}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (\left (4032 A +4704 C \right ) a^{4}+28224 b^{2} \left (A +\frac {25 C}{28}\right ) a^{2}+4200 b^{4} \left (A +\frac {9 C}{10}\right )\right ) \sin \left (3 d x +3 c \right )+\left (\left (1344 A +1568 C \right ) a^{4}+9408 a^{2} \left (A +\frac {5 C}{4}\right ) b^{2}+1960 \left (A +\frac {15 C}{14}\right ) b^{4}\right ) \sin \left (5 d x +5 c \right )+\left (\left (192 A +224 C \right ) a^{4}+1344 a^{2} \left (A +\frac {5 C}{4}\right ) b^{2}+280 \left (A +\frac {3 C}{2}\right ) b^{4}\right ) \sin \left (7 d x +7 c \right )+19810 b \left (\left (A +\frac {186 C}{283}\right ) a^{2}+\frac {186 b^{2} \left (A +\frac {20 C}{31}\right )}{283}\right ) a \sin \left (2 d x +2 c \right )+7000 b \left (\left (A +\frac {6 C}{5}\right ) a^{2}+\frac {6 b^{2} \left (A +\frac {4 C}{5}\right )}{5}\right ) a \sin \left (4 d x +4 c \right )+1050 b a \left (\left (A +\frac {6 C}{5}\right ) a^{2}+\frac {6 \left (A +\frac {4 C}{3}\right ) b^{2}}{5}\right ) \sin \left (6 d x +6 c \right )+6720 \left (a^{4} \left (A +\frac {C}{2}\right )+3 \left (A +\frac {3 C}{4}\right ) a^{2} b^{2}+\frac {3 \left (A +\frac {5 C}{6}\right ) b^{4}}{8}\right ) \sin \left (d x +c \right )}{2940 \left (\cos \left (5 d x +5 c \right )+\frac {\cos \left (7 d x +7 c \right )}{7}+3 \cos \left (3 d x +3 c \right )+5 \cos \left (d x +c \right )\right ) d}\) | \(476\) |
| risch | \(\text {Expression too large to display}\) | \(1064\) |
-a^4*A/d*(-16/35-1/7*sec(d*x+c)^6-6/35*sec(d*x+c)^4-8/35*sec(d*x+c)^2)*tan (d*x+c)-(A*b^4+6*C*a^2*b^2)/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+(4*A*a*b^ 3+4*C*a^3*b)/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)))-(6*A*a^2*b^2+C*a^4)/d*(-8/15-1/5*sec(d*x+c)^4-4/15*sec (d*x+c)^2)*tan(d*x+c)+C*b^4/d*tan(d*x+c)+4*A*a^3*b/d*(-(-1/6*sec(d*x+c)^5- 5/24*sec(d*x+c)^3-5/16*sec(d*x+c))*tan(d*x+c)+5/16*ln(sec(d*x+c)+tan(d*x+c )))+4*C*a*b^3/d*(1/2*sec(d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))
Time = 0.32 (sec) , antiderivative size = 325, normalized size of antiderivative = 0.92 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {105 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} b + 2 \, {\left (3 \, A + 4 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} b + 2 \, {\left (3 \, A + 4 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (4 \, {\left (8 \, {\left (6 \, A + 7 \, C\right )} a^{4} + 84 \, {\left (4 \, A + 5 \, C\right )} a^{2} b^{2} + 35 \, {\left (2 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} + 280 \, A a^{3} b \cos \left (d x + c\right ) + 105 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} b + 2 \, {\left (3 \, A + 4 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{5} + 60 \, A a^{4} + 4 \, {\left (4 \, {\left (6 \, A + 7 \, C\right )} a^{4} + 42 \, {\left (4 \, A + 5 \, C\right )} a^{2} b^{2} + 35 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} + 70 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} b + 6 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 12 \, {\left ({\left (6 \, A + 7 \, C\right )} a^{4} + 42 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{840 \, d \cos \left (d x + c\right )^{7}} \]
1/840*(105*((5*A + 6*C)*a^3*b + 2*(3*A + 4*C)*a*b^3)*cos(d*x + c)^7*log(si n(d*x + c) + 1) - 105*((5*A + 6*C)*a^3*b + 2*(3*A + 4*C)*a*b^3)*cos(d*x + c)^7*log(-sin(d*x + c) + 1) + 2*(4*(8*(6*A + 7*C)*a^4 + 84*(4*A + 5*C)*a^2 *b^2 + 35*(2*A + 3*C)*b^4)*cos(d*x + c)^6 + 280*A*a^3*b*cos(d*x + c) + 105 *((5*A + 6*C)*a^3*b + 2*(3*A + 4*C)*a*b^3)*cos(d*x + c)^5 + 60*A*a^4 + 4*( 4*(6*A + 7*C)*a^4 + 42*(4*A + 5*C)*a^2*b^2 + 35*A*b^4)*cos(d*x + c)^4 + 70 *((5*A + 6*C)*a^3*b + 6*A*a*b^3)*cos(d*x + c)^3 + 12*((6*A + 7*C)*a^4 + 42 *A*a^2*b^2)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^7)
Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.33 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {24 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} A a^{4} + 56 \, {\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 )} C a^{4} + 336 \, {\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^{2} b^{2} + 1680 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{2} b^{2} + 280 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A b^{4} - 35 \, A a^{3} b {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 210 \, C 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 )} - 210 \, A a b^{3} {\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 )} - 840 \, C 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 )} + 840 \, C b^{4} \tan \left (d x + c\right )}{840 \, d} \]
1/840*(24*(5*tan(d*x + c)^7 + 21*tan(d*x + c)^5 + 35*tan(d*x + c)^3 + 35*t an(d*x + c))*A*a^4 + 56*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c))*C*a^4 + 336*(3*tan(d*x + c)^5 + 10*tan(d*x + c)^3 + 15*tan(d*x + c) )*A*a^2*b^2 + 1680*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a^2*b^2 + 280*(tan( d*x + c)^3 + 3*tan(d*x + c))*A*b^4 - 35*A*a^3*b*(2*(15*sin(d*x + c)^5 - 40 *sin(d*x + c)^3 + 33*sin(d*x + c))/(sin(d*x + c)^6 - 3*sin(d*x + c)^4 + 3* sin(d*x + c)^2 - 1) - 15*log(sin(d*x + c) + 1) + 15*log(sin(d*x + c) - 1)) - 210*C*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)) - 210*A*a*b^3*(2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*si n(d*x + c)^2 + 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 8 40*C*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)) + 840*C*b^4*tan(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 1280 vs. \(2 (339) = 678\).
Time = 0.38 (sec) , antiderivative size = 1280, normalized size of antiderivative = 3.61 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\text {Too large to display} \]
1/420*(105*(5*A*a^3*b + 6*C*a^3*b + 6*A*a*b^3 + 8*C*a*b^3)*log(abs(tan(1/2 *d*x + 1/2*c) + 1)) - 105*(5*A*a^3*b + 6*C*a^3*b + 6*A*a*b^3 + 8*C*a*b^3)* log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(420*A*a^4*tan(1/2*d*x + 1/2*c)^13 + 420*C*a^4*tan(1/2*d*x + 1/2*c)^13 - 1155*A*a^3*b*tan(1/2*d*x + 1/2*c)^13 - 1050*C*a^3*b*tan(1/2*d*x + 1/2*c)^13 + 2520*A*a^2*b^2*tan(1/2*d*x + 1/2 *c)^13 + 2520*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^13 - 1050*A*a*b^3*tan(1/2*d*x + 1/2*c)^13 - 840*C*a*b^3*tan(1/2*d*x + 1/2*c)^13 + 420*A*b^4*tan(1/2*d*x + 1/2*c)^13 + 420*C*b^4*tan(1/2*d*x + 1/2*c)^13 - 840*A*a^4*tan(1/2*d*x + 1/2*c)^11 - 1400*C*a^4*tan(1/2*d*x + 1/2*c)^11 + 980*A*a^3*b*tan(1/2*d*x + 1/2*c)^11 + 2520*C*a^3*b*tan(1/2*d*x + 1/2*c)^11 - 8400*A*a^2*b^2*tan(1/ 2*d*x + 1/2*c)^11 - 11760*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 + 2520*A*a*b^3 *tan(1/2*d*x + 1/2*c)^11 + 3360*C*a*b^3*tan(1/2*d*x + 1/2*c)^11 - 1960*A*b ^4*tan(1/2*d*x + 1/2*c)^11 - 2520*C*b^4*tan(1/2*d*x + 1/2*c)^11 + 3612*A*a ^4*tan(1/2*d*x + 1/2*c)^9 + 3164*C*a^4*tan(1/2*d*x + 1/2*c)^9 - 2975*A*a^3 *b*tan(1/2*d*x + 1/2*c)^9 - 1890*C*a^3*b*tan(1/2*d*x + 1/2*c)^9 + 18984*A* a^2*b^2*tan(1/2*d*x + 1/2*c)^9 + 24360*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 1890*A*a*b^3*tan(1/2*d*x + 1/2*c)^9 - 4200*C*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 4060*A*b^4*tan(1/2*d*x + 1/2*c)^9 + 6300*C*b^4*tan(1/2*d*x + 1/2*c)^9 - 2544*A*a^4*tan(1/2*d*x + 1/2*c)^7 - 4368*C*a^4*tan(1/2*d*x + 1/2*c)^7 - 26 208*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 30240*C*a^2*b^2*tan(1/2*d*x + 1/...
Time = 5.72 (sec) , antiderivative size = 755, normalized size of antiderivative = 2.13 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^8(c+d x) \, dx=\frac {a\,b\,\mathrm {atanh}\left (\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,A\,a^2+6\,A\,b^2+6\,C\,a^2+8\,C\,b^2\right )}{6\,A\,a\,b^3+5\,A\,a^3\,b+8\,C\,a\,b^3+6\,C\,a^3\,b}\right )\,\left (5\,A\,a^2+6\,A\,b^2+6\,C\,a^2+8\,C\,b^2\right )}{2\,d}-\frac {\left (2\,A\,a^4+2\,A\,b^4+2\,C\,a^4+2\,C\,b^4+12\,A\,a^2\,b^2+12\,C\,a^2\,b^2-5\,A\,a\,b^3-\frac {11\,A\,a^3\,b}{2}-4\,C\,a\,b^3-5\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (12\,A\,a\,b^3-\frac {28\,A\,b^4}{3}-\frac {20\,C\,a^4}{3}-12\,C\,b^4-40\,A\,a^2\,b^2-56\,C\,a^2\,b^2-4\,A\,a^4+\frac {14\,A\,a^3\,b}{3}+16\,C\,a\,b^3+12\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {86\,A\,a^4}{5}+\frac {58\,A\,b^4}{3}+\frac {226\,C\,a^4}{15}+30\,C\,b^4+\frac {452\,A\,a^2\,b^2}{5}+116\,C\,a^2\,b^2-9\,A\,a\,b^3-\frac {85\,A\,a^3\,b}{6}-20\,C\,a\,b^3-9\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-\frac {424\,A\,a^4}{35}-24\,A\,b^4-\frac {104\,C\,a^4}{5}-40\,C\,b^4-\frac {624\,A\,a^2\,b^2}{5}-144\,C\,a^2\,b^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {86\,A\,a^4}{5}+\frac {58\,A\,b^4}{3}+\frac {226\,C\,a^4}{15}+30\,C\,b^4+\frac {452\,A\,a^2\,b^2}{5}+116\,C\,a^2\,b^2+9\,A\,a\,b^3+\frac {85\,A\,a^3\,b}{6}+20\,C\,a\,b^3+9\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-4\,A\,a^4-\frac {28\,A\,b^4}{3}-\frac {20\,C\,a^4}{3}-12\,C\,b^4-40\,A\,a^2\,b^2-56\,C\,a^2\,b^2-12\,A\,a\,b^3-\frac {14\,A\,a^3\,b}{3}-16\,C\,a\,b^3-12\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a^4+2\,A\,b^4+2\,C\,a^4+2\,C\,b^4+12\,A\,a^2\,b^2+12\,C\,a^2\,b^2+5\,A\,a\,b^3+\frac {11\,A\,a^3\,b}{2}+4\,C\,a\,b^3+5\,C\,a^3\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
(a*b*atanh((a*b*tan(c/2 + (d*x)/2)*(5*A*a^2 + 6*A*b^2 + 6*C*a^2 + 8*C*b^2) )/(6*A*a*b^3 + 5*A*a^3*b + 8*C*a*b^3 + 6*C*a^3*b))*(5*A*a^2 + 6*A*b^2 + 6* C*a^2 + 8*C*b^2))/(2*d) - (tan(c/2 + (d*x)/2)*(2*A*a^4 + 2*A*b^4 + 2*C*a^4 + 2*C*b^4 + 12*A*a^2*b^2 + 12*C*a^2*b^2 + 5*A*a*b^3 + (11*A*a^3*b)/2 + 4* C*a*b^3 + 5*C*a^3*b) - tan(c/2 + (d*x)/2)^7*((424*A*a^4)/35 + 24*A*b^4 + ( 104*C*a^4)/5 + 40*C*b^4 + (624*A*a^2*b^2)/5 + 144*C*a^2*b^2) + tan(c/2 + ( d*x)/2)^13*(2*A*a^4 + 2*A*b^4 + 2*C*a^4 + 2*C*b^4 + 12*A*a^2*b^2 + 12*C*a^ 2*b^2 - 5*A*a*b^3 - (11*A*a^3*b)/2 - 4*C*a*b^3 - 5*C*a^3*b) - tan(c/2 + (d *x)/2)^3*(4*A*a^4 + (28*A*b^4)/3 + (20*C*a^4)/3 + 12*C*b^4 + 40*A*a^2*b^2 + 56*C*a^2*b^2 + 12*A*a*b^3 + (14*A*a^3*b)/3 + 16*C*a*b^3 + 12*C*a^3*b) - tan(c/2 + (d*x)/2)^11*(4*A*a^4 + (28*A*b^4)/3 + (20*C*a^4)/3 + 12*C*b^4 + 40*A*a^2*b^2 + 56*C*a^2*b^2 - 12*A*a*b^3 - (14*A*a^3*b)/3 - 16*C*a*b^3 - 1 2*C*a^3*b) + tan(c/2 + (d*x)/2)^5*((86*A*a^4)/5 + (58*A*b^4)/3 + (226*C*a^ 4)/15 + 30*C*b^4 + (452*A*a^2*b^2)/5 + 116*C*a^2*b^2 + 9*A*a*b^3 + (85*A*a ^3*b)/6 + 20*C*a*b^3 + 9*C*a^3*b) + tan(c/2 + (d*x)/2)^9*((86*A*a^4)/5 + ( 58*A*b^4)/3 + (226*C*a^4)/15 + 30*C*b^4 + (452*A*a^2*b^2)/5 + 116*C*a^2*b^ 2 - 9*A*a*b^3 - (85*A*a^3*b)/6 - 20*C*a*b^3 - 9*C*a^3*b))/(d*(7*tan(c/2 + (d*x)/2)^2 - 21*tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 - 35*tan(c/ 2 + (d*x)/2)^8 + 21*tan(c/2 + (d*x)/2)^10 - 7*tan(c/2 + (d*x)/2)^12 + tan( c/2 + (d*x)/2)^14 - 1))