Integrand size = 21, antiderivative size = 404 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx=b^8 x+\frac {4 a b \left (24 a^6-88 a^4 b^2+125 a^2 b^4-96 b^6\right ) \cos (c+d x)}{105 d}+\frac {b^2 \left (48 a^6-152 a^4 b^2+174 a^2 b^4-105 b^6\right ) \cos (c+d x) \sin (c+d x)}{105 d}+\frac {2 a b \left (24 a^4-40 a^2 b^2+9 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{105 d}+\frac {2 b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{105 d}+\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^7}{7 d}-\frac {2 \sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{105 d}-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{35 d}-\frac {2 \sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{105 d} \]
b^8*x+4/105*a*b*(24*a^6-88*a^4*b^2+125*a^2*b^4-96*b^6)*cos(d*x+c)/d+1/105* b^2*(48*a^6-152*a^4*b^2+174*a^2*b^4-105*b^6)*cos(d*x+c)*sin(d*x+c)/d+2/105 *a*b*(24*a^4-40*a^2*b^2+9*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^2/d+2/105*b*(24 *a^4+8*a^2*b^2-35*b^4)*cos(d*x+c)*(a+b*sin(d*x+c))^3/d+1/7*sec(d*x+c)^7*(b +a*sin(d*x+c))*(a+b*sin(d*x+c))^7/d-2/105*sec(d*x+c)^3*(a+b*sin(d*x+c))^5* (b*(6*a^2-7*b^2)-a*(12*a^2-11*b^2)*sin(d*x+c))/d-1/35*sec(d*x+c)^5*(a+b*si n(d*x+c))^6*(a*b-(6*a^2-7*b^2)*sin(d*x+c))/d-2/105*sec(d*x+c)*(a+b*sin(d*x +c))^4*(3*a*b*(12*a^2-11*b^2)-(24*a^4+8*a^2*b^2-35*b^4)*sin(d*x+c))/d
Time = 4.26 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.19 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {\sec ^7(c+d x) \left (7680 a^7 b+16128 a^5 b^3+25536 a^3 b^5-5088 a b^7+3675 b^8 (c+d x) \cos (c+d x)-37632 a^5 b^3 \cos (2 (c+d x))-12544 a^3 b^5 \cos (2 (c+d x))-14448 a b^7 \cos (2 (c+d x))+2205 b^8 (c+d x) \cos (3 (c+d x))+15680 a^3 b^5 \cos (4 (c+d x))-3360 a b^7 \cos (4 (c+d x))+735 b^8 (c+d x) \cos (5 (c+d x))-1680 a b^7 \cos (6 (c+d x))+105 b^8 (c+d x) \cos (7 (c+d x))+1680 a^8 \sin (c+d x)+23520 a^6 b^2 \sin (c+d x)+44100 a^4 b^4 \sin (c+d x)+14700 a^2 b^6 \sin (c+d x)+1008 a^8 \sin (3 (c+d x))-4704 a^6 b^2 \sin (3 (c+d x))-20580 a^4 b^4 \sin (3 (c+d x))-8820 a^2 b^6 \sin (3 (c+d x))-1176 b^8 \sin (3 (c+d x))+336 a^8 \sin (5 (c+d x))-1568 a^6 b^2 \sin (5 (c+d x))+2940 a^4 b^4 \sin (5 (c+d x))+2940 a^2 b^6 \sin (5 (c+d x))-392 b^8 \sin (5 (c+d x))+48 a^8 \sin (7 (c+d x))-224 a^6 b^2 \sin (7 (c+d x))+420 a^4 b^4 \sin (7 (c+d x))-420 a^2 b^6 \sin (7 (c+d x))-176 b^8 \sin (7 (c+d x))\right )}{6720 d} \]
(Sec[c + d*x]^7*(7680*a^7*b + 16128*a^5*b^3 + 25536*a^3*b^5 - 5088*a*b^7 + 3675*b^8*(c + d*x)*Cos[c + d*x] - 37632*a^5*b^3*Cos[2*(c + d*x)] - 12544* a^3*b^5*Cos[2*(c + d*x)] - 14448*a*b^7*Cos[2*(c + d*x)] + 2205*b^8*(c + d* x)*Cos[3*(c + d*x)] + 15680*a^3*b^5*Cos[4*(c + d*x)] - 3360*a*b^7*Cos[4*(c + d*x)] + 735*b^8*(c + d*x)*Cos[5*(c + d*x)] - 1680*a*b^7*Cos[6*(c + d*x) ] + 105*b^8*(c + d*x)*Cos[7*(c + d*x)] + 1680*a^8*Sin[c + d*x] + 23520*a^6 *b^2*Sin[c + d*x] + 44100*a^4*b^4*Sin[c + d*x] + 14700*a^2*b^6*Sin[c + d*x ] + 1008*a^8*Sin[3*(c + d*x)] - 4704*a^6*b^2*Sin[3*(c + d*x)] - 20580*a^4* b^4*Sin[3*(c + d*x)] - 8820*a^2*b^6*Sin[3*(c + d*x)] - 1176*b^8*Sin[3*(c + d*x)] + 336*a^8*Sin[5*(c + d*x)] - 1568*a^6*b^2*Sin[5*(c + d*x)] + 2940*a ^4*b^4*Sin[5*(c + d*x)] + 2940*a^2*b^6*Sin[5*(c + d*x)] - 392*b^8*Sin[5*(c + d*x)] + 48*a^8*Sin[7*(c + d*x)] - 224*a^6*b^2*Sin[7*(c + d*x)] + 420*a^ 4*b^4*Sin[7*(c + d*x)] - 420*a^2*b^6*Sin[7*(c + d*x)] - 176*b^8*Sin[7*(c + d*x)]))/(6720*d)
Time = 1.96 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.07, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.905, Rules used = {3042, 3170, 25, 3042, 3340, 27, 3042, 3340, 25, 3042, 3340, 27, 3042, 3232, 27, 3042, 3232, 3042, 3213}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \sin (c+d x))^8}{\cos (c+d x)^8}dx\) |
| \(\Big \downarrow \) 3170 |
| \(\displaystyle \frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}-\frac {1}{7} \int -\sec ^6(c+d x) (a+b \sin (c+d x))^6 \left (6 a^2-b \sin (c+d x) a-7 b^2\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{7} \int \sec ^6(c+d x) (a+b \sin (c+d x))^6 \left (6 a^2-b \sin (c+d x) a-7 b^2\right )dx+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \frac {(a+b \sin (c+d x))^6 \left (6 a^2-b \sin (c+d x) a-7 b^2\right )}{\cos (c+d x)^6}dx+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3340 |
| \(\displaystyle \frac {1}{7} \left (-\frac {1}{5} \int -2 \sec ^4(c+d x) (a+b \sin (c+d x))^5 \left (a \left (12 a^2-11 b^2\right )-b \left (6 a^2-7 b^2\right ) \sin (c+d x)\right )dx-\frac {\sec ^5(c+d x) \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right ) (a+b \sin (c+d x))^6}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \sec ^4(c+d x) (a+b \sin (c+d x))^5 \left (a \left (12 a^2-11 b^2\right )-b \left (6 a^2-7 b^2\right ) \sin (c+d x)\right )dx-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \int \frac {(a+b \sin (c+d x))^5 \left (a \left (12 a^2-11 b^2\right )-b \left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{\cos (c+d x)^4}dx-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3340 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (-\frac {1}{3} \int -\sec ^2(c+d x) (a+b \sin (c+d x))^4 \left (24 a^4+8 b^2 a^2-3 b \left (12 a^2-11 b^2\right ) \sin (c+d x) a-35 b^4\right )dx-\frac {\sec ^3(c+d x) \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right ) (a+b \sin (c+d x))^5}{3 d}\right )-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (\frac {1}{3} \int \sec ^2(c+d x) (a+b \sin (c+d x))^4 \left (24 a^4+8 b^2 a^2-3 b \left (12 a^2-11 b^2\right ) \sin (c+d x) a-35 b^4\right )dx-\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{3 d}\right )-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (\frac {1}{3} \int \frac {(a+b \sin (c+d x))^4 \left (24 a^4+8 b^2 a^2-3 b \left (12 a^2-11 b^2\right ) \sin (c+d x) a-35 b^4\right )}{\cos (c+d x)^2}dx-\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{3 d}\right )-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3340 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (\frac {1}{3} \left (-\int -4 (a+b \sin (c+d x))^3 \left (3 a b^2 \left (12 a^2-11 b^2\right )-b \left (24 a^4+8 b^2 a^2-35 b^4\right ) \sin (c+d x)\right )dx-\frac {\sec (c+d x) \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right ) (a+b \sin (c+d x))^4}{d}\right )-\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{3 d}\right )-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (\frac {1}{3} \left (4 \int (a+b \sin (c+d x))^3 \left (3 a b^2 \left (12 a^2-11 b^2\right )-b \left (24 a^4+8 b^2 a^2-35 b^4\right ) \sin (c+d x)\right )dx-\frac {\sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{d}\right )-\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{3 d}\right )-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (\frac {1}{3} \left (4 \int (a+b \sin (c+d x))^3 \left (3 a b^2 \left (12 a^2-11 b^2\right )-b \left (24 a^4+8 b^2 a^2-35 b^4\right ) \sin (c+d x)\right )dx-\frac {\sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{d}\right )-\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{3 d}\right )-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (\frac {1}{3} \left (4 \left (\frac {1}{4} \int 3 (a+b \sin (c+d x))^2 \left (b^2 \left (24 a^4-52 b^2 a^2+35 b^4\right )-a b \left (24 a^4-40 b^2 a^2+9 b^4\right ) \sin (c+d x)\right )dx+\frac {b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 d}\right )-\frac {\sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{d}\right )-\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{3 d}\right )-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (\frac {1}{3} \left (4 \left (\frac {3}{4} \int (a+b \sin (c+d x))^2 \left (b^2 \left (24 a^4-52 b^2 a^2+35 b^4\right )-a b \left (24 a^4-40 b^2 a^2+9 b^4\right ) \sin (c+d x)\right )dx+\frac {b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 d}\right )-\frac {\sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{d}\right )-\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{3 d}\right )-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (\frac {1}{3} \left (4 \left (\frac {3}{4} \int (a+b \sin (c+d x))^2 \left (b^2 \left (24 a^4-52 b^2 a^2+35 b^4\right )-a b \left (24 a^4-40 b^2 a^2+9 b^4\right ) \sin (c+d x)\right )dx+\frac {b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 d}\right )-\frac {\sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{d}\right )-\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{3 d}\right )-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (\frac {1}{3} \left (4 \left (\frac {3}{4} \left (\frac {1}{3} \int (a+b \sin (c+d x)) \left (a b^2 \left (24 a^4-76 b^2 a^2+87 b^4\right )-b \left (48 a^6-152 b^2 a^4+174 b^4 a^2-105 b^6\right ) \sin (c+d x)\right )dx+\frac {a b \left (24 a^4-40 a^2 b^2+9 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{3 d}\right )+\frac {b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 d}\right )-\frac {\sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{d}\right )-\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{3 d}\right )-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (\frac {1}{3} \left (4 \left (\frac {3}{4} \left (\frac {1}{3} \int (a+b \sin (c+d x)) \left (a b^2 \left (24 a^4-76 b^2 a^2+87 b^4\right )-b \left (48 a^6-152 b^2 a^4+174 b^4 a^2-105 b^6\right ) \sin (c+d x)\right )dx+\frac {a b \left (24 a^4-40 a^2 b^2+9 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{3 d}\right )+\frac {b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 d}\right )-\frac {\sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{d}\right )-\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{3 d}\right )-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}\) |
| \(\Big \downarrow \) 3213 |
| \(\displaystyle \frac {1}{7} \left (\frac {2}{5} \left (\frac {1}{3} \left (4 \left (\frac {b \left (24 a^4+8 a^2 b^2-35 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^3}{4 d}+\frac {3}{4} \left (\frac {a b \left (24 a^4-40 a^2 b^2+9 b^4\right ) \cos (c+d x) (a+b \sin (c+d x))^2}{3 d}+\frac {1}{3} \left (\frac {2 a b \left (24 a^6-88 a^4 b^2+125 a^2 b^4-96 b^6\right ) \cos (c+d x)}{d}+\frac {b^2 \left (48 a^6-152 a^4 b^2+174 a^2 b^4-105 b^6\right ) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {105 b^8 x}{2}\right )\right )\right )-\frac {\sec (c+d x) (a+b \sin (c+d x))^4 \left (3 a b \left (12 a^2-11 b^2\right )-\left (24 a^4+8 a^2 b^2-35 b^4\right ) \sin (c+d x)\right )}{d}\right )-\frac {\sec ^3(c+d x) (a+b \sin (c+d x))^5 \left (b \left (6 a^2-7 b^2\right )-a \left (12 a^2-11 b^2\right ) \sin (c+d x)\right )}{3 d}\right )-\frac {\sec ^5(c+d x) (a+b \sin (c+d x))^6 \left (a b-\left (6 a^2-7 b^2\right ) \sin (c+d x)\right )}{5 d}\right )+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^7}{7 d}\) |
(Sec[c + d*x]^7*(b + a*Sin[c + d*x])*(a + b*Sin[c + d*x])^7)/(7*d) + (-1/5 *(Sec[c + d*x]^5*(a + b*Sin[c + d*x])^6*(a*b - (6*a^2 - 7*b^2)*Sin[c + d*x ]))/d + (2*(-1/3*(Sec[c + d*x]^3*(a + b*Sin[c + d*x])^5*(b*(6*a^2 - 7*b^2) - a*(12*a^2 - 11*b^2)*Sin[c + d*x]))/d + (-((Sec[c + d*x]*(a + b*Sin[c + d*x])^4*(3*a*b*(12*a^2 - 11*b^2) - (24*a^4 + 8*a^2*b^2 - 35*b^4)*Sin[c + d *x]))/d) + 4*((b*(24*a^4 + 8*a^2*b^2 - 35*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^3)/(4*d) + (3*((a*b*(24*a^4 - 40*a^2*b^2 + 9*b^4)*Cos[c + d*x]*(a + b*Sin[c + d*x])^2)/(3*d) + ((105*b^8*x)/2 + (2*a*b*(24*a^6 - 88*a^4*b^2 + 125*a^2*b^4 - 96*b^6)*Cos[c + d*x])/d + (b^2*(48*a^6 - 152*a^4*b^2 + 174* a^2*b^4 - 105*b^6)*Cos[c + d*x]*Sin[c + d*x])/(2*d))/3))/4))/3))/5)/7
3.5.23.3.1 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[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-(g*Cos[e + f*x])^(p + 1))*(a + b*Sin[e + f*x ])^(m - 1)*((b + a*Sin[e + f*x])/(f*g*(p + 1))), x] + Simp[1/(g^2*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(p + 2) + a*b*(m + p + 1)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g }, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && LtQ[p, -1] && (IntegersQ[2*m, 2* p] || IntegerQ[m])
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.) *(x_)]), x_Symbol] :> Simp[(2*a*c + b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Co s[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /; Free Q[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[1/(m + 1) Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ [{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(g* Cos[e + f*x])^(p + 1))*(a + b*Sin[e + f*x])^m*((d + c*Sin[e + f*x])/(f*g*(p + 1))), x] + Simp[1/(g^2*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Si n[e + f*x])^(m - 1)*Simp[a*c*(p + 2) + b*d*m + b*c*(m + p + 2)*Sin[e + f*x] , x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[a^2 - b^2, 0] && GtQ [m, 0] && LtQ[p, -1] && IntegerQ[2*m] && !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && SimplerQ[c + d*x, a + b*x])
Time = 2.33 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.34
| method | result | size |
| parallelrisch | \(\frac {\left (105 b^{8} d x +1120 a^{5} b^{3}\right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-210 a^{8}+210 b^{8}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-735 b^{8} d x -1680 a^{7} b -7840 a^{5} b^{3}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (420 a^{8}-7840 a^{6} b^{2}-1540 b^{8}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2205 x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{8} d +\left (-1806 a^{8}-6272 a^{6} b^{2}-47040 a^{4} b^{4}+4942 b^{8}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-3675 b^{8} d x -8400 a^{7} b -62720 a^{5} b^{3}-62720 a^{3} b^{5}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (1272 a^{8}-25536 a^{6} b^{2}-40320 a^{4} b^{4}-53760 a^{2} b^{6}-9144 b^{8}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7840 \left (-\frac {15}{32} b^{5} d x +a^{5}+4 a^{3} b^{2}+\frac {24}{7} a \,b^{4}\right ) b^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-1806 a^{8}-6272 a^{6} b^{2}-47040 a^{4} b^{4}+4942 b^{8}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-2205 b^{8} d x -5040 a^{7} b -32928 a^{5} b^{3}-18816 a^{3} b^{5}+16128 a \,b^{7}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (420 a^{8}-7840 a^{6} b^{2}-1540 b^{8}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (735 b^{8} d x +3136 a^{5} b^{3}+6272 a^{3} b^{5}-5376 a \,b^{7}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-210 a^{8}+210 b^{8}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-105 b^{8} d x -240 a^{7} b -448 a^{5} b^{3}-896 a^{3} b^{5}+768 a \,b^{7}}{105 d \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{7}}\) | \(541\) |
| derivativedivides | \(\frac {-a^{8} \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 )+\frac {8 a^{7} b}{7 \cos \left (d x +c \right )^{7}}+28 a^{6} b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+56 a^{5} b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{35}\right )+70 a^{4} b^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )+56 a^{3} b^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{6}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{6}\left (d x +c \right )}{105 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{6}\left (d x +c \right )}{35 \cos \left (d x +c \right )}+\frac {\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )}{35}\right )+\frac {4 a^{2} b^{6} \left (\sin ^{7}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{7}}+8 a \,b^{7} \left (\frac {\sin ^{8}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}-\frac {\sin ^{8}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{8}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{8}\left (d x +c \right )}{7 \cos \left (d x +c \right )}-\frac {\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{7}\right )+b^{8} \left (\frac {\left (\tan ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )}{d}\) | \(567\) |
| default | \(\frac {-a^{8} \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 )+\frac {8 a^{7} b}{7 \cos \left (d x +c \right )^{7}}+28 a^{6} b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {4 \left (\sin ^{3}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {8 \left (\sin ^{3}\left (d x +c \right )\right )}{105 \cos \left (d x +c \right )^{3}}\right )+56 a^{5} b^{3} \left (\frac {\sin ^{4}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {3 \left (\sin ^{4}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{4}\left (d x +c \right )}{35 \cos \left (d x +c \right )}-\frac {\left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{35}\right )+70 a^{4} b^{4} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )+56 a^{3} b^{5} \left (\frac {\sin ^{6}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{6}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{6}\left (d x +c \right )}{105 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{6}\left (d x +c \right )}{35 \cos \left (d x +c \right )}+\frac {\left (\frac {8}{3}+\sin ^{4}\left (d x +c \right )+\frac {4 \left (\sin ^{2}\left (d x +c \right )\right )}{3}\right ) \cos \left (d x +c \right )}{35}\right )+\frac {4 a^{2} b^{6} \left (\sin ^{7}\left (d x +c \right )\right )}{\cos \left (d x +c \right )^{7}}+8 a \,b^{7} \left (\frac {\sin ^{8}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}-\frac {\sin ^{8}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{5}}+\frac {\sin ^{8}\left (d x +c \right )}{35 \cos \left (d x +c \right )^{3}}-\frac {\sin ^{8}\left (d x +c \right )}{7 \cos \left (d x +c \right )}-\frac {\left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{7}\right )+b^{8} \left (\frac {\left (\tan ^{7}\left (d x +c \right )\right )}{7}-\frac {\left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\tan ^{3}\left (d x +c \right )\right )}{3}-\tan \left (d x +c \right )+d x +c \right )}{d}\) | \(567\) |
| risch | \(b^{8} x -\frac {8 \left (-4032 a^{5} b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+770 i b^{8} {\mathrm e}^{8 i \left (d x +c \right )}+770 i b^{8} {\mathrm e}^{6 i \left (d x +c \right )}-252 i a^{8} {\mathrm e}^{4 i \left (d x +c \right )}+1568 a^{3} b^{5} {\mathrm e}^{5 i \left (d x +c \right )}+1806 a \,b^{7} {\mathrm e}^{5 i \left (d x +c \right )}+210 a \,b^{7} {\mathrm e}^{i \left (d x +c \right )}-6384 a^{3} b^{5} {\mathrm e}^{7 i \left (d x +c \right )}+1272 a \,b^{7} {\mathrm e}^{7 i \left (d x +c \right )}+4704 a^{5} b^{3} {\mathrm e}^{5 i \left (d x +c \right )}-105 i a^{4} b^{4}-735 i a^{4} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+735 i a^{2} b^{6} {\mathrm e}^{12 i \left (d x +c \right )}-3675 i a^{4} b^{4} {\mathrm e}^{10 i \left (d x +c \right )}+3920 i a^{6} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}-7350 i a^{4} b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+1470 i a^{4} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+2205 i a^{2} b^{6} {\mathrm e}^{4 i \left (d x +c \right )}+392 i a^{6} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-12 i a^{8}+44 i b^{8}-1960 a^{3} b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+420 a \,b^{7} {\mathrm e}^{3 i \left (d x +c \right )}+609 i b^{8} {\mathrm e}^{4 i \left (d x +c \right )}+1568 a^{3} b^{5} {\mathrm e}^{9 i \left (d x +c \right )}+56 i a^{6} b^{2}+105 i a^{2} b^{6}-84 i a^{8} {\mathrm e}^{2 i \left (d x +c \right )}+203 i b^{8} {\mathrm e}^{2 i \left (d x +c \right )}+1806 a \,b^{7} {\mathrm e}^{9 i \left (d x +c \right )}-1960 a^{3} b^{5} {\mathrm e}^{11 i \left (d x +c \right )}+315 i b^{8} {\mathrm e}^{10 i \left (d x +c \right )}-420 i a^{8} {\mathrm e}^{6 i \left (d x +c \right )}+4704 a^{5} b^{3} {\mathrm e}^{9 i \left (d x +c \right )}-1920 a^{7} b \,{\mathrm e}^{7 i \left (d x +c \right )}+210 a \,b^{7} {\mathrm e}^{13 i \left (d x +c \right )}+420 a \,b^{7} {\mathrm e}^{11 i \left (d x +c \right )}+105 i b^{8} {\mathrm e}^{12 i \left (d x +c \right )}-1960 i a^{6} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+1176 i a^{6} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+3675 i a^{4} b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+3675 i a^{2} b^{6} {\mathrm e}^{8 i \left (d x +c \right )}\right )}{105 d \left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )^{7}}\) | \(672\) |
1/105*((105*b^8*d*x+1120*a^5*b^3)*tan(1/2*d*x+1/2*c)^14+(-210*a^8+210*b^8) *tan(1/2*d*x+1/2*c)^13+(-735*b^8*d*x-1680*a^7*b-7840*a^5*b^3)*tan(1/2*d*x+ 1/2*c)^12+(420*a^8-7840*a^6*b^2-1540*b^8)*tan(1/2*d*x+1/2*c)^11+2205*x*tan (1/2*d*x+1/2*c)^10*b^8*d+(-1806*a^8-6272*a^6*b^2-47040*a^4*b^4+4942*b^8)*t an(1/2*d*x+1/2*c)^9+(-3675*b^8*d*x-8400*a^7*b-62720*a^5*b^3-62720*a^3*b^5) *tan(1/2*d*x+1/2*c)^8+(1272*a^8-25536*a^6*b^2-40320*a^4*b^4-53760*a^2*b^6- 9144*b^8)*tan(1/2*d*x+1/2*c)^7-7840*(-15/32*b^5*d*x+a^5+4*a^3*b^2+24/7*a*b ^4)*b^3*tan(1/2*d*x+1/2*c)^6+(-1806*a^8-6272*a^6*b^2-47040*a^4*b^4+4942*b^ 8)*tan(1/2*d*x+1/2*c)^5+(-2205*b^8*d*x-5040*a^7*b-32928*a^5*b^3-18816*a^3* b^5+16128*a*b^7)*tan(1/2*d*x+1/2*c)^4+(420*a^8-7840*a^6*b^2-1540*b^8)*tan( 1/2*d*x+1/2*c)^3+(735*b^8*d*x+3136*a^5*b^3+6272*a^3*b^5-5376*a*b^7)*tan(1/ 2*d*x+1/2*c)^2+(-210*a^8+210*b^8)*tan(1/2*d*x+1/2*c)-105*b^8*d*x-240*a^7*b -448*a^5*b^3-896*a^3*b^5+768*a*b^7)/d/(tan(1/2*d*x+1/2*c)^2-1)^7
Time = 0.29 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.76 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {105 \, b^{8} d x \cos \left (d x + c\right )^{7} - 840 \, a b^{7} \cos \left (d x + c\right )^{6} + 120 \, a^{7} b + 840 \, a^{5} b^{3} + 840 \, a^{3} b^{5} + 120 \, a b^{7} + 280 \, {\left (7 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{4} - 168 \, {\left (7 \, a^{5} b^{3} + 14 \, a^{3} b^{5} + 3 \, a b^{7}\right )} \cos \left (d x + c\right )^{2} + {\left (15 \, a^{8} + 420 \, a^{6} b^{2} + 1050 \, a^{4} b^{4} + 420 \, a^{2} b^{6} + 15 \, b^{8} + 4 \, {\left (12 \, a^{8} - 56 \, a^{6} b^{2} + 105 \, a^{4} b^{4} - 105 \, a^{2} b^{6} - 44 \, b^{8}\right )} \cos \left (d x + c\right )^{6} + 2 \, {\left (12 \, a^{8} - 56 \, a^{6} b^{2} + 105 \, a^{4} b^{4} + 630 \, a^{2} b^{6} + 61 \, b^{8}\right )} \cos \left (d x + c\right )^{4} + 6 \, {\left (3 \, a^{8} - 14 \, a^{6} b^{2} - 280 \, a^{4} b^{4} - 210 \, a^{2} b^{6} - 11 \, b^{8}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{105 \, d \cos \left (d x + c\right )^{7}} \]
1/105*(105*b^8*d*x*cos(d*x + c)^7 - 840*a*b^7*cos(d*x + c)^6 + 120*a^7*b + 840*a^5*b^3 + 840*a^3*b^5 + 120*a*b^7 + 280*(7*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^4 - 168*(7*a^5*b^3 + 14*a^3*b^5 + 3*a*b^7)*cos(d*x + c)^2 + (15*a^8 + 420*a^6*b^2 + 1050*a^4*b^4 + 420*a^2*b^6 + 15*b^8 + 4*(12*a^8 - 56*a^6*b ^2 + 105*a^4*b^4 - 105*a^2*b^6 - 44*b^8)*cos(d*x + c)^6 + 2*(12*a^8 - 56*a ^6*b^2 + 105*a^4*b^4 + 630*a^2*b^6 + 61*b^8)*cos(d*x + c)^4 + 6*(3*a^8 - 1 4*a^6*b^2 - 280*a^4*b^4 - 210*a^2*b^6 - 11*b^8)*cos(d*x + c)^2)*sin(d*x + c))/(d*cos(d*x + c)^7)
Timed out. \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx=\text {Timed out} \]
Time = 0.27 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.77 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx=\frac {420 \, a^{2} b^{6} \tan \left (d x + c\right )^{7} + 3 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} a^{8} + 28 \, {\left (15 \, \tan \left (d x + c\right )^{7} + 42 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3}\right )} a^{6} b^{2} + 210 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 7 \, \tan \left (d x + c\right )^{5}\right )} a^{4} b^{4} + {\left (15 \, \tan \left (d x + c\right )^{7} - 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 105 \, d x + 105 \, c - 105 \, \tan \left (d x + c\right )\right )} b^{8} - \frac {168 \, {\left (7 \, \cos \left (d x + c\right )^{2} - 5\right )} a^{5} b^{3}}{\cos \left (d x + c\right )^{7}} + \frac {56 \, {\left (35 \, \cos \left (d x + c\right )^{4} - 42 \, \cos \left (d x + c\right )^{2} + 15\right )} a^{3} b^{5}}{\cos \left (d x + c\right )^{7}} - \frac {24 \, {\left (35 \, \cos \left (d x + c\right )^{6} - 35 \, \cos \left (d x + c\right )^{4} + 21 \, \cos \left (d x + c\right )^{2} - 5\right )} a b^{7}}{\cos \left (d x + c\right )^{7}} + \frac {120 \, a^{7} b}{\cos \left (d x + c\right )^{7}}}{105 \, d} \]
1/105*(420*a^2*b^6*tan(d*x + c)^7 + 3*(5*tan(d*x + c)^7 + 21*tan(d*x + c)^ 5 + 35*tan(d*x + c)^3 + 35*tan(d*x + c))*a^8 + 28*(15*tan(d*x + c)^7 + 42* tan(d*x + c)^5 + 35*tan(d*x + c)^3)*a^6*b^2 + 210*(5*tan(d*x + c)^7 + 7*ta n(d*x + c)^5)*a^4*b^4 + (15*tan(d*x + c)^7 - 21*tan(d*x + c)^5 + 35*tan(d* x + c)^3 + 105*d*x + 105*c - 105*tan(d*x + c))*b^8 - 168*(7*cos(d*x + c)^2 - 5)*a^5*b^3/cos(d*x + c)^7 + 56*(35*cos(d*x + c)^4 - 42*cos(d*x + c)^2 + 15)*a^3*b^5/cos(d*x + c)^7 - 24*(35*cos(d*x + c)^6 - 35*cos(d*x + c)^4 + 21*cos(d*x + c)^2 - 5)*a*b^7/cos(d*x + c)^7 + 120*a^7*b/cos(d*x + c)^7)/d
Time = 0.45 (sec) , antiderivative size = 726, normalized size of antiderivative = 1.80 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx=\text {Too large to display} \]
1/105*(105*(d*x + c)*b^8 - 2*(105*a^8*tan(1/2*d*x + 1/2*c)^13 - 105*b^8*ta n(1/2*d*x + 1/2*c)^13 + 840*a^7*b*tan(1/2*d*x + 1/2*c)^12 - 210*a^8*tan(1/ 2*d*x + 1/2*c)^11 + 3920*a^6*b^2*tan(1/2*d*x + 1/2*c)^11 + 770*b^8*tan(1/2 *d*x + 1/2*c)^11 + 11760*a^5*b^3*tan(1/2*d*x + 1/2*c)^10 + 903*a^8*tan(1/2 *d*x + 1/2*c)^9 + 3136*a^6*b^2*tan(1/2*d*x + 1/2*c)^9 + 23520*a^4*b^4*tan( 1/2*d*x + 1/2*c)^9 - 2471*b^8*tan(1/2*d*x + 1/2*c)^9 + 4200*a^7*b*tan(1/2* d*x + 1/2*c)^8 + 11760*a^5*b^3*tan(1/2*d*x + 1/2*c)^8 + 31360*a^3*b^5*tan( 1/2*d*x + 1/2*c)^8 - 636*a^8*tan(1/2*d*x + 1/2*c)^7 + 12768*a^6*b^2*tan(1/ 2*d*x + 1/2*c)^7 + 20160*a^4*b^4*tan(1/2*d*x + 1/2*c)^7 + 26880*a^2*b^6*ta n(1/2*d*x + 1/2*c)^7 + 4572*b^8*tan(1/2*d*x + 1/2*c)^7 + 23520*a^5*b^3*tan (1/2*d*x + 1/2*c)^6 + 15680*a^3*b^5*tan(1/2*d*x + 1/2*c)^6 + 13440*a*b^7*t an(1/2*d*x + 1/2*c)^6 + 903*a^8*tan(1/2*d*x + 1/2*c)^5 + 3136*a^6*b^2*tan( 1/2*d*x + 1/2*c)^5 + 23520*a^4*b^4*tan(1/2*d*x + 1/2*c)^5 - 2471*b^8*tan(1 /2*d*x + 1/2*c)^5 + 2520*a^7*b*tan(1/2*d*x + 1/2*c)^4 + 4704*a^5*b^3*tan(1 /2*d*x + 1/2*c)^4 + 9408*a^3*b^5*tan(1/2*d*x + 1/2*c)^4 - 8064*a*b^7*tan(1 /2*d*x + 1/2*c)^4 - 210*a^8*tan(1/2*d*x + 1/2*c)^3 + 3920*a^6*b^2*tan(1/2* d*x + 1/2*c)^3 + 770*b^8*tan(1/2*d*x + 1/2*c)^3 + 2352*a^5*b^3*tan(1/2*d*x + 1/2*c)^2 - 3136*a^3*b^5*tan(1/2*d*x + 1/2*c)^2 + 2688*a*b^7*tan(1/2*d*x + 1/2*c)^2 + 105*a^8*tan(1/2*d*x + 1/2*c) - 105*b^8*tan(1/2*d*x + 1/2*c) + 120*a^7*b - 336*a^5*b^3 + 448*a^3*b^5 - 384*a*b^7)/(tan(1/2*d*x + 1/2...
Time = 8.66 (sec) , antiderivative size = 546, normalized size of antiderivative = 1.35 \[ \int \sec ^8(c+d x) (a+b \sin (c+d x))^8 \, dx=b^8\,x-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (-4\,a^8+\frac {224\,a^6\,b^2}{3}+\frac {44\,b^8}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (-4\,a^8+\frac {224\,a^6\,b^2}{3}+\frac {44\,b^8}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (2\,a^8-2\,b^8\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {224\,a^5\,b^3}{5}-\frac {896\,a^3\,b^5}{15}+\frac {256\,a\,b^7}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (448\,a^5\,b^3+\frac {896\,a^3\,b^5}{3}+256\,a\,b^7\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (80\,a^7\,b+224\,a^5\,b^3+\frac {1792\,a^3\,b^5}{3}\right )-\frac {256\,a\,b^7}{35}+\frac {16\,a^7\,b}{7}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (48\,a^7\,b+\frac {448\,a^5\,b^3}{5}+\frac {896\,a^3\,b^5}{5}-\frac {768\,a\,b^7}{5}\right )+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^8-2\,b^8\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (-\frac {424\,a^8}{35}+\frac {1216\,a^6\,b^2}{5}+384\,a^4\,b^4+512\,a^2\,b^6+\frac {3048\,b^8}{35}\right )+\frac {128\,a^3\,b^5}{15}-\frac {32\,a^5\,b^3}{5}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {86\,a^8}{5}+\frac {896\,a^6\,b^2}{15}+448\,a^4\,b^4-\frac {706\,b^8}{15}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {86\,a^8}{5}+\frac {896\,a^6\,b^2}{15}+448\,a^4\,b^4-\frac {706\,b^8}{15}\right )+224\,a^5\,b^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+16\,a^7\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{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 )} \]
b^8*x - (tan(c/2 + (d*x)/2)^3*((44*b^8)/3 - 4*a^8 + (224*a^6*b^2)/3) + tan (c/2 + (d*x)/2)^11*((44*b^8)/3 - 4*a^8 + (224*a^6*b^2)/3) + tan(c/2 + (d*x )/2)^13*(2*a^8 - 2*b^8) + tan(c/2 + (d*x)/2)^2*((256*a*b^7)/5 - (896*a^3*b ^5)/15 + (224*a^5*b^3)/5) + tan(c/2 + (d*x)/2)^6*(256*a*b^7 + (896*a^3*b^5 )/3 + 448*a^5*b^3) + tan(c/2 + (d*x)/2)^8*(80*a^7*b + (1792*a^3*b^5)/3 + 2 24*a^5*b^3) - (256*a*b^7)/35 + (16*a^7*b)/7 + tan(c/2 + (d*x)/2)^4*(48*a^7 *b - (768*a*b^7)/5 + (896*a^3*b^5)/5 + (448*a^5*b^3)/5) + tan(c/2 + (d*x)/ 2)*(2*a^8 - 2*b^8) + tan(c/2 + (d*x)/2)^7*((3048*b^8)/35 - (424*a^8)/35 + 512*a^2*b^6 + 384*a^4*b^4 + (1216*a^6*b^2)/5) + (128*a^3*b^5)/15 - (32*a^5 *b^3)/5 + tan(c/2 + (d*x)/2)^5*((86*a^8)/5 - (706*b^8)/15 + 448*a^4*b^4 + (896*a^6*b^2)/15) + tan(c/2 + (d*x)/2)^9*((86*a^8)/5 - (706*b^8)/15 + 448* a^4*b^4 + (896*a^6*b^2)/15) + 224*a^5*b^3*tan(c/2 + (d*x)/2)^10 + 16*a^7*b *tan(c/2 + (d*x)/2)^12)/(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))