Integrand size = 41, antiderivative size = 303 \[ \int \cos ^8(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{128} a^4 (323 A+352 B+392 C) x+\frac {a^4 (208 A+227 B+252 C) \sin (c+d x)}{35 d}+\frac {a^4 (323 A+352 B+392 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a^4 (2007 A+2208 B+2408 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac {a (A+2 B) \cos ^6(c+d x) (a+a \sec (c+d x))^3 \sin (c+d x)}{14 d}+\frac {A \cos ^7(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(61 A+80 B+56 C) \cos ^5(c+d x) \left (a^2+a^2 \sec (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (7 A+8 (B+C)) \cos ^4(c+d x) \left (a^4+a^4 \sec (c+d x)\right ) \sin (c+d x)}{120 d}-\frac {a^4 (208 A+227 B+252 C) \sin ^3(c+d x)}{105 d} \] Output:
1/128*a^4*(323*A+352*B+392*C)*x+1/35*a^4*(208*A+227*B+252*C)*sin(d*x+c)/d+ 1/128*a^4*(323*A+352*B+392*C)*cos(d*x+c)*sin(d*x+c)/d+1/2240*a^4*(2007*A+2 208*B+2408*C)*cos(d*x+c)^3*sin(d*x+c)/d+1/14*a*(A+2*B)*cos(d*x+c)^6*(a+a*s ec(d*x+c))^3*sin(d*x+c)/d+1/8*A*cos(d*x+c)^7*(a+a*sec(d*x+c))^4*sin(d*x+c) /d+1/336*(61*A+80*B+56*C)*cos(d*x+c)^5*(a^2+a^2*sec(d*x+c))^2*sin(d*x+c)/d +7/120*(7*A+8*B+8*C)*cos(d*x+c)^4*(a^4+a^4*sec(d*x+c))*sin(d*x+c)/d-1/105* a^4*(208*A+227*B+252*C)*sin(d*x+c)^3/d
Time = 1.10 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.78 \[ \int \cos ^8(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 (106680 A c+295680 B c+271320 A d x+295680 B d x+329280 C d x+1680 (300 A+323 B+352 C) \sin (c+d x)+1680 (120 A+124 B+127 C) \sin (2 (c+d x))+91840 A \sin (3 (c+d x))+87920 B \sin (3 (c+d x))+80640 C \sin (3 (c+d x))+39480 A \sin (4 (c+d x))+33600 B \sin (4 (c+d x))+25200 C \sin (4 (c+d x))+14784 A \sin (5 (c+d x))+10416 B \sin (5 (c+d x))+5376 C \sin (5 (c+d x))+4480 A \sin (6 (c+d x))+2240 B \sin (6 (c+d x))+560 C \sin (6 (c+d x))+960 A \sin (7 (c+d x))+240 B \sin (7 (c+d x))+105 A \sin (8 (c+d x)))}{107520 d} \] Input:
Integrate[Cos[c + d*x]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Se c[c + d*x]^2),x]
Output:
(a^4*(106680*A*c + 295680*B*c + 271320*A*d*x + 295680*B*d*x + 329280*C*d*x + 1680*(300*A + 323*B + 352*C)*Sin[c + d*x] + 1680*(120*A + 124*B + 127*C )*Sin[2*(c + d*x)] + 91840*A*Sin[3*(c + d*x)] + 87920*B*Sin[3*(c + d*x)] + 80640*C*Sin[3*(c + d*x)] + 39480*A*Sin[4*(c + d*x)] + 33600*B*Sin[4*(c + d*x)] + 25200*C*Sin[4*(c + d*x)] + 14784*A*Sin[5*(c + d*x)] + 10416*B*Sin[ 5*(c + d*x)] + 5376*C*Sin[5*(c + d*x)] + 4480*A*Sin[6*(c + d*x)] + 2240*B* Sin[6*(c + d*x)] + 560*C*Sin[6*(c + d*x)] + 960*A*Sin[7*(c + d*x)] + 240*B *Sin[7*(c + d*x)] + 105*A*Sin[8*(c + d*x)]))/(107520*d)
Time = 2.02 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.01, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.463, Rules used = {3042, 4574, 3042, 4505, 3042, 4505, 3042, 4505, 27, 3042, 4484, 25, 3042, 4274, 3042, 3113, 2009, 3115, 24}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \cos ^8(c+d x) (a \sec (c+d x)+a)^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^8}dx\) |
| \(\Big \downarrow \) 4574 |
| \(\displaystyle \frac {\int \cos ^7(c+d x) (\sec (c+d x) a+a)^4 (4 a (A+2 B)+a (3 A+8 C) \sec (c+d x))dx}{8 a}+\frac {A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (4 a (A+2 B)+a (3 A+8 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^7}dx}{8 a}+\frac {A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {1}{7} \int \cos ^6(c+d x) (\sec (c+d x) a+a)^3 \left ((61 A+80 B+56 C) a^2+(33 A+24 B+56 C) \sec (c+d x) a^2\right )dx+\frac {4 a^2 (A+2 B) \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}}{8 a}+\frac {A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left ((61 A+80 B+56 C) a^2+(33 A+24 B+56 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx+\frac {4 a^2 (A+2 B) \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}}{8 a}+\frac {A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \int \cos ^5(c+d x) (\sec (c+d x) a+a)^2 \left (98 (7 A+8 (B+C)) a^3+3 (127 A+128 B+168 C) \sec (c+d x) a^3\right )dx+\frac {a^3 (61 A+80 B+56 C) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (A+2 B) \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}}{8 a}+\frac {A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (98 (7 A+8 (B+C)) a^3+3 (127 A+128 B+168 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {a^3 (61 A+80 B+56 C) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (A+2 B) \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}}{8 a}+\frac {A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \int 3 \cos ^4(c+d x) (\sec (c+d x) a+a) \left ((2007 A+2208 B+2408 C) a^4+(1321 A+1424 B+1624 C) \sec (c+d x) a^4\right )dx+\frac {98 (7 A+8 (B+C)) \sin (c+d x) \cos ^4(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (61 A+80 B+56 C) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (A+2 B) \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}}{8 a}+\frac {A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \int \cos ^4(c+d x) (\sec (c+d x) a+a) \left ((2007 A+2208 B+2408 C) a^4+(1321 A+1424 B+1624 C) \sec (c+d x) a^4\right )dx+\frac {98 (7 A+8 (B+C)) \sin (c+d x) \cos ^4(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (61 A+80 B+56 C) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (A+2 B) \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}}{8 a}+\frac {A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((2007 A+2208 B+2408 C) a^4+(1321 A+1424 B+1624 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^4\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {98 (7 A+8 (B+C)) \sin (c+d x) \cos ^4(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (61 A+80 B+56 C) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (A+2 B) \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}}{8 a}+\frac {A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 4484 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {a^5 (2007 A+2208 B+2408 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac {1}{4} \int -\cos ^3(c+d x) \left (64 (208 A+227 B+252 C) a^5+35 (323 A+352 B+392 C) \sec (c+d x) a^5\right )dx\right )+\frac {98 (7 A+8 (B+C)) \sin (c+d x) \cos ^4(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (61 A+80 B+56 C) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (A+2 B) \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}}{8 a}+\frac {A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \int \cos ^3(c+d x) \left (64 (208 A+227 B+252 C) a^5+35 (323 A+352 B+392 C) \sec (c+d x) a^5\right )dx+\frac {a^5 (2007 A+2208 B+2408 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (7 A+8 (B+C)) \sin (c+d x) \cos ^4(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (61 A+80 B+56 C) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (A+2 B) \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}}{8 a}+\frac {A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \int \frac {64 (208 A+227 B+252 C) a^5+35 (323 A+352 B+392 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^5}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a^5 (2007 A+2208 B+2408 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (7 A+8 (B+C)) \sin (c+d x) \cos ^4(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (61 A+80 B+56 C) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (A+2 B) \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}}{8 a}+\frac {A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (64 a^5 (208 A+227 B+252 C) \int \cos ^3(c+d x)dx+35 a^5 (323 A+352 B+392 C) \int \cos ^2(c+d x)dx\right )+\frac {a^5 (2007 A+2208 B+2408 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (7 A+8 (B+C)) \sin (c+d x) \cos ^4(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (61 A+80 B+56 C) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (A+2 B) \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}}{8 a}+\frac {A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^5 (323 A+352 B+392 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+64 a^5 (208 A+227 B+252 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx\right )+\frac {a^5 (2007 A+2208 B+2408 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (7 A+8 (B+C)) \sin (c+d x) \cos ^4(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (61 A+80 B+56 C) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (A+2 B) \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}}{8 a}+\frac {A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^5 (323 A+352 B+392 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {64 a^5 (208 A+227 B+252 C) \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}\right )+\frac {a^5 (2007 A+2208 B+2408 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (7 A+8 (B+C)) \sin (c+d x) \cos ^4(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (61 A+80 B+56 C) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (A+2 B) \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}}{8 a}+\frac {A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^5 (323 A+352 B+392 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {64 a^5 (208 A+227 B+252 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {a^5 (2007 A+2208 B+2408 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (7 A+8 (B+C)) \sin (c+d x) \cos ^4(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (61 A+80 B+56 C) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (A+2 B) \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}}{8 a}+\frac {A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^5 (323 A+352 B+392 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {64 a^5 (208 A+227 B+252 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {a^5 (2007 A+2208 B+2408 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (7 A+8 (B+C)) \sin (c+d x) \cos ^4(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (61 A+80 B+56 C) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (A+2 B) \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}}{8 a}+\frac {A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {4 a^2 (A+2 B) \sin (c+d x) \cos ^6(c+d x) (a \sec (c+d x)+a)^3}{7 d}+\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {a^5 (2007 A+2208 B+2408 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {1}{4} \left (35 a^5 (323 A+352 B+392 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {64 a^5 (208 A+227 B+252 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )\right )+\frac {98 (7 A+8 (B+C)) \sin (c+d x) \cos ^4(c+d x) \left (a^5 \sec (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (61 A+80 B+56 C) \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d}\right )}{8 a}+\frac {A \sin (c+d x) \cos ^7(c+d x) (a \sec (c+d x)+a)^4}{8 d}\) |
Input:
Int[Cos[c + d*x]^8*(a + a*Sec[c + d*x])^4*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Output:
(A*Cos[c + d*x]^7*(a + a*Sec[c + d*x])^4*Sin[c + d*x])/(8*d) + ((4*a^2*(A + 2*B)*Cos[c + d*x]^6*(a + a*Sec[c + d*x])^3*Sin[c + d*x])/(7*d) + ((a^3*( 61*A + 80*B + 56*C)*Cos[c + d*x]^5*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(6 *d) + ((98*(7*A + 8*(B + C))*Cos[c + d*x]^4*(a^5 + a^5*Sec[c + d*x])*Sin[c + d*x])/(5*d) + (3*((a^5*(2007*A + 2208*B + 2408*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (35*a^5*(323*A + 352*B + 392*C)*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)) - (64*a^5*(208*A + 227*B + 252*C)*(-Sin[c + d*x] + Sin[c + d*x]^3/3))/d)/4))/5)/6)/7)/(8*a)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Simp[1/(d*n) Int[(d*Csc[e + f*x])^( n + 1)*Simp[n*(B*a + A*b) + (B*b*n + A*a*(n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && LeQ[n, -1]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[a*A*Cot [e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*n)), x] - Sim p[b/(a*d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Sim p[a*A*(m - n - 1) - b*B*n - (a*B*n + A*b*(m + n))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0 ] && GtQ[m, 1/2] && LtQ[n, -1]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. ))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a _))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e + f*x])^n/(f*n)), x] - Simp[1/(b*d*n) Int[(a + b*Csc[e + f*x])^m*(d*Csc[ e + f*x])^(n + 1)*Simp[a*A*m - b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x] , x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
Time = 5.96 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.55
| method | result | size |
| parallelrisch | \(\frac {41 \left (\frac {3 \left (30 A +31 B +\frac {127 C}{4}\right ) \sin \left (2 d x +2 c \right )}{41}+\left (A +\frac {157 B}{164}+\frac {36 C}{41}\right ) \sin \left (3 d x +3 c \right )+\frac {3 \left (\frac {47 A}{8}+5 B +\frac {15 C}{4}\right ) \sin \left (4 d x +4 c \right )}{41}+\frac {3 \left (11 A +\frac {31 B}{4}+4 C \right ) \sin \left (5 d x +5 c \right )}{205}+\frac {\left (2 A +B +\frac {C}{4}\right ) \sin \left (6 d x +6 c \right )}{41}+\frac {3 \left (A +\frac {B}{4}\right ) \sin \left (7 d x +7 c \right )}{287}+\frac {3 A \sin \left (8 d x +8 c \right )}{2624}+\frac {3 \left (75 A +\frac {323 B}{4}+88 C \right ) \sin \left (d x +c \right )}{41}+\frac {969 x \left (A +\frac {352 B}{323}+\frac {392 C}{323}\right ) d}{328}\right ) a^{4}}{48 d}\) | \(168\) |
| risch | \(\frac {323 a^{4} A x}{128}+\frac {11 a^{4} x B}{4}+\frac {49 a^{4} x C}{16}+\frac {75 \sin \left (d x +c \right ) a^{4} A}{16 d}+\frac {323 \sin \left (d x +c \right ) B \,a^{4}}{64 d}+\frac {11 \sin \left (d x +c \right ) a^{4} C}{2 d}+\frac {a^{4} A \sin \left (8 d x +8 c \right )}{1024 d}+\frac {a^{4} A \sin \left (7 d x +7 c \right )}{112 d}+\frac {\sin \left (7 d x +7 c \right ) B \,a^{4}}{448 d}+\frac {a^{4} A \sin \left (6 d x +6 c \right )}{24 d}+\frac {\sin \left (6 d x +6 c \right ) B \,a^{4}}{48 d}+\frac {\sin \left (6 d x +6 c \right ) a^{4} C}{192 d}+\frac {11 a^{4} A \sin \left (5 d x +5 c \right )}{80 d}+\frac {31 \sin \left (5 d x +5 c \right ) B \,a^{4}}{320 d}+\frac {\sin \left (5 d x +5 c \right ) a^{4} C}{20 d}+\frac {47 a^{4} A \sin \left (4 d x +4 c \right )}{128 d}+\frac {5 \sin \left (4 d x +4 c \right ) B \,a^{4}}{16 d}+\frac {15 \sin \left (4 d x +4 c \right ) a^{4} C}{64 d}+\frac {41 a^{4} A \sin \left (3 d x +3 c \right )}{48 d}+\frac {157 \sin \left (3 d x +3 c \right ) B \,a^{4}}{192 d}+\frac {3 \sin \left (3 d x +3 c \right ) a^{4} C}{4 d}+\frac {15 \sin \left (2 d x +2 c \right ) a^{4} A}{8 d}+\frac {31 \sin \left (2 d x +2 c \right ) B \,a^{4}}{16 d}+\frac {127 \sin \left (2 d x +2 c \right ) a^{4} C}{64 d}\) | \(392\) |
| derivativedivides | \(\frac {a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{4} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{4} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+6 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {6 B \,a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+6 a^{4} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{4} A \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}+4 B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 a^{4} C \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )+\frac {B \,a^{4} \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}+a^{4} C \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(577\) |
| default | \(\frac {a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {B \,a^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{4} C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{4} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+6 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {6 B \,a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+6 a^{4} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{4} A \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}+4 B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 a^{4} C \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{7}+\frac {7 \cos \left (d x +c \right )^{5}}{6}+\frac {35 \cos \left (d x +c \right )^{3}}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )+\frac {B \,a^{4} \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}+a^{4} C \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(577\) |
Input:
int(cos(d*x+c)^8*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x,meth od=_RETURNVERBOSE)
Output:
41/48*(3/41*(30*A+31*B+127/4*C)*sin(2*d*x+2*c)+(A+157/164*B+36/41*C)*sin(3 *d*x+3*c)+3/41*(47/8*A+5*B+15/4*C)*sin(4*d*x+4*c)+3/205*(11*A+31/4*B+4*C)* sin(5*d*x+5*c)+1/41*(2*A+B+1/4*C)*sin(6*d*x+6*c)+3/287*(A+1/4*B)*sin(7*d*x +7*c)+3/2624*A*sin(8*d*x+8*c)+3/41*(75*A+323/4*B+88*C)*sin(d*x+c)+969/328* x*(A+352/323*B+392/323*C)*d)*a^4/d
Time = 0.09 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.63 \[ \int \cos ^8(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (323 \, A + 352 \, B + 392 \, C\right )} a^{4} d x + {\left (1680 \, A a^{4} \cos \left (d x + c\right )^{7} + 1920 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{6} + 280 \, {\left (55 \, A + 32 \, B + 8 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 1536 \, {\left (13 \, A + 12 \, B + 7 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (323 \, A + 352 \, B + 328 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 128 \, {\left (208 \, A + 227 \, B + 252 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \, {\left (323 \, A + 352 \, B + 392 \, C\right )} a^{4} \cos \left (d x + c\right ) + 256 \, {\left (208 \, A + 227 \, B + 252 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{13440 \, d} \] Input:
integrate(cos(d*x+c)^8*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="fricas")
Output:
1/13440*(105*(323*A + 352*B + 392*C)*a^4*d*x + (1680*A*a^4*cos(d*x + c)^7 + 1920*(4*A + B)*a^4*cos(d*x + c)^6 + 280*(55*A + 32*B + 8*C)*a^4*cos(d*x + c)^5 + 1536*(13*A + 12*B + 7*C)*a^4*cos(d*x + c)^4 + 70*(323*A + 352*B + 328*C)*a^4*cos(d*x + c)^3 + 128*(208*A + 227*B + 252*C)*a^4*cos(d*x + c)^ 2 + 105*(323*A + 352*B + 392*C)*a^4*cos(d*x + c) + 256*(208*A + 227*B + 25 2*C)*a^4)*sin(d*x + c))/d
Timed out. \[ \int \cos ^8(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**8*(a+a*sec(d*x+c))**4*(A+B*sec(d*x+c)+C*sec(d*x+c)** 2),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (286) = 572\).
Time = 0.06 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.91 \[ \int \cos ^8(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^8*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="maxima")
Output:
-1/107520*(12288*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*A*a^4 - 28672*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^4 + 35*(128*sin(2*d*x + 2*c)^3 - 840*d*x - 840*c - 3* sin(8*d*x + 8*c) - 168*sin(4*d*x + 4*c) - 768*sin(2*d*x + 2*c))*A*a^4 + 33 60*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d *x + 2*c))*A*a^4 - 3360*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 + 3072*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c) ^3 - 35*sin(d*x + c))*B*a^4 - 43008*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a^4 + 2240*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9* sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*B*a^4 + 35840*(sin(d*x + c)^3 - 3* sin(d*x + c))*B*a^4 - 13440*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d* x + 2*c))*B*a^4 - 28672*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*a^4 + 560*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4 *c) - 48*sin(2*d*x + 2*c))*C*a^4 + 143360*(sin(d*x + c)^3 - 3*sin(d*x + c) )*C*a^4 - 20160*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C* a^4 - 26880*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^4)/d
Time = 0.37 (sec) , antiderivative size = 452, normalized size of antiderivative = 1.49 \[ \int \cos ^8(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^8*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="giac")
Output:
1/13440*(105*(323*A*a^4 + 352*B*a^4 + 392*C*a^4)*(d*x + c) + 2*(33915*A*a^ 4*tan(1/2*d*x + 1/2*c)^15 + 36960*B*a^4*tan(1/2*d*x + 1/2*c)^15 + 41160*C* a^4*tan(1/2*d*x + 1/2*c)^15 + 260015*A*a^4*tan(1/2*d*x + 1/2*c)^13 + 28336 0*B*a^4*tan(1/2*d*x + 1/2*c)^13 + 315560*C*a^4*tan(1/2*d*x + 1/2*c)^13 + 8 65963*A*a^4*tan(1/2*d*x + 1/2*c)^11 + 943712*B*a^4*tan(1/2*d*x + 1/2*c)^11 + 1050952*C*a^4*tan(1/2*d*x + 1/2*c)^11 + 1632119*A*a^4*tan(1/2*d*x + 1/2 *c)^9 + 1778656*B*a^4*tan(1/2*d*x + 1/2*c)^9 + 1980776*C*a^4*tan(1/2*d*x + 1/2*c)^9 + 1872009*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 2090016*B*a^4*tan(1/2*d *x + 1/2*c)^7 + 2277016*C*a^4*tan(1/2*d*x + 1/2*c)^7 + 1442133*A*a^4*tan(1 /2*d*x + 1/2*c)^5 + 1479072*B*a^4*tan(1/2*d*x + 1/2*c)^5 + 1658552*C*a^4*t an(1/2*d*x + 1/2*c)^5 + 528465*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 648480*B*a^4 *tan(1/2*d*x + 1/2*c)^3 + 759640*C*a^4*tan(1/2*d*x + 1/2*c)^3 + 181125*A*a ^4*tan(1/2*d*x + 1/2*c) + 178080*B*a^4*tan(1/2*d*x + 1/2*c) + 173880*C*a^4 *tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^8)/d
Time = 14.59 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.39 \[ \int \cos ^8(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (\frac {323\,A\,a^4}{64}+\frac {11\,B\,a^4}{2}+\frac {49\,C\,a^4}{8}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+\left (\frac {7429\,A\,a^4}{192}+\frac {253\,B\,a^4}{6}+\frac {1127\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {123709\,A\,a^4}{960}+\frac {4213\,B\,a^4}{30}+\frac {18767\,C\,a^4}{120}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {1632119\,A\,a^4}{6720}+\frac {55583\,B\,a^4}{210}+\frac {35371\,C\,a^4}{120}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {624003\,A\,a^4}{2240}+\frac {21771\,B\,a^4}{70}+\frac {40661\,C\,a^4}{120}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {68673\,A\,a^4}{320}+\frac {2201\,B\,a^4}{10}+\frac {29617\,C\,a^4}{120}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {5033\,A\,a^4}{64}+\frac {193\,B\,a^4}{2}+\frac {2713\,C\,a^4}{24}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {1725\,A\,a^4}{64}+\frac {53\,B\,a^4}{2}+\frac {207\,C\,a^4}{8}\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 )}^{16}+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+70\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+56\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+28\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^4\,\mathrm {atan}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (323\,A+352\,B+392\,C\right )}{64\,\left (\frac {323\,A\,a^4}{64}+\frac {11\,B\,a^4}{2}+\frac {49\,C\,a^4}{8}\right )}\right )\,\left (323\,A+352\,B+392\,C\right )}{64\,d} \] Input:
int(cos(c + d*x)^8*(a + a/cos(c + d*x))^4*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)
Output:
(tan(c/2 + (d*x)/2)^15*((323*A*a^4)/64 + (11*B*a^4)/2 + (49*C*a^4)/8) + ta n(c/2 + (d*x)/2)^3*((5033*A*a^4)/64 + (193*B*a^4)/2 + (2713*C*a^4)/24) + t an(c/2 + (d*x)/2)^13*((7429*A*a^4)/192 + (253*B*a^4)/6 + (1127*C*a^4)/24) + tan(c/2 + (d*x)/2)^5*((68673*A*a^4)/320 + (2201*B*a^4)/10 + (29617*C*a^4 )/120) + tan(c/2 + (d*x)/2)^11*((123709*A*a^4)/960 + (4213*B*a^4)/30 + (18 767*C*a^4)/120) + tan(c/2 + (d*x)/2)^7*((624003*A*a^4)/2240 + (21771*B*a^4 )/70 + (40661*C*a^4)/120) + tan(c/2 + (d*x)/2)^9*((1632119*A*a^4)/6720 + ( 55583*B*a^4)/210 + (35371*C*a^4)/120) + tan(c/2 + (d*x)/2)*((1725*A*a^4)/6 4 + (53*B*a^4)/2 + (207*C*a^4)/8))/(d*(8*tan(c/2 + (d*x)/2)^2 + 28*tan(c/2 + (d*x)/2)^4 + 56*tan(c/2 + (d*x)/2)^6 + 70*tan(c/2 + (d*x)/2)^8 + 56*tan (c/2 + (d*x)/2)^10 + 28*tan(c/2 + (d*x)/2)^12 + 8*tan(c/2 + (d*x)/2)^14 + tan(c/2 + (d*x)/2)^16 + 1)) + (a^4*atan((a^4*tan(c/2 + (d*x)/2)*(323*A + 3 52*B + 392*C))/(64*((323*A*a^4)/64 + (11*B*a^4)/2 + (49*C*a^4)/8)))*(323*A + 352*B + 392*C))/(64*d)
Time = 0.17 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.00 \[ \int \cos ^8(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^{4} \left (-1680 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} a +20440 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a +8960 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b +2240 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} c -58450 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a -42560 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b -27440 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} c +73605 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +70560 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b +66360 \cos \left (d x +c \right ) \sin \left (d x +c \right ) c -7680 \sin \left (d x +c \right )^{7} a -1920 \sin \left (d x +c \right )^{7} b +43008 \sin \left (d x +c \right )^{5} a +24192 \sin \left (d x +c \right )^{5} b +10752 \sin \left (d x +c \right )^{5} c -89600 \sin \left (d x +c \right )^{3} a -71680 \sin \left (d x +c \right )^{3} b -53760 \sin \left (d x +c \right )^{3} c +107520 \sin \left (d x +c \right ) a +107520 \sin \left (d x +c \right ) b +107520 \sin \left (d x +c \right ) c +33915 a d x +36960 b d x +41160 c d x \right )}{13440 d} \] Input:
int(cos(d*x+c)^8*(a+a*sec(d*x+c))^4*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
Output:
(a**4*( - 1680*cos(c + d*x)*sin(c + d*x)**7*a + 20440*cos(c + d*x)*sin(c + d*x)**5*a + 8960*cos(c + d*x)*sin(c + d*x)**5*b + 2240*cos(c + d*x)*sin(c + d*x)**5*c - 58450*cos(c + d*x)*sin(c + d*x)**3*a - 42560*cos(c + d*x)*s in(c + d*x)**3*b - 27440*cos(c + d*x)*sin(c + d*x)**3*c + 73605*cos(c + d* x)*sin(c + d*x)*a + 70560*cos(c + d*x)*sin(c + d*x)*b + 66360*cos(c + d*x) *sin(c + d*x)*c - 7680*sin(c + d*x)**7*a - 1920*sin(c + d*x)**7*b + 43008* sin(c + d*x)**5*a + 24192*sin(c + d*x)**5*b + 10752*sin(c + d*x)**5*c - 89 600*sin(c + d*x)**3*a - 71680*sin(c + d*x)**3*b - 53760*sin(c + d*x)**3*c + 107520*sin(c + d*x)*a + 107520*sin(c + d*x)*b + 107520*sin(c + d*x)*c + 33915*a*d*x + 36960*b*d*x + 41160*c*d*x))/(13440*d)