Integrand size = 33, antiderivative size = 279 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{128} a^4 (392 A+323 C) x+\frac {4 a^4 (63 A+52 C) \sin (c+d x)}{35 d}+\frac {a^4 (392 A+323 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a^4 (2408 A+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac {a C \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}-\frac {4 a^4 (63 A+52 C) \sin ^3(c+d x)}{105 d} \]
1/128*a^4*(392*A+323*C)*x+4/35*a^4*(63*A+52*C)*sin(d*x+c)/d+1/128*a^4*(392 *A+323*C)*cos(d*x+c)*sin(d*x+c)/d+1/2240*a^4*(2408*A+2007*C)*cos(d*x+c)^3* sin(d*x+c)/d+1/14*a*C*cos(d*x+c)^3*(a+a*cos(d*x+c))^3*sin(d*x+c)/d+1/8*C*c os(d*x+c)^3*(a+a*cos(d*x+c))^4*sin(d*x+c)/d+1/336*(56*A+61*C)*cos(d*x+c)^3 *(a^2+a^2*cos(d*x+c))^2*sin(d*x+c)/d+7/120*(8*A+7*C)*cos(d*x+c)^3*(a^4+a^4 *cos(d*x+c))*sin(d*x+c)/d-4/105*a^4*(63*A+52*C)*sin(d*x+c)^3/d
Time = 0.56 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.60 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {a^4 (164640 c C+329280 A d x+271320 C d x+6720 (88 A+75 C) \sin (c+d x)+1680 (127 A+120 C) \sin (2 (c+d x))+80640 A \sin (3 (c+d x))+91840 C \sin (3 (c+d x))+25200 A \sin (4 (c+d x))+39480 C \sin (4 (c+d x))+5376 A \sin (5 (c+d x))+14784 C \sin (5 (c+d x))+560 A \sin (6 (c+d x))+4480 C \sin (6 (c+d x))+960 C \sin (7 (c+d x))+105 C \sin (8 (c+d x)))}{107520 d} \]
(a^4*(164640*c*C + 329280*A*d*x + 271320*C*d*x + 6720*(88*A + 75*C)*Sin[c + d*x] + 1680*(127*A + 120*C)*Sin[2*(c + d*x)] + 80640*A*Sin[3*(c + d*x)] + 91840*C*Sin[3*(c + d*x)] + 25200*A*Sin[4*(c + d*x)] + 39480*C*Sin[4*(c + d*x)] + 5376*A*Sin[5*(c + d*x)] + 14784*C*Sin[5*(c + d*x)] + 560*A*Sin[6* (c + d*x)] + 4480*C*Sin[6*(c + d*x)] + 960*C*Sin[7*(c + d*x)] + 105*C*Sin[ 8*(c + d*x)]))/(107520*d)
Time = 1.91 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.04, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.606, Rules used = {3042, 3525, 3042, 3455, 3042, 3455, 3042, 3455, 27, 3042, 3447, 3042, 3502, 3042, 3227, 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.
| \(\displaystyle \int \cos ^2(c+d x) (a \cos (c+d x)+a)^4 \left (A+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3525 |
| \(\displaystyle \frac {\int \cos ^2(c+d x) (\cos (c+d x) a+a)^4 (a (8 A+3 C)+4 a C \cos (c+d x))dx}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (a (8 A+3 C)+4 a C \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{7} \int \cos ^2(c+d x) (\cos (c+d x) a+a)^3 \left ((56 A+33 C) a^2+(56 A+61 C) \cos (c+d x) a^2\right )dx+\frac {4 a^2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left ((56 A+33 C) a^2+(56 A+61 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {4 a^2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \int \cos ^2(c+d x) (\cos (c+d x) a+a)^2 \left (3 (168 A+127 C) a^3+98 (8 A+7 C) \cos (c+d x) a^3\right )dx+\frac {a^3 (56 A+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (3 (168 A+127 C) a^3+98 (8 A+7 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {a^3 (56 A+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \int 3 \cos ^2(c+d x) (\cos (c+d x) a+a) \left ((1624 A+1321 C) a^4+(2408 A+2007 C) \cos (c+d x) a^4\right )dx+\frac {98 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \int \cos ^2(c+d x) (\cos (c+d x) a+a) \left ((1624 A+1321 C) a^4+(2408 A+2007 C) \cos (c+d x) a^4\right )dx+\frac {98 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((1624 A+1321 C) a^4+(2408 A+2007 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )dx+\frac {98 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \int \cos ^2(c+d x) \left ((2408 A+2007 C) \cos ^2(c+d x) a^5+(1624 A+1321 C) a^5+\left ((1624 A+1321 C) a^5+(2408 A+2007 C) a^5\right ) \cos (c+d x)\right )dx+\frac {98 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left ((2408 A+2007 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^5+(1624 A+1321 C) a^5+\left ((1624 A+1321 C) a^5+(2408 A+2007 C) a^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {98 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \int \cos ^2(c+d x) \left (35 (392 A+323 C) a^5+256 (63 A+52 C) \cos (c+d x) a^5\right )dx+\frac {a^5 (2408 A+2007 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (35 (392 A+323 C) a^5+256 (63 A+52 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^5\right )dx+\frac {a^5 (2408 A+2007 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (256 a^5 (63 A+52 C) \int \cos ^3(c+d x)dx+35 a^5 (392 A+323 C) \int \cos ^2(c+d x)dx\right )+\frac {a^5 (2408 A+2007 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^5 (392 A+323 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+256 a^5 (63 A+52 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx\right )+\frac {a^5 (2408 A+2007 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^5 (392 A+323 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {256 a^5 (63 A+52 C) \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}\right )+\frac {a^5 (2408 A+2007 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^5 (392 A+323 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {256 a^5 (63 A+52 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {a^5 (2408 A+2007 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^5 (392 A+323 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {256 a^5 (63 A+52 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {a^5 (2408 A+2007 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {4 a^2 C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}+\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {a^5 (2408 A+2007 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {1}{4} \left (35 a^5 (392 A+323 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {256 a^5 (63 A+52 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )\right )+\frac {98 (8 A+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
(C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(8*d) + ((4*a^2*C*C os[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d) + ((a^3*(56*A + 6 1*C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(6*d) + ((98*(8*A + 7*C)*Cos[c + d*x]^3*(a^5 + a^5*Cos[c + d*x])*Sin[c + d*x])/(5*d) + (3*( (a^5*(2408*A + 2007*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (35*a^5*(392*A + 323*C)*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)) - (256*a^5*(63*A + 52* C)*(-Sin[c + d*x] + Sin[c + d*x]^3/3))/d)/4))/5)/6)/7)/(8*a)
3.1.28.3.1 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[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[((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_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim p[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 ) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] && !LtQ[n, -1 ] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m + 2)) Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && !LtQ[m, -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)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 1 )/(d*f*(m + n + 2))), x] + Simp[1/(b*d*(m + n + 2)) Int[(a + b*Sin[e + f* x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m + n + 2) + C*(a*c*m + b*d*(n + 1 )) + C*(a*d*m - b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !LtQ[m, -2^(-1)] && NeQ[m + n + 2, 0]
Time = 10.95 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.51
| method | result | size |
| parallelrisch | \(\frac {15 \left (\left (\frac {127 A}{15}+8 C \right ) \sin \left (2 d x +2 c \right )+\frac {4 \left (4 A +\frac {41 C}{9}\right ) \sin \left (3 d x +3 c \right )}{5}+\left (A +\frac {47 C}{30}\right ) \sin \left (4 d x +4 c \right )+\frac {4 \left (4 A +11 C \right ) \sin \left (5 d x +5 c \right )}{75}+\frac {\left (A +8 C \right ) \sin \left (6 d x +6 c \right )}{45}+\frac {4 \sin \left (7 d x +7 c \right ) C}{105}+\frac {C \sin \left (8 d x +8 c \right )}{240}+4 \left (\frac {88 A}{15}+5 C \right ) \sin \left (d x +c \right )+\frac {196 \left (A +\frac {323 C}{392}\right ) x d}{15}\right ) a^{4}}{64 d}\) | \(142\) |
| risch | \(\frac {49 a^{4} x A}{16}+\frac {323 a^{4} C x}{128}+\frac {11 \sin \left (d x +c \right ) a^{4} A}{2 d}+\frac {75 \sin \left (d x +c \right ) C \,a^{4}}{16 d}+\frac {C \,a^{4} \sin \left (8 d x +8 c \right )}{1024 d}+\frac {C \,a^{4} \sin \left (7 d x +7 c \right )}{112 d}+\frac {\sin \left (6 d x +6 c \right ) a^{4} A}{192 d}+\frac {\sin \left (6 d x +6 c \right ) C \,a^{4}}{24 d}+\frac {\sin \left (5 d x +5 c \right ) a^{4} A}{20 d}+\frac {11 \sin \left (5 d x +5 c \right ) C \,a^{4}}{80 d}+\frac {15 \sin \left (4 d x +4 c \right ) a^{4} A}{64 d}+\frac {47 \sin \left (4 d x +4 c \right ) C \,a^{4}}{128 d}+\frac {3 \sin \left (3 d x +3 c \right ) a^{4} A}{4 d}+\frac {41 \sin \left (3 d x +3 c \right ) C \,a^{4}}{48 d}+\frac {127 \sin \left (2 d x +2 c \right ) a^{4} A}{64 d}+\frac {15 \sin \left (2 d x +2 c \right ) C \,a^{4}}{8 d}\) | \(262\) |
| parts | \(\frac {\left (a^{4} A +6 C \,a^{4}\right ) \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {\left (4 a^{4} A +4 C \,a^{4}\right ) \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {\left (6 a^{4} A +C \,a^{4}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {C \,a^{4} \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}+\frac {4 C \,a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7 d}+\frac {4 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}\) | \(315\) |
| derivativedivides | \(\frac {a^{4} A \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+C \,a^{4} \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )+\frac {4 a^{4} A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {4 C \,a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+6 a^{4} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+6 C \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {4 C \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(393\) |
| default | \(\frac {a^{4} A \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+C \,a^{4} \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )+\frac {4 a^{4} A \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {4 C \,a^{4} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}+6 a^{4} A \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+6 C \,a^{4} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 a^{4} A \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+\frac {4 C \,a^{4} \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,a^{4} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(393\) |
| norman | \(\frac {\frac {a^{4} \left (392 A +323 C \right ) x}{128}+\frac {69 a^{4} \left (24 A +25 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}+\frac {5053 a^{4} \left (392 A +323 C \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6720 d}+\frac {383 a^{4} \left (392 A +323 C \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{960 d}+\frac {23 a^{4} \left (392 A +323 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {a^{4} \left (392 A +323 C \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {a^{4} \left (392 A +323 C \right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {7 a^{4} \left (392 A +323 C \right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {7 a^{4} \left (392 A +323 C \right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a^{4} \left (392 A +323 C \right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64}+\frac {7 a^{4} \left (392 A +323 C \right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {7 a^{4} \left (392 A +323 C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {a^{4} \left (392 A +323 C \right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {a^{4} \left (392 A +323 C \right ) x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\frac {a^{4} \left (21704 A +15099 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {a^{4} \left (236936 A +206019 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{960 d}+\frac {a^{4} \left (2277016 A +1872009 C \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6720 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}\) | \(429\) |
15/64*((127/15*A+8*C)*sin(2*d*x+2*c)+4/5*(4*A+41/9*C)*sin(3*d*x+3*c)+(A+47 /30*C)*sin(4*d*x+4*c)+4/75*(4*A+11*C)*sin(5*d*x+5*c)+1/45*(A+8*C)*sin(6*d* x+6*c)+4/105*sin(7*d*x+7*c)*C+1/240*C*sin(8*d*x+8*c)+4*(88/15*A+5*C)*sin(d *x+c)+196/15*(A+323/392*C)*x*d)*a^4/d
Time = 0.29 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.59 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (392 \, A + 323 \, C\right )} a^{4} d x + {\left (1680 \, C a^{4} \cos \left (d x + c\right )^{7} + 7680 \, C a^{4} \cos \left (d x + c\right )^{6} + 280 \, {\left (8 \, A + 55 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 1536 \, {\left (7 \, A + 13 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (328 \, A + 323 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 512 \, {\left (63 \, A + 52 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \, {\left (392 \, A + 323 \, C\right )} a^{4} \cos \left (d x + c\right ) + 1024 \, {\left (63 \, A + 52 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{13440 \, d} \]
1/13440*(105*(392*A + 323*C)*a^4*d*x + (1680*C*a^4*cos(d*x + c)^7 + 7680*C *a^4*cos(d*x + c)^6 + 280*(8*A + 55*C)*a^4*cos(d*x + c)^5 + 1536*(7*A + 13 *C)*a^4*cos(d*x + c)^4 + 70*(328*A + 323*C)*a^4*cos(d*x + c)^3 + 512*(63*A + 52*C)*a^4*cos(d*x + c)^2 + 105*(392*A + 323*C)*a^4*cos(d*x + c) + 1024* (63*A + 52*C)*a^4)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 1149 vs. \(2 (262) = 524\).
Time = 0.77 (sec) , antiderivative size = 1149, normalized size of antiderivative = 4.12 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
Piecewise((5*A*a**4*x*sin(c + d*x)**6/16 + 15*A*a**4*x*sin(c + d*x)**4*cos (c + d*x)**2/16 + 9*A*a**4*x*sin(c + d*x)**4/4 + 15*A*a**4*x*sin(c + d*x)* *2*cos(c + d*x)**4/16 + 9*A*a**4*x*sin(c + d*x)**2*cos(c + d*x)**2/2 + A*a **4*x*sin(c + d*x)**2/2 + 5*A*a**4*x*cos(c + d*x)**6/16 + 9*A*a**4*x*cos(c + d*x)**4/4 + A*a**4*x*cos(c + d*x)**2/2 + 5*A*a**4*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 32*A*a**4*sin(c + d*x)**5/(15*d) + 5*A*a**4*sin(c + d*x)* *3*cos(c + d*x)**3/(6*d) + 16*A*a**4*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + 9*A*a**4*sin(c + d*x)**3*cos(c + d*x)/(4*d) + 8*A*a**4*sin(c + d*x)**3/ (3*d) + 11*A*a**4*sin(c + d*x)*cos(c + d*x)**5/(16*d) + 4*A*a**4*sin(c + d *x)*cos(c + d*x)**4/d + 15*A*a**4*sin(c + d*x)*cos(c + d*x)**3/(4*d) + 4*A *a**4*sin(c + d*x)*cos(c + d*x)**2/d + A*a**4*sin(c + d*x)*cos(c + d*x)/(2 *d) + 35*C*a**4*x*sin(c + d*x)**8/128 + 35*C*a**4*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 15*C*a**4*x*sin(c + d*x)**6/8 + 105*C*a**4*x*sin(c + d*x)** 4*cos(c + d*x)**4/64 + 45*C*a**4*x*sin(c + d*x)**4*cos(c + d*x)**2/8 + 3*C *a**4*x*sin(c + d*x)**4/8 + 35*C*a**4*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 45*C*a**4*x*sin(c + d*x)**2*cos(c + d*x)**4/8 + 3*C*a**4*x*sin(c + d*x) **2*cos(c + d*x)**2/4 + 35*C*a**4*x*cos(c + d*x)**8/128 + 15*C*a**4*x*cos( c + d*x)**6/8 + 3*C*a**4*x*cos(c + d*x)**4/8 + 35*C*a**4*sin(c + d*x)**7*c os(c + d*x)/(128*d) + 64*C*a**4*sin(c + d*x)**7/(35*d) + 385*C*a**4*sin(c + d*x)**5*cos(c + d*x)**3/(384*d) + 32*C*a**4*sin(c + d*x)**5*cos(c + d...
Time = 0.22 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.41 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {28672 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} - 560 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 143360 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 20160 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 26880 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 12288 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} C a^{4} + 28672 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{4} - 35 \, {\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} - 3360 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 3360 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4}}{107520 \, d} \]
1/107520*(28672*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A *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))*A*a^4 - 143360*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^4 + 20160*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^4 + 26 880*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*a^4 - 12288*(5*sin(d*x + c)^7 - 21* sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*C*a^4 + 28672*(3*sin (d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*C*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))*C*a^4 - 3360*(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 + 3360*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^4)/d
Time = 0.34 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.76 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{4} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {C a^{4} \sin \left (7 \, d x + 7 \, c\right )}{112 \, d} + \frac {1}{128} \, {\left (392 \, A a^{4} + 323 \, C a^{4}\right )} x + \frac {{\left (A a^{4} + 8 \, C a^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (4 \, A a^{4} + 11 \, C a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {{\left (30 \, A a^{4} + 47 \, C a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (36 \, A a^{4} + 41 \, C a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {{\left (127 \, A a^{4} + 120 \, C a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (88 \, A a^{4} + 75 \, C a^{4}\right )} \sin \left (d x + c\right )}{16 \, d} \]
1/1024*C*a^4*sin(8*d*x + 8*c)/d + 1/112*C*a^4*sin(7*d*x + 7*c)/d + 1/128*( 392*A*a^4 + 323*C*a^4)*x + 1/192*(A*a^4 + 8*C*a^4)*sin(6*d*x + 6*c)/d + 1/ 80*(4*A*a^4 + 11*C*a^4)*sin(5*d*x + 5*c)/d + 1/128*(30*A*a^4 + 47*C*a^4)*s in(4*d*x + 4*c)/d + 1/48*(36*A*a^4 + 41*C*a^4)*sin(3*d*x + 3*c)/d + 1/64*( 127*A*a^4 + 120*C*a^4)*sin(2*d*x + 2*c)/d + 1/16*(88*A*a^4 + 75*C*a^4)*sin (d*x + c)/d
Time = 1.54 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.40 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (\frac {49\,A\,a^4}{8}+\frac {323\,C\,a^4}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}+\left (\frac {1127\,A\,a^4}{24}+\frac {7429\,C\,a^4}{192}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (\frac {18767\,A\,a^4}{120}+\frac {123709\,C\,a^4}{960}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {35371\,A\,a^4}{120}+\frac {1632119\,C\,a^4}{6720}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {40661\,A\,a^4}{120}+\frac {624003\,C\,a^4}{2240}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {29617\,A\,a^4}{120}+\frac {68673\,C\,a^4}{320}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {2713\,A\,a^4}{24}+\frac {5033\,C\,a^4}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {207\,A\,a^4}{8}+\frac {1725\,C\,a^4}{64}\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\,\left (392\,A+323\,C\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{64\,d}+\frac {a^4\,\mathrm {atan}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (392\,A+323\,C\right )}{64\,\left (\frac {49\,A\,a^4}{8}+\frac {323\,C\,a^4}{64}\right )}\right )\,\left (392\,A+323\,C\right )}{64\,d} \]
(tan(c/2 + (d*x)/2)*((207*A*a^4)/8 + (1725*C*a^4)/64) + tan(c/2 + (d*x)/2) ^15*((49*A*a^4)/8 + (323*C*a^4)/64) + tan(c/2 + (d*x)/2)^3*((2713*A*a^4)/2 4 + (5033*C*a^4)/64) + tan(c/2 + (d*x)/2)^13*((1127*A*a^4)/24 + (7429*C*a^ 4)/192) + tan(c/2 + (d*x)/2)^5*((29617*A*a^4)/120 + (68673*C*a^4)/320) + t an(c/2 + (d*x)/2)^11*((18767*A*a^4)/120 + (123709*C*a^4)/960) + tan(c/2 + (d*x)/2)^7*((40661*A*a^4)/120 + (624003*C*a^4)/2240) + tan(c/2 + (d*x)/2)^ 9*((35371*A*a^4)/120 + (1632119*C*a^4)/6720))/(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*(392*A + 323*C)*(atan(tan(c/ 2 + (d*x)/2)) - (d*x)/2))/(64*d) + (a^4*atan((a^4*tan(c/2 + (d*x)/2)*(392* A + 323*C))/(64*((49*A*a^4)/8 + (323*C*a^4)/64)))*(392*A + 323*C))/(64*d)