Integrand size = 41, antiderivative size = 304 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {1}{128} a^4 (392 A+352 B+323 C) x+\frac {a^4 (252 A+227 B+208 C) \sin (c+d x)}{35 d}+\frac {a^4 (392 A+352 B+323 C) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {a^4 (2408 A+2208 B+2007 C) \cos ^3(c+d x) \sin (c+d x)}{2240 d}+\frac {a (2 B+C) \cos ^3(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{14 d}+\frac {C \cos ^3(c+d x) (a+a \cos (c+d x))^4 \sin (c+d x)}{8 d}+\frac {(56 A+80 B+61 C) \cos ^3(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{336 d}+\frac {7 (8 A+8 B+7 C) \cos ^3(c+d x) \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{120 d}-\frac {a^4 (252 A+227 B+208 C) \sin ^3(c+d x)}{105 d} \]
1/128*a^4*(392*A+352*B+323*C)*x+1/35*a^4*(252*A+227*B+208*C)*sin(d*x+c)/d+ 1/128*a^4*(392*A+352*B+323*C)*cos(d*x+c)*sin(d*x+c)/d+1/2240*a^4*(2408*A+2 208*B+2007*C)*cos(d*x+c)^3*sin(d*x+c)/d+1/14*a*(2*B+C)*cos(d*x+c)^3*(a+a*c os(d*x+c))^3*sin(d*x+c)/d+1/8*C*cos(d*x+c)^3*(a+a*cos(d*x+c))^4*sin(d*x+c) /d+1/336*(56*A+80*B+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+8*B+7*C)*cos(d*x+c)^3*(a^4+a^4*cos(d*x+c))*sin(d*x+c)/d-1/105* a^4*(252*A+227*B+208*C)*sin(d*x+c)^3/d
Time = 1.25 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.78 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {a^4 (295680 B c+164640 c C+329280 A d x+295680 B d x+271320 C d x+1680 (352 A+323 B+300 C) \sin (c+d x)+1680 (127 A+124 B+120 C) \sin (2 (c+d x))+80640 A \sin (3 (c+d x))+87920 B \sin (3 (c+d x))+91840 C \sin (3 (c+d x))+25200 A \sin (4 (c+d x))+33600 B \sin (4 (c+d x))+39480 C \sin (4 (c+d x))+5376 A \sin (5 (c+d x))+10416 B \sin (5 (c+d x))+14784 C \sin (5 (c+d x))+560 A \sin (6 (c+d x))+2240 B \sin (6 (c+d x))+4480 C \sin (6 (c+d x))+240 B \sin (7 (c+d x))+960 C \sin (7 (c+d x))+105 C \sin (8 (c+d x)))}{107520 d} \]
(a^4*(295680*B*c + 164640*c*C + 329280*A*d*x + 295680*B*d*x + 271320*C*d*x + 1680*(352*A + 323*B + 300*C)*Sin[c + d*x] + 1680*(127*A + 124*B + 120*C )*Sin[2*(c + d*x)] + 80640*A*Sin[3*(c + d*x)] + 87920*B*Sin[3*(c + d*x)] + 91840*C*Sin[3*(c + d*x)] + 25200*A*Sin[4*(c + d*x)] + 33600*B*Sin[4*(c + d*x)] + 39480*C*Sin[4*(c + d*x)] + 5376*A*Sin[5*(c + d*x)] + 10416*B*Sin[5 *(c + d*x)] + 14784*C*Sin[5*(c + d*x)] + 560*A*Sin[6*(c + d*x)] + 2240*B*S in[6*(c + d*x)] + 4480*C*Sin[6*(c + d*x)] + 240*B*Sin[7*(c + d*x)] + 960*C *Sin[7*(c + d*x)] + 105*C*Sin[8*(c + d*x)]))/(107520*d)
Time = 2.03 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.01, number of steps used = 21, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.488, Rules used = {3042, 3524, 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.
| \(\displaystyle \int \cos ^2(c+d x) (a \cos (c+d x)+a)^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^4 \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx\) |
| \(\Big \downarrow \) 3524 |
| \(\displaystyle \frac {\int \cos ^2(c+d x) (\cos (c+d x) a+a)^4 (a (8 A+3 C)+4 a (2 B+C) \cos (c+d x))dx}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (a (8 A+3 C)+4 a (2 B+C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{7} \int \cos ^2(c+d x) (\cos (c+d x) a+a)^3 \left ((56 A+24 B+33 C) a^2+(56 A+80 B+61 C) \cos (c+d x) a^2\right )dx+\frac {4 a^2 (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left ((56 A+24 B+33 C) a^2+(56 A+80 B+61 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {4 a^2 (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \int \cos ^2(c+d x) (\cos (c+d x) a+a)^2 \left (3 (168 A+128 B+127 C) a^3+98 (8 A+8 B+7 C) \cos (c+d x) a^3\right )dx+\frac {a^3 (56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (3 (168 A+128 B+127 C) a^3+98 (8 A+8 B+7 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {a^3 (56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3455 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {1}{5} \int 3 \cos ^2(c+d x) (\cos (c+d x) a+a) \left ((1624 A+1424 B+1321 C) a^4+(2408 A+2208 B+2007 C) \cos (c+d x) a^4\right )dx+\frac {98 (8 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \int \cos ^2(c+d x) (\cos (c+d x) a+a) \left ((1624 A+1424 B+1321 C) a^4+(2408 A+2208 B+2007 C) \cos (c+d x) a^4\right )dx+\frac {98 (8 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((1624 A+1424 B+1321 C) a^4+(2408 A+2208 B+2007 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )dx+\frac {98 (8 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3447 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \int \cos ^2(c+d x) \left ((2408 A+2208 B+2007 C) \cos ^2(c+d x) a^5+(1624 A+1424 B+1321 C) a^5+\left ((1624 A+1424 B+1321 C) a^5+(2408 A+2208 B+2007 C) a^5\right ) \cos (c+d x)\right )dx+\frac {98 (8 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left ((2408 A+2208 B+2007 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^5+(1624 A+1424 B+1321 C) a^5+\left ((1624 A+1424 B+1321 C) a^5+(2408 A+2208 B+2007 C) a^5\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {98 (8 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \int \cos ^2(c+d x) \left (35 (392 A+352 B+323 C) a^5+64 (252 A+227 B+208 C) \cos (c+d x) a^5\right )dx+\frac {a^5 (2408 A+2208 B+2007 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (8 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2 \left (35 (392 A+352 B+323 C) a^5+64 (252 A+227 B+208 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^5\right )dx+\frac {a^5 (2408 A+2208 B+2007 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (8 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (64 a^5 (252 A+227 B+208 C) \int \cos ^3(c+d x)dx+35 a^5 (392 A+352 B+323 C) \int \cos ^2(c+d x)dx\right )+\frac {a^5 (2408 A+2208 B+2007 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (8 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^5 (392 A+352 B+323 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+64 a^5 (252 A+227 B+208 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^3dx\right )+\frac {a^5 (2408 A+2208 B+2007 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (8 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^5 (392 A+352 B+323 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {64 a^5 (252 A+227 B+208 C) \int \left (1-\sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}\right )+\frac {a^5 (2408 A+2208 B+2007 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (8 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^5 (392 A+352 B+323 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx-\frac {64 a^5 (252 A+227 B+208 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {a^5 (2408 A+2208 B+2007 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (8 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {1}{4} \left (35 a^5 (392 A+352 B+323 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )-\frac {64 a^5 (252 A+227 B+208 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )+\frac {a^5 (2408 A+2208 B+2007 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {98 (8 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )+\frac {4 a^2 (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {4 a^2 (2 B+C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^3}{7 d}+\frac {1}{7} \left (\frac {1}{6} \left (\frac {3}{5} \left (\frac {a^5 (2408 A+2208 B+2007 C) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {1}{4} \left (35 a^5 (392 A+352 B+323 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {64 a^5 (252 A+227 B+208 C) \left (\frac {1}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\right )\right )+\frac {98 (8 A+8 B+7 C) \sin (c+d x) \cos ^3(c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{5 d}\right )+\frac {a^3 (56 A+80 B+61 C) \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^2}{6 d}\right )}{8 a}+\frac {C \sin (c+d x) \cos ^3(c+d x) (a \cos (c+d x)+a)^4}{8 d}\) |
(C*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^4*Sin[c + d*x])/(8*d) + ((4*a^2*(2* B + C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(7*d) + ((a^3*( 56*A + 80*B + 61*C)*Cos[c + d*x]^3*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(6 *d) + ((98*(8*A + 8*B + 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 + 2208*B + 2007*C)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + (35*a^5*(392*A + 352*B + 323*C)*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)) - (64*a^5*(252*A + 227*B + 208*C)*(-Sin[c + d*x] + Sin[c + d*x]^3/3))/d)/4))/5)/6)/7)/(8*a)
3.4.28.3.1 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_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (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)) + b*B*d*(m + n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n} , x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && !Lt Q[m, -2^(-1)] && NeQ[m + n + 2, 0]
Time = 12.18 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.54
| method | result | size |
| parallelrisch | \(\frac {3 \left (\frac {\left (\frac {127 A}{24}+\frac {31 B}{6}+5 C \right ) \sin \left (2 d x +2 c \right )}{2}+\left (A +\frac {157 B}{144}+\frac {41 C}{36}\right ) \sin \left (3 d x +3 c \right )+\frac {\left (\frac {5 A}{4}+\frac {5 B}{3}+\frac {47 C}{24}\right ) \sin \left (4 d x +4 c \right )}{4}+\frac {\left (A +\frac {31 B}{16}+\frac {11 C}{4}\right ) \sin \left (5 d x +5 c \right )}{15}+\frac {\left (\frac {A}{8}+\frac {B}{2}+C \right ) \sin \left (6 d x +6 c \right )}{18}+\frac {\left (\frac {B}{4}+C \right ) \sin \left (7 d x +7 c \right )}{84}+\frac {C \sin \left (8 d x +8 c \right )}{768}+\left (\frac {22 A}{3}+\frac {323 B}{48}+\frac {25 C}{4}\right ) \sin \left (d x +c \right )+\frac {49 x \left (A +\frac {44 B}{49}+\frac {323 C}{392}\right ) d}{12}\right ) a^{4}}{4 d}\) | \(165\) |
| parts | \(\frac {\left (4 a^{4} A +B \,a^{4}\right ) \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (B \,a^{4}+4 C \,a^{4}\right ) \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7 d}+\frac {\left (a^{4} A +4 B \,a^{4}+6 C \,a^{4}\right ) \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {\left (4 a^{4} A +6 B \,a^{4}+4 C \,a^{4}\right ) \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}+\frac {\left (6 a^{4} A +4 B \,a^{4}+C \,a^{4}\right ) \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {a^{4} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {C \,a^{4} \left (\frac {\left (\cos ^{7}\left (d x +c \right )+\frac {7 \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {35 \left (\cos ^{3}\left (d x +c \right )\right )}{24}+\frac {35 \cos \left (d x +c \right )}{16}\right ) \sin \left (d x +c \right )}{8}+\frac {35 d x}{128}+\frac {35 c}{128}\right )}{d}\) | \(349\) |
| risch | \(\frac {49 a^{4} x A}{16}+\frac {11 a^{4} B x}{4}+\frac {323 a^{4} C x}{128}+\frac {11 \sin \left (d x +c \right ) a^{4} A}{2 d}+\frac {323 \sin \left (d x +c \right ) B \,a^{4}}{64 d}+\frac {75 \sin \left (d x +c \right ) C \,a^{4}}{16 d}+\frac {C \,a^{4} \sin \left (8 d x +8 c \right )}{1024 d}+\frac {\sin \left (7 d x +7 c \right ) B \,a^{4}}{448 d}+\frac {\sin \left (7 d x +7 c \right ) C \,a^{4}}{112 d}+\frac {\sin \left (6 d x +6 c \right ) a^{4} A}{192 d}+\frac {\sin \left (6 d x +6 c \right ) B \,a^{4}}{48 d}+\frac {\sin \left (6 d x +6 c \right ) C \,a^{4}}{24 d}+\frac {\sin \left (5 d x +5 c \right ) a^{4} A}{20 d}+\frac {31 \sin \left (5 d x +5 c \right ) B \,a^{4}}{320 d}+\frac {11 \sin \left (5 d x +5 c \right ) C \,a^{4}}{80 d}+\frac {15 \sin \left (4 d x +4 c \right ) a^{4} A}{64 d}+\frac {5 \sin \left (4 d x +4 c \right ) B \,a^{4}}{16 d}+\frac {47 \sin \left (4 d x +4 c \right ) C \,a^{4}}{128 d}+\frac {3 \sin \left (3 d x +3 c \right ) a^{4} A}{4 d}+\frac {157 \sin \left (3 d x +3 c \right ) B \,a^{4}}{192 d}+\frac {41 \sin \left (3 d x +3 c \right ) C \,a^{4}}{48 d}+\frac {127 \sin \left (2 d x +2 c \right ) a^{4} A}{64 d}+\frac {31 \sin \left (2 d x +2 c \right ) B \,a^{4}}{16 d}+\frac {15 \sin \left (2 d x +2 c \right ) C \,a^{4}}{8 d}\) | \(392\) |
| norman | \(\frac {\frac {a^{4} \left (392 A +352 B +323 C \right ) x}{128}+\frac {5053 a^{4} \left (392 A +352 B +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 +352 B +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 +352 B +323 C \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {a^{4} \left (392 A +352 B +323 C \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {a^{4} \left (392 A +352 B +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 +352 B +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 +352 B +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 +352 B +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 +352 B +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 +352 B +323 C \right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32}+\frac {a^{4} \left (392 A +352 B +323 C \right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {a^{4} \left (392 A +352 B +323 C \right ) x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128}+\frac {a^{4} \left (1656 A +1696 B +1725 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}+\frac {a^{4} \left (21704 A +18528 B +15099 C \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\frac {a^{4} \left (236936 A +211296 B +206019 C \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{960 d}+\frac {a^{4} \left (2277016 A +2090016 B +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}}\) | \(480\) |
| 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 )+\frac {B \,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}+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}+4 B \,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 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 )+\frac {6 B \,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}+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}+4 B \,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 )+\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 )+\frac {B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+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}\) | \(577\) |
| 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 )+\frac {B \,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}+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}+4 B \,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 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 )+\frac {6 B \,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}+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}+4 B \,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 )+\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 )+\frac {B \,a^{4} \left (2+\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3}+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}\) | \(577\) |
3/4*(1/2*(127/24*A+31/6*B+5*C)*sin(2*d*x+2*c)+(A+157/144*B+41/36*C)*sin(3* d*x+3*c)+1/4*(5/4*A+5/3*B+47/24*C)*sin(4*d*x+4*c)+1/15*(A+31/16*B+11/4*C)* sin(5*d*x+5*c)+1/18*(1/8*A+1/2*B+C)*sin(6*d*x+6*c)+1/84*(1/4*B+C)*sin(7*d* x+7*c)+1/768*C*sin(8*d*x+8*c)+(22/3*A+323/48*B+25/4*C)*sin(d*x+c)+49/12*x* (A+44/49*B+323/392*C)*d)*a^4/d
Time = 0.29 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.63 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (392 \, A + 352 \, B + 323 \, C\right )} a^{4} d x + {\left (1680 \, C a^{4} \cos \left (d x + c\right )^{7} + 1920 \, {\left (B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{6} + 280 \, {\left (8 \, A + 32 \, B + 55 \, C\right )} a^{4} \cos \left (d x + c\right )^{5} + 1536 \, {\left (7 \, A + 12 \, B + 13 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} + 70 \, {\left (328 \, A + 352 \, B + 323 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 128 \, {\left (252 \, A + 227 \, B + 208 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 105 \, {\left (392 \, A + 352 \, B + 323 \, C\right )} a^{4} \cos \left (d x + c\right ) + 256 \, {\left (252 \, A + 227 \, B + 208 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{13440 \, d} \]
integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2), x, algorithm="fricas")
1/13440*(105*(392*A + 352*B + 323*C)*a^4*d*x + (1680*C*a^4*cos(d*x + c)^7 + 1920*(B + 4*C)*a^4*cos(d*x + c)^6 + 280*(8*A + 32*B + 55*C)*a^4*cos(d*x + c)^5 + 1536*(7*A + 12*B + 13*C)*a^4*cos(d*x + c)^4 + 70*(328*A + 352*B + 323*C)*a^4*cos(d*x + c)^3 + 128*(252*A + 227*B + 208*C)*a^4*cos(d*x + c)^ 2 + 105*(392*A + 352*B + 323*C)*a^4*cos(d*x + c) + 256*(252*A + 227*B + 20 8*C)*a^4)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 1640 vs. \(2 (289) = 578\).
Time = 0.86 (sec) , antiderivative size = 1640, normalized size of antiderivative = 5.39 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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) + 5*B*a**4*x*sin(c + d*x)**6/4 + 15*B*a**4*x*sin(c + d*x)**4*cos(c + d *x)**2/4 + 3*B*a**4*x*sin(c + d*x)**4/2 + 15*B*a**4*x*sin(c + d*x)**2*cos( c + d*x)**4/4 + 3*B*a**4*x*sin(c + d*x)**2*cos(c + d*x)**2 + 5*B*a**4*x*co s(c + d*x)**6/4 + 3*B*a**4*x*cos(c + d*x)**4/2 + 16*B*a**4*sin(c + d*x)**7 /(35*d) + 8*B*a**4*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 5*B*a**4*sin(c + d*x)**5*cos(c + d*x)/(4*d) + 16*B*a**4*sin(c + d*x)**5/(5*d) + 2*B*a**4* sin(c + d*x)**3*cos(c + d*x)**4/d + 10*B*a**4*sin(c + d*x)**3*cos(c + d*x) **3/(3*d) + 8*B*a**4*sin(c + d*x)**3*cos(c + d*x)**2/d + 3*B*a**4*sin(c + d*x)**3*cos(c + d*x)/(2*d) + 2*B*a**4*sin(c + d*x)**3/(3*d) + B*a**4*si...
Leaf count of result is larger than twice the leaf count of optimal. 579 vs. \(2 (286) = 572\).
Time = 0.22 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.90 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+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} - 3072 \, {\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 )} B a^{4} + 43008 \, {\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 )} B a^{4} - 2240 \, {\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 )} B a^{4} - 35840 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 13440 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B 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} \]
integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2), x, algorithm="maxima")
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 - 3072*(5*sin(d*x + c)^7 - 21*s in(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 - 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 = 0.39 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.86 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {C a^{4} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {1}{128} \, {\left (392 \, A a^{4} + 352 \, B a^{4} + 323 \, C a^{4}\right )} x + \frac {{\left (B a^{4} + 4 \, C a^{4}\right )} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {{\left (A a^{4} + 4 \, B a^{4} + 8 \, C a^{4}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (16 \, A a^{4} + 31 \, B a^{4} + 44 \, C a^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (30 \, A a^{4} + 40 \, B a^{4} + 47 \, C a^{4}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (144 \, A a^{4} + 157 \, B a^{4} + 164 \, C a^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {{\left (127 \, A a^{4} + 124 \, B a^{4} + 120 \, C a^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {{\left (352 \, A a^{4} + 323 \, B a^{4} + 300 \, C a^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \]
integrate(cos(d*x+c)^2*(a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2), x, algorithm="giac")
1/1024*C*a^4*sin(8*d*x + 8*c)/d + 1/128*(392*A*a^4 + 352*B*a^4 + 323*C*a^4 )*x + 1/448*(B*a^4 + 4*C*a^4)*sin(7*d*x + 7*c)/d + 1/192*(A*a^4 + 4*B*a^4 + 8*C*a^4)*sin(6*d*x + 6*c)/d + 1/320*(16*A*a^4 + 31*B*a^4 + 44*C*a^4)*sin (5*d*x + 5*c)/d + 1/128*(30*A*a^4 + 40*B*a^4 + 47*C*a^4)*sin(4*d*x + 4*c)/ d + 1/192*(144*A*a^4 + 157*B*a^4 + 164*C*a^4)*sin(3*d*x + 3*c)/d + 1/64*(1 27*A*a^4 + 124*B*a^4 + 120*C*a^4)*sin(2*d*x + 2*c)/d + 1/64*(352*A*a^4 + 3 23*B*a^4 + 300*C*a^4)*sin(d*x + c)/d
Time = 3.61 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.49 \[ \int \cos ^2(c+d x) (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\left (\frac {49\,A\,a^4}{8}+\frac {11\,B\,a^4}{2}+\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 {253\,B\,a^4}{6}+\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 {4213\,B\,a^4}{30}+\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 {55583\,B\,a^4}{210}+\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 {21771\,B\,a^4}{70}+\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 {2201\,B\,a^4}{10}+\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 {193\,B\,a^4}{2}+\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 {53\,B\,a^4}{2}+\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\,\mathrm {atan}\left (\frac {a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (392\,A+352\,B+323\,C\right )}{64\,\left (\frac {49\,A\,a^4}{8}+\frac {11\,B\,a^4}{2}+\frac {323\,C\,a^4}{64}\right )}\right )\,\left (392\,A+352\,B+323\,C\right )}{64\,d}-\frac {a^4\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )\,\left (392\,A+352\,B+323\,C\right )}{64\,d} \]
(tan(c/2 + (d*x)/2)^15*((49*A*a^4)/8 + (11*B*a^4)/2 + (323*C*a^4)/64) + ta n(c/2 + (d*x)/2)^3*((2713*A*a^4)/24 + (193*B*a^4)/2 + (5033*C*a^4)/64) + t an(c/2 + (d*x)/2)^13*((1127*A*a^4)/24 + (253*B*a^4)/6 + (7429*C*a^4)/192) + tan(c/2 + (d*x)/2)^5*((29617*A*a^4)/120 + (2201*B*a^4)/10 + (68673*C*a^4 )/320) + tan(c/2 + (d*x)/2)^11*((18767*A*a^4)/120 + (4213*B*a^4)/30 + (123 709*C*a^4)/960) + tan(c/2 + (d*x)/2)^7*((40661*A*a^4)/120 + (21771*B*a^4)/ 70 + (624003*C*a^4)/2240) + tan(c/2 + (d*x)/2)^9*((35371*A*a^4)/120 + (555 83*B*a^4)/210 + (1632119*C*a^4)/6720) + tan(c/2 + (d*x)/2)*((207*A*a^4)/8 + (53*B*a^4)/2 + (1725*C*a^4)/64))/(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)*(392*A + 3 52*B + 323*C))/(64*((49*A*a^4)/8 + (11*B*a^4)/2 + (323*C*a^4)/64)))*(392*A + 352*B + 323*C))/(64*d) - (a^4*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2)*(392 *A + 352*B + 323*C))/(64*d)