Integrand size = 43, antiderivative size = 444 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\frac {2 \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (308 a^3 b B+220 a b^3 B+77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 b \left (1353 a^2 b B+539 b^3 B+192 a^3 C+2 a b^2 (891 A+673 C)\right ) \sin (c+d x)}{3465 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (682 a^3 b B+660 a b^3 B+64 a^4 C+15 b^4 (11 A+9 C)+9 a^2 b^2 (143 A+101 C)\right ) \sin (c+d x)}{693 d \sqrt {\sec (c+d x)}}+\frac {2 \left (33 A b^2+55 a b B+16 a^2 C+27 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 (11 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{99 d \sqrt {\sec (c+d x)}}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{11 d \sqrt {\sec (c+d x)}} \]
2/3465*b*(1353*B*a^2*b+539*B*b^3+192*a^3*C+2*a*b^2*(891*A+673*C))*sin(d*x+ c)/d/sec(d*x+c)^(3/2)+2/693*(682*B*a^3*b+660*B*a*b^3+64*a^4*C+15*b^4*(11*A +9*C)+9*a^2*b^2*(143*A+101*C))*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2/231*(33*A*b ^2+55*B*a*b+16*C*a^2+27*C*b^2)*(a+b*cos(d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^ (1/2)+2/99*(11*B*b+8*C*a)*(a+b*cos(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)^(1/2) +2/11*C*(a+b*cos(d*x+c))^4*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2/15*(15*B*a^4+54 *B*a^2*b^2+7*B*b^4+12*a^3*b*(5*A+3*C)+4*a*b^3*(9*A+7*C))*(cos(1/2*d*x+1/2* c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d *x+c)^(1/2)*sec(d*x+c)^(1/2)/d+2/231*(308*B*a^3*b+220*B*a*b^3+77*a^4*(3*A+ C)+66*a^2*b^2*(7*A+5*C)+5*b^4*(11*A+9*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos (1/2*d*x+1/2*c)*EllipticF(sin(1/2*d*x+1/2*c),2^(1/2))*cos(d*x+c)^(1/2)*sec (d*x+c)^(1/2)/d
Time = 6.56 (sec) , antiderivative size = 338, normalized size of antiderivative = 0.76 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\frac {\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (3696 \left (15 a^4 B+54 a^2 b^2 B+7 b^4 B+12 a^3 b (5 A+3 C)+4 a b^3 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+240 \left (308 a^3 b B+220 a b^3 B+77 a^4 (3 A+C)+66 a^2 b^2 (7 A+5 C)+5 b^4 (11 A+9 C)\right ) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\frac {\left (154 b \left (216 a^2 b B+43 b^3 B+144 a^3 C+4 a b^2 (36 A+43 C)\right ) \cos (c+d x)+5 \left (7392 a^3 b B+6864 a b^3 B+1848 a^4 C+792 a^2 b^2 (14 A+13 C)+3 b^4 (572 A+531 C)+36 b^2 \left (11 A b^2+44 a b B+66 a^2 C+16 b^2 C\right ) \cos (2 (c+d x))+154 b^3 (b B+4 a C) \cos (3 (c+d x))+63 b^4 C \cos (4 (c+d x))\right )\right ) \sin (2 (c+d x))}{\sqrt {\cos (c+d x)}}\right )}{27720 d} \]
Integrate[(a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*S qrt[Sec[c + d*x]],x]
(Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*(3696*(15*a^4*B + 54*a^2*b^2*B + 7* b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2] + 240*(308*a^3*b*B + 220*a*b^3*B + 77*a^4*(3*A + C) + 66*a^2*b^2*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2] + ((154*b*(216*a^2* b*B + 43*b^3*B + 144*a^3*C + 4*a*b^2*(36*A + 43*C))*Cos[c + d*x] + 5*(7392 *a^3*b*B + 6864*a*b^3*B + 1848*a^4*C + 792*a^2*b^2*(14*A + 13*C) + 3*b^4*( 572*A + 531*C) + 36*b^2*(11*A*b^2 + 44*a*b*B + 66*a^2*C + 16*b^2*C)*Cos[2* (c + d*x)] + 154*b^3*(b*B + 4*a*C)*Cos[3*(c + d*x)] + 63*b^4*C*Cos[4*(c + d*x)]))*Sin[2*(c + d*x)])/Sqrt[Cos[c + d*x]]))/(27720*d)
Time = 2.64 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.512, Rules used = {3042, 4709, 3042, 3528, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3512, 27, 3042, 3502, 27, 3042, 3227, 3042, 3119, 3120}
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.
| \(\displaystyle \int \sqrt {\sec (c+d x)} (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\sec (c+d x)} (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos (c+d x)^2\right )dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {(a+b \cos (c+d x))^4 \left (C \cos ^2(c+d x)+B \cos (c+d x)+A\right )}{\sqrt {\cos (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+A\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{11} \int \frac {(a+b \cos (c+d x))^3 \left ((11 b B+8 a C) \cos ^2(c+d x)+(11 A b+9 C b+11 a B) \cos (c+d x)+a (11 A+C)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \int \frac {(a+b \cos (c+d x))^3 \left ((11 b B+8 a C) \cos ^2(c+d x)+(11 A b+9 C b+11 a B) \cos (c+d x)+a (11 A+C)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left ((11 b B+8 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(11 A b+9 C b+11 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+a (11 A+C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {2}{9} \int \frac {(a+b \cos (c+d x))^2 \left (3 \left (16 C a^2+55 b B a+33 A b^2+27 b^2 C\right ) \cos ^2(c+d x)+\left (99 B a^2+198 A b a+146 b C a+77 b^2 B\right ) \cos (c+d x)+a (99 a A+11 b B+17 a C)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \int \frac {(a+b \cos (c+d x))^2 \left (3 \left (16 C a^2+55 b B a+33 A b^2+27 b^2 C\right ) \cos ^2(c+d x)+\left (99 B a^2+198 A b a+146 b C a+77 b^2 B\right ) \cos (c+d x)+a (99 a A+11 b B+17 a C)\right )}{\sqrt {\cos (c+d x)}}dx+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (3 \left (16 C a^2+55 b B a+33 A b^2+27 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (99 B a^2+198 A b a+146 b C a+77 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a (99 a A+11 b B+17 a C)\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {(a+b \cos (c+d x)) \left (\left (192 C a^3+1353 b B a^2+2 b^2 (891 A+673 C) a+539 b^3 B\right ) \cos ^2(c+d x)+\left (693 B a^3+b (2079 A+1381 C) a^2+1441 b^2 B a+45 b^3 (11 A+9 C)\right ) \cos (c+d x)+a \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right )\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {(a+b \cos (c+d x)) \left (\left (192 C a^3+1353 b B a^2+2 b^2 (891 A+673 C) a+539 b^3 B\right ) \cos ^2(c+d x)+\left (693 B a^3+b (2079 A+1381 C) a^2+1441 b^2 B a+45 b^3 (11 A+9 C)\right ) \cos (c+d x)+a \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right )\right )}{\sqrt {\cos (c+d x)}}dx+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (\left (192 C a^3+1353 b B a^2+2 b^2 (891 A+673 C) a+539 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (693 B a^3+b (2079 A+1381 C) a^2+1441 b^2 B a+45 b^3 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right )\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {5 \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right ) a^2+15 \left (64 C a^4+682 b B a^3+9 b^2 (143 A+101 C) a^2+660 b^3 B a+15 b^4 (11 A+9 C)\right ) \cos ^2(c+d x)+231 \left (15 B a^4+12 b (5 A+3 C) a^3+54 b^2 B a^2+4 b^3 (9 A+7 C) a+7 b^4 B\right ) \cos (c+d x)}{2 \sqrt {\cos (c+d x)}}dx+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right ) a^2+15 \left (64 C a^4+682 b B a^3+9 b^2 (143 A+101 C) a^2+660 b^3 B a+15 b^4 (11 A+9 C)\right ) \cos ^2(c+d x)+231 \left (15 B a^4+12 b (5 A+3 C) a^3+54 b^2 B a^2+4 b^3 (9 A+7 C) a+7 b^4 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \int \frac {5 \left ((693 A+167 C) a^2+242 b B a+9 b^2 (11 A+9 C)\right ) a^2+15 \left (64 C a^4+682 b B a^3+9 b^2 (143 A+101 C) a^2+660 b^3 B a+15 b^4 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+231 \left (15 B a^4+12 b (5 A+3 C) a^3+54 b^2 B a^2+4 b^3 (9 A+7 C) a+7 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (\frac {2}{3} \int \frac {9 \left (5 \left (77 (3 A+C) a^4+308 b B a^3+66 b^2 (7 A+5 C) a^2+220 b^3 B a+5 b^4 (11 A+9 C)\right )+77 \left (15 B a^4+12 b (5 A+3 C) a^3+54 b^2 B a^2+4 b^3 (9 A+7 C) a+7 b^4 B\right ) \cos (c+d x)\right )}{2 \sqrt {\cos (c+d x)}}dx+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 \left (77 (3 A+C) a^4+308 b B a^3+66 b^2 (7 A+5 C) a^2+220 b^3 B a+5 b^4 (11 A+9 C)\right )+77 \left (15 B a^4+12 b (5 A+3 C) a^3+54 b^2 B a^2+4 b^3 (9 A+7 C) a+7 b^4 B\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}}dx+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \int \frac {5 \left (77 (3 A+C) a^4+308 b B a^3+66 b^2 (7 A+5 C) a^2+220 b^3 B a+5 b^4 (11 A+9 C)\right )+77 \left (15 B a^4+12 b (5 A+3 C) a^3+54 b^2 B a^2+4 b^3 (9 A+7 C) a+7 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}}dx+77 \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+77 \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx\right )+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {1}{5} \left (3 \left (5 \left (77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right ) \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {154 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right )}{d}\right )+\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}\right )+\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {6 \sin (c+d x) \sqrt {\cos (c+d x)} \left (16 a^2 C+55 a b B+33 A b^2+27 b^2 C\right ) (a+b \cos (c+d x))^2}{7 d}+\frac {1}{7} \left (\frac {2 b \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^3 C+1353 a^2 b B+2 a b^2 (891 A+673 C)+539 b^3 B\right )}{5 d}+\frac {1}{5} \left (\frac {10 \sin (c+d x) \sqrt {\cos (c+d x)} \left (64 a^4 C+682 a^3 b B+9 a^2 b^2 (143 A+101 C)+660 a b^3 B+15 b^4 (11 A+9 C)\right )}{d}+3 \left (\frac {10 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \left (77 a^4 (3 A+C)+308 a^3 b B+66 a^2 b^2 (7 A+5 C)+220 a b^3 B+5 b^4 (11 A+9 C)\right )}{d}+\frac {154 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (15 a^4 B+12 a^3 b (5 A+3 C)+54 a^2 b^2 B+4 a b^3 (9 A+7 C)+7 b^4 B\right )}{d}\right )\right )\right )\right )+\frac {2 (8 a C+11 b B) \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3}{9 d}\right )+\frac {2 C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4}{11 d}\right )\) |
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*C*Sqrt[Cos[c + d*x]]*(a + b*Cos[ c + d*x])^4*Sin[c + d*x])/(11*d) + ((2*(11*b*B + 8*a*C)*Sqrt[Cos[c + d*x]] *(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(9*d) + ((6*(33*A*b^2 + 55*a*b*B + 1 6*a^2*C + 27*b^2*C)*Sqrt[Cos[c + d*x]]*(a + b*Cos[c + d*x])^2*Sin[c + d*x] )/(7*d) + ((2*b*(1353*a^2*b*B + 539*b^3*B + 192*a^3*C + 2*a*b^2*(891*A + 6 73*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (3*((154*(15*a^4*B + 54*a^ 2*b^2*B + 7*b^4*B + 12*a^3*b*(5*A + 3*C) + 4*a*b^3*(9*A + 7*C))*EllipticE[ (c + d*x)/2, 2])/d + (10*(308*a^3*b*B + 220*a*b^3*B + 77*a^4*(3*A + C) + 6 6*a^2*b^2*(7*A + 5*C) + 5*b^4*(11*A + 9*C))*EllipticF[(c + d*x)/2, 2])/d) + (10*(682*a^3*b*B + 660*a*b^3*B + 64*a^4*C + 15*b^4*(11*A + 9*C) + 9*a^2* b^2*(143*A + 101*C))*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/d)/5)/7)/9)/11)
3.15.80.3.1 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[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)* (c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[1/Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticF[(1/2 )*(c - Pi/2 + d*x), 2], x] /; FreeQ[{c, d}, x]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[c Int[(b*Sin[e + f*x])^m, x], x] + Simp[d/b Int [(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-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_)])*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f _.)*(x_)]^2), x_Symbol] :> Simp[(-C)*d*Cos[e + f*x]*Sin[e + f*x]*((a + b*Si n[e + f*x])^(m + 1)/(b*f*(m + 3))), x] + Simp[1/(b*(m + 3)) Int[(a + b*Si n[e + f*x])^m*Simp[a*C*d + A*b*c*(m + 3) + b*(B*c*(m + 3) + d*(C*(m + 2) + A*(m + 3)))*Sin[e + f*x] - (2*a*C*d - b*(c*C + B*d)*(m + 3))*Sin[e + f*x]^2 , x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && !LtQ[m, -1]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ .) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*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/(d*(m + n + 2)) Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a *c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} , x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Simp[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m Int[ActivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] && !IntegerQ[m] && KnownSineIntegrandQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(1272\) vs. \(2(464)=928\).
Time = 13.31 (sec) , antiderivative size = 1273, normalized size of antiderivative = 2.87
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1273\) |
| parts | \(\text {Expression too large to display}\) | \(1380\) |
int((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(1/2),x, method=_RETURNVERBOSE)
-2/3465*((-1+2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(20160*C* b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+(-12320*B*b^4-49280*C*a*b^3-5 0400*C*b^4)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(7920*A*b^4+31680*B*a *b^3+24640*B*b^4+47520*C*a^2*b^2+98560*C*a*b^3+56880*C*b^4)*sin(1/2*d*x+1/ 2*c)^8*cos(1/2*d*x+1/2*c)+(-22176*A*a*b^3-11880*A*b^4-33264*B*a^2*b^2-4752 0*B*a*b^3-22792*B*b^4-22176*C*a^3*b-71280*C*a^2*b^2-91168*C*a*b^3-34920*C* b^4)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(27720*A*a^2*b^2+22176*A*a*b^ 3+9240*A*b^4+18480*B*a^3*b+33264*B*a^2*b^2+36960*B*a*b^3+10472*B*b^4+4620* C*a^4+22176*C*a^3*b+55440*C*a^2*b^2+41888*C*a*b^3+13860*C*b^4)*sin(1/2*d*x +1/2*c)^4*cos(1/2*d*x+1/2*c)+(-13860*A*a^2*b^2-5544*A*a*b^3-2640*A*b^4-924 0*B*a^3*b-8316*B*a^2*b^2-10560*B*a*b^3-1848*B*b^4-2310*C*a^4-5544*C*a^3*b- 15840*C*a^2*b^2-7392*C*a*b^3-2790*C*b^4)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+ 1/2*c)+3465*A*a^4*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^ (1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))+6930*A*a^2*b^2*(sin(1/2*d*x+1/ 2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c ),2^(1/2))+825*A*b^4*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2- 1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))-13860*A*(sin(1/2*d*x+1/2*c) ^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^ (1/2))*a^3*b-8316*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1 )^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^3+4620*B*a^3*b*(sin(1...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.17 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.12 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=-\frac {15 \, \sqrt {2} {\left (77 i \, {\left (3 \, A + C\right )} a^{4} + 308 i \, B a^{3} b + 66 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b^{2} + 220 i \, B a b^{3} + 5 i \, {\left (11 \, A + 9 \, C\right )} b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 15 \, \sqrt {2} {\left (-77 i \, {\left (3 \, A + C\right )} a^{4} - 308 i \, B a^{3} b - 66 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b^{2} - 220 i \, B a b^{3} - 5 i \, {\left (11 \, A + 9 \, C\right )} b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 \, \sqrt {2} {\left (-15 i \, B a^{4} - 12 i \, {\left (5 \, A + 3 \, C\right )} a^{3} b - 54 i \, B a^{2} b^{2} - 4 i \, {\left (9 \, A + 7 \, C\right )} a b^{3} - 7 i \, B b^{4}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 \, \sqrt {2} {\left (15 i \, B a^{4} + 12 i \, {\left (5 \, A + 3 \, C\right )} a^{3} b + 54 i \, B a^{2} b^{2} + 4 i \, {\left (9 \, A + 7 \, C\right )} a b^{3} + 7 i \, B b^{4}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (315 \, C b^{4} \cos \left (d x + c\right )^{5} + 385 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{4} + 45 \, {\left (66 \, C a^{2} b^{2} + 44 \, B a b^{3} + {\left (11 \, A + 9 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{3} + 77 \, {\left (36 \, C a^{3} b + 54 \, B a^{2} b^{2} + 4 \, {\left (9 \, A + 7 \, C\right )} a b^{3} + 7 \, B b^{4}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (77 \, C a^{4} + 308 \, B a^{3} b + 66 \, {\left (7 \, A + 5 \, C\right )} a^{2} b^{2} + 220 \, B a b^{3} + 5 \, {\left (11 \, A + 9 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{3465 \, d} \]
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(1 /2),x, algorithm="fricas")
-1/3465*(15*sqrt(2)*(77*I*(3*A + C)*a^4 + 308*I*B*a^3*b + 66*I*(7*A + 5*C) *a^2*b^2 + 220*I*B*a*b^3 + 5*I*(11*A + 9*C)*b^4)*weierstrassPInverse(-4, 0 , cos(d*x + c) + I*sin(d*x + c)) + 15*sqrt(2)*(-77*I*(3*A + C)*a^4 - 308*I *B*a^3*b - 66*I*(7*A + 5*C)*a^2*b^2 - 220*I*B*a*b^3 - 5*I*(11*A + 9*C)*b^4 )*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 231*sqrt(2)* (-15*I*B*a^4 - 12*I*(5*A + 3*C)*a^3*b - 54*I*B*a^2*b^2 - 4*I*(9*A + 7*C)*a *b^3 - 7*I*B*b^4)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d* x + c) + I*sin(d*x + c))) + 231*sqrt(2)*(15*I*B*a^4 + 12*I*(5*A + 3*C)*a^3 *b + 54*I*B*a^2*b^2 + 4*I*(9*A + 7*C)*a*b^3 + 7*I*B*b^4)*weierstrassZeta(- 4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(315* C*b^4*cos(d*x + c)^5 + 385*(4*C*a*b^3 + B*b^4)*cos(d*x + c)^4 + 45*(66*C*a ^2*b^2 + 44*B*a*b^3 + (11*A + 9*C)*b^4)*cos(d*x + c)^3 + 77*(36*C*a^3*b + 54*B*a^2*b^2 + 4*(9*A + 7*C)*a*b^3 + 7*B*b^4)*cos(d*x + c)^2 + 15*(77*C*a^ 4 + 308*B*a^3*b + 66*(7*A + 5*C)*a^2*b^2 + 220*B*a*b^3 + 5*(11*A + 9*C)*b^ 4)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\text {Timed out} \]
\[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sqrt {\sec \left (d x + c\right )} \,d x } \]
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(1 /2),x, algorithm="maxima")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^4*s qrt(sec(d*x + c)), x)
\[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sqrt {\sec \left (d x + c\right )} \,d x } \]
integrate((a+b*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^(1 /2),x, algorithm="giac")
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^4*s qrt(sec(d*x + c)), x)
Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sqrt {\sec (c+d x)} \, dx=\int \sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]