Integrand size = 43, antiderivative size = 517 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {2 \left (468 a^3 b B+364 a b^3 B+39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {2 \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 b \left (2171 a^2 b B+1053 b^3 B+192 a^3 C+2 a b^2 (1573 A+1259 C)\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (3458 a^3 b B+4004 a b^3 B+192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (143 A b^2+221 a b B+48 a^2 C+121 b^2 C\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 (13 b B+8 a C) (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}} \]
2/9009*b*(2171*B*a^2*b+1053*B*b^3+192*a^3*C+2*a*b^2*(1573*A+1259*C))*sin(d *x+c)/d/sec(d*x+c)^(5/2)+2/6435*(3458*B*a^3*b+4004*B*a*b^3+192*a^4*C+77*b^ 4*(13*A+11*C)+11*a^2*b^2*(637*A+491*C))*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/12 87*(143*A*b^2+221*B*a*b+48*C*a^2+121*C*b^2)*(a+b*cos(d*x+c))^2*sin(d*x+c)/ d/sec(d*x+c)^(3/2)+2/143*(13*B*b+8*C*a)*(a+b*cos(d*x+c))^3*sin(d*x+c)/d/se c(d*x+c)^(3/2)+2/13*C*(a+b*cos(d*x+c))^4*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/2 31*(77*B*a^4+330*B*a^2*b^2+45*B*b^4+44*a^3*b*(7*A+5*C)+20*a*b^3*(11*A+9*C) )*sin(d*x+c)/d/sec(d*x+c)^(1/2)+2/195*(468*B*a^3*b+364*B*a*b^3+39*a^4*(5*A +3*C)+78*a^2*b^2*(9*A+7*C)+7*b^4*(13*A+11*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*(77*B*a^4+330*B*a^2*b^2+45*B*b^4+44*a^3*b*(7*A+5 *C)+20*a*b^3*(11*A+9*C))*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*E llipticF(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 = 7.23 (sec) , antiderivative size = 400, normalized size of antiderivative = 0.77 \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {\sqrt {\sec (c+d x)} \left (7392 \left (468 a^3 b B+364 a b^3 B+39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+6240 \left (77 a^4 B+330 a^2 b^2 B+45 b^4 B+44 a^3 b (7 A+5 C)+20 a b^3 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\left (154 \left (3744 a^3 b B+4472 a b^3 B+936 a^4 C+156 a^2 b^2 (36 A+43 C)+b^4 (1118 A+1171 C)\right ) \cos (c+d x)+5 \left (78 \left (616 a^4 B+3432 a^2 b^2 B+531 b^4 B+176 a^3 b (14 A+13 C)+4 a b^3 (572 A+531 C)\right )+1872 b \left (33 a^2 b B+8 b^3 B+22 a^3 C+2 a b^2 (11 A+16 C)\right ) \cos (2 (c+d x))+77 b^2 \left (52 A b^2+208 a b B+312 a^2 C+89 b^2 C\right ) \cos (3 (c+d x))+1638 b^3 (b B+4 a C) \cos (4 (c+d x))+693 b^4 C \cos (5 (c+d x))\right )\right ) \sin (2 (c+d x))\right )}{720720 d} \]
Integrate[((a + b*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)) /Sqrt[Sec[c + d*x]],x]
(Sqrt[Sec[c + d*x]]*(7392*(468*a^3*b*B + 364*a*b^3*B + 39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11*C))*Sqrt[Cos[c + d*x]]*Ellipti cE[(c + d*x)/2, 2] + 6240*(77*a^4*B + 330*a^2*b^2*B + 45*b^4*B + 44*a^3*b* (7*A + 5*C) + 20*a*b^3*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x )/2, 2] + (154*(3744*a^3*b*B + 4472*a*b^3*B + 936*a^4*C + 156*a^2*b^2*(36* A + 43*C) + b^4*(1118*A + 1171*C))*Cos[c + d*x] + 5*(78*(616*a^4*B + 3432* a^2*b^2*B + 531*b^4*B + 176*a^3*b*(14*A + 13*C) + 4*a*b^3*(572*A + 531*C)) + 1872*b*(33*a^2*b*B + 8*b^3*B + 22*a^3*C + 2*a*b^2*(11*A + 16*C))*Cos[2* (c + d*x)] + 77*b^2*(52*A*b^2 + 208*a*b*B + 312*a^2*C + 89*b^2*C)*Cos[3*(c + d*x)] + 1638*b^3*(b*B + 4*a*C)*Cos[4*(c + d*x)] + 693*b^4*C*Cos[5*(c + d*x)]))*Sin[2*(c + d*x)]))/(720720*d)
Time = 2.92 (sec) , antiderivative size = 476, normalized size of antiderivative = 0.92, number of steps used = 24, number of rules used = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.558, Rules used = {3042, 4709, 3042, 3528, 27, 3042, 3528, 27, 3042, 3528, 27, 3042, 3512, 27, 3042, 3502, 27, 3042, 3227, 3042, 3115, 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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos (c+d x)^2\right )}{\sqrt {\sec (c+d x)}}dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (C \cos ^2(c+d x)+B \cos (c+d x)+A\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+A\right )dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{13} \int \frac {1}{2} \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left ((13 b B+8 a C) \cos ^2(c+d x)+(13 A b+11 C b+13 a B) \cos (c+d x)+a (13 A+3 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left ((13 b B+8 a C) \cos ^2(c+d x)+(13 A b+11 C b+13 a B) \cos (c+d x)+a (13 A+3 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left ((13 b B+8 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(13 A b+11 C b+13 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+a (13 A+3 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {2}{11} \int \frac {1}{2} \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \left (\left (48 C a^2+221 b B a+143 A b^2+121 b^2 C\right ) \cos ^2(c+d x)+\left (143 B a^2+286 A b a+226 b C a+117 b^2 B\right ) \cos (c+d x)+a (143 a A+39 b B+57 a C)\right )dx+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \left (\left (48 C a^2+221 b B a+143 A b^2+121 b^2 C\right ) \cos ^2(c+d x)+\left (143 B a^2+286 A b a+226 b C a+117 b^2 B\right ) \cos (c+d x)+a (143 a A+39 b B+57 a C)\right )dx+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (\left (48 C a^2+221 b B a+143 A b^2+121 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (143 B a^2+286 A b a+226 b C a+117 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a (143 a A+39 b B+57 a C)\right )dx+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {2}{9} \int \frac {1}{2} \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (\left (192 C a^3+2171 b B a^2+2 b^2 (1573 A+1259 C) a+1053 b^3 B\right ) \cos ^2(c+d x)+\left (1287 B a^3+3 b (1287 A+961 C) a^2+2951 b^2 B a+77 b^3 (13 A+11 C)\right ) \cos (c+d x)+3 a \left (3 (143 A+73 C) a^2+338 b B a+11 b^2 (13 A+11 C)\right )\right )dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (\left (192 C a^3+2171 b B a^2+2 b^2 (1573 A+1259 C) a+1053 b^3 B\right ) \cos ^2(c+d x)+\left (1287 B a^3+3 b (1287 A+961 C) a^2+2951 b^2 B a+77 b^3 (13 A+11 C)\right ) \cos (c+d x)+3 a \left (3 (143 A+73 C) a^2+338 b B a+11 b^2 (13 A+11 C)\right )\right )dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (\left (192 C a^3+2171 b B a^2+2 b^2 (1573 A+1259 C) a+1053 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (1287 B a^3+3 b (1287 A+961 C) a^2+2951 b^2 B a+77 b^3 (13 A+11 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+3 a \left (3 (143 A+73 C) a^2+338 b B a+11 b^2 (13 A+11 C)\right )\right )dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {1}{2} \sqrt {\cos (c+d x)} \left (21 \left (3 (143 A+73 C) a^2+338 b B a+11 b^2 (13 A+11 C)\right ) a^2+7 \left (192 C a^4+3458 b B a^3+11 b^2 (637 A+491 C) a^2+4004 b^3 B a+77 b^4 (13 A+11 C)\right ) \cos ^2(c+d x)+117 \left (77 B a^4+44 b (7 A+5 C) a^3+330 b^2 B a^2+20 b^3 (11 A+9 C) a+45 b^4 B\right ) \cos (c+d x)\right )dx+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \sqrt {\cos (c+d x)} \left (21 \left (3 (143 A+73 C) a^2+338 b B a+11 b^2 (13 A+11 C)\right ) a^2+7 \left (192 C a^4+3458 b B a^3+11 b^2 (637 A+491 C) a^2+4004 b^3 B a+77 b^4 (13 A+11 C)\right ) \cos ^2(c+d x)+117 \left (77 B a^4+44 b (7 A+5 C) a^3+330 b^2 B a^2+20 b^3 (11 A+9 C) a+45 b^4 B\right ) \cos (c+d x)\right )dx+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (21 \left (3 (143 A+73 C) a^2+338 b B a+11 b^2 (13 A+11 C)\right ) a^2+7 \left (192 C a^4+3458 b B a^3+11 b^2 (637 A+491 C) a^2+4004 b^3 B a+77 b^4 (13 A+11 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+117 \left (77 B a^4+44 b (7 A+5 C) a^3+330 b^2 B a^2+20 b^3 (11 A+9 C) a+45 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {3}{2} \sqrt {\cos (c+d x)} \left (77 \left (39 (5 A+3 C) a^4+468 b B a^3+78 b^2 (9 A+7 C) a^2+364 b^3 B a+7 b^4 (13 A+11 C)\right )+195 \left (77 B a^4+44 b (7 A+5 C) a^3+330 b^2 B a^2+20 b^3 (11 A+9 C) a+45 b^4 B\right ) \cos (c+d x)\right )dx+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \int \sqrt {\cos (c+d x)} \left (77 \left (39 (5 A+3 C) a^4+468 b B a^3+78 b^2 (9 A+7 C) a^2+364 b^3 B a+7 b^4 (13 A+11 C)\right )+195 \left (77 B a^4+44 b (7 A+5 C) a^3+330 b^2 B a^2+20 b^3 (11 A+9 C) a+45 b^4 B\right ) \cos (c+d x)\right )dx+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (77 \left (39 (5 A+3 C) a^4+468 b B a^3+78 b^2 (9 A+7 C) a^2+364 b^3 B a+7 b^4 (13 A+11 C)\right )+195 \left (77 B a^4+44 b (7 A+5 C) a^3+330 b^2 B a^2+20 b^3 (11 A+9 C) a+45 b^4 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (195 \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right ) \int \cos ^{\frac {3}{2}}(c+d x)dx+77 \left (39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (77 \left (39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+195 \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (77 \left (39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+195 \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (77 \left (39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+195 \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (\frac {3}{5} \left (195 \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {154 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right )}{d}\right )+\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{5 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}\right )+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (48 a^2 C+221 a b B+143 A b^2+121 b^2 C\right ) (a+b \cos (c+d x))^2}{9 d}+\frac {1}{9} \left (\frac {2 b \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (192 a^3 C+2171 a^2 b B+2 a b^2 (1573 A+1259 C)+1053 b^3 B\right )}{7 d}+\frac {1}{7} \left (\frac {14 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (192 a^4 C+3458 a^3 b B+11 a^2 b^2 (637 A+491 C)+4004 a b^3 B+77 b^4 (13 A+11 C)\right )}{5 d}+\frac {3}{5} \left (\frac {154 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (39 a^4 (5 A+3 C)+468 a^3 b B+78 a^2 b^2 (9 A+7 C)+364 a b^3 B+7 b^4 (13 A+11 C)\right )}{d}+195 \left (77 a^4 B+44 a^3 b (7 A+5 C)+330 a^2 b^2 B+20 a b^3 (11 A+9 C)+45 b^4 B\right ) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )\right )\right )\right )+\frac {2 (8 a C+13 b B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^4}{13 d}\right )\) |
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[ c + d*x])^4*Sin[c + d*x])/(13*d) + ((2*(13*b*B + 8*a*C)*Cos[c + d*x]^(3/2) *(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(11*d) + ((2*(143*A*b^2 + 221*a*b*B + 48*a^2*C + 121*b^2*C)*Cos[c + d*x]^(3/2)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(9*d) + ((2*b*(2171*a^2*b*B + 1053*b^3*B + 192*a^3*C + 2*a*b^2*(1573 *A + 1259*C))*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + ((14*(3458*a^3*b*B + 4004*a*b^3*B + 192*a^4*C + 77*b^4*(13*A + 11*C) + 11*a^2*b^2*(637*A + 49 1*C))*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (3*((154*(468*a^3*b*B + 364 *a*b^3*B + 39*a^4*(5*A + 3*C) + 78*a^2*b^2*(9*A + 7*C) + 7*b^4*(13*A + 11* C))*EllipticE[(c + d*x)/2, 2])/d + 195*(77*a^4*B + 330*a^2*b^2*B + 45*b^4* B + 44*a^3*b*(7*A + 5*C) + 20*a*b^3*(11*A + 9*C))*((2*EllipticF[(c + d*x)/ 2, 2])/(3*d) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d))))/5)/7)/9)/11)/1 3)
3.15.81.3.1 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[((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[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. \(1406\) vs. \(2(533)=1066\).
Time = 21.78 (sec) , antiderivative size = 1407, normalized size of antiderivative = 2.72
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1407\) |
| parts | \(\text {Expression too large to display}\) | \(1485\) |
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)
-2/45045*((-1+2*cos(1/2*d*x+1/2*c)^2)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-443520 *C*b^4*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^14+(262080*B*b^4+1048320*C*a* b^3+1330560*C*b^4)*sin(1/2*d*x+1/2*c)^12*cos(1/2*d*x+1/2*c)+(-160160*A*b^4 -640640*B*a*b^3-655200*B*b^4-960960*C*a^2*b^2-2620800*C*a*b^3-1798720*C*b^ 4)*sin(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(411840*A*a*b^3+320320*A*b^4+6 17760*B*a^2*b^2+1281280*B*a*b^3+739440*B*b^4+411840*C*a^3*b+1921920*C*a^2* b^2+2957760*C*a*b^3+1379840*C*b^4)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c) +(-432432*A*a^2*b^2-617760*A*a*b^3-296296*A*b^4-288288*B*a^3*b-926640*B*a^ 2*b^2-1185184*B*a*b^3-453960*B*b^4-72072*C*a^4-617760*C*a^3*b-1777776*C*a^ 2*b^2-1815840*C*a*b^3-666512*C*b^4)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c )+(240240*A*a^3*b+432432*A*a^2*b^2+480480*A*a*b^3+136136*A*b^4+60060*B*a^4 +288288*B*a^3*b+720720*B*a^2*b^2+544544*B*a*b^3+180180*B*b^4+72072*C*a^4+4 80480*C*a^3*b+816816*C*a^2*b^2+720720*C*a*b^3+198352*C*b^4)*sin(1/2*d*x+1/ 2*c)^4*cos(1/2*d*x+1/2*c)+(-120120*A*a^3*b-108108*A*a^2*b^2-137280*A*a*b^3 -24024*A*b^4-30030*B*a^4-72072*B*a^3*b-205920*B*a^2*b^2-96096*B*a*b^3-3627 0*B*b^4-18018*C*a^4-137280*C*a^3*b-144144*C*a^2*b^2-145080*C*a*b^3-27258*C *b^4)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+60060*A*a^3*b*(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))+42900*a*A*b^3*(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))-45045*A*(sin(1/2*d*...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.18 (sec) , antiderivative size = 560, normalized size of antiderivative = 1.08 \[ \int \frac {(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 {195 \, \sqrt {2} {\left (77 i \, B a^{4} + 44 i \, {\left (7 \, A + 5 \, C\right )} a^{3} b + 330 i \, B a^{2} b^{2} + 20 i \, {\left (11 \, A + 9 \, C\right )} a b^{3} + 45 i \, B b^{4}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 195 \, \sqrt {2} {\left (-77 i \, B a^{4} - 44 i \, {\left (7 \, A + 5 \, C\right )} a^{3} b - 330 i \, B a^{2} b^{2} - 20 i \, {\left (11 \, A + 9 \, C\right )} a b^{3} - 45 i \, B 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 (-39 i \, {\left (5 \, A + 3 \, C\right )} a^{4} - 468 i \, B a^{3} b - 78 i \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} - 364 i \, B a b^{3} - 7 i \, {\left (13 \, A + 11 \, C\right )} 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 (39 i \, {\left (5 \, A + 3 \, C\right )} a^{4} + 468 i \, B a^{3} b + 78 i \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} + 364 i \, B a b^{3} + 7 i \, {\left (13 \, A + 11 \, C\right )} 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 (3465 \, C b^{4} \cos \left (d x + c\right )^{6} + 4095 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{5} + 385 \, {\left (78 \, C a^{2} b^{2} + 52 \, B a b^{3} + {\left (13 \, A + 11 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 585 \, {\left (44 \, C a^{3} b + 66 \, B a^{2} b^{2} + 4 \, {\left (11 \, A + 9 \, C\right )} a b^{3} + 9 \, B b^{4}\right )} \cos \left (d x + c\right )^{3} + 77 \, {\left (117 \, C a^{4} + 468 \, B a^{3} b + 78 \, {\left (9 \, A + 7 \, C\right )} a^{2} b^{2} + 364 \, B a b^{3} + 7 \, {\left (13 \, A + 11 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 195 \, {\left (77 \, B a^{4} + 44 \, {\left (7 \, A + 5 \, C\right )} a^{3} b + 330 \, B a^{2} b^{2} + 20 \, {\left (11 \, A + 9 \, C\right )} a b^{3} + 45 \, B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{45045 \, 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")
-1/45045*(195*sqrt(2)*(77*I*B*a^4 + 44*I*(7*A + 5*C)*a^3*b + 330*I*B*a^2*b ^2 + 20*I*(11*A + 9*C)*a*b^3 + 45*I*B*b^4)*weierstrassPInverse(-4, 0, cos( d*x + c) + I*sin(d*x + c)) + 195*sqrt(2)*(-77*I*B*a^4 - 44*I*(7*A + 5*C)*a ^3*b - 330*I*B*a^2*b^2 - 20*I*(11*A + 9*C)*a*b^3 - 45*I*B*b^4)*weierstrass PInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) + 231*sqrt(2)*(-39*I*(5*A + 3*C)*a^4 - 468*I*B*a^3*b - 78*I*(9*A + 7*C)*a^2*b^2 - 364*I*B*a*b^3 - 7*I *(13*A + 11*C)*b^4)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos( d*x + c) + I*sin(d*x + c))) + 231*sqrt(2)*(39*I*(5*A + 3*C)*a^4 + 468*I*B* a^3*b + 78*I*(9*A + 7*C)*a^2*b^2 + 364*I*B*a*b^3 + 7*I*(13*A + 11*C)*b^4)* weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(3465*C*b^4*cos(d*x + c)^6 + 4095*(4*C*a*b^3 + B*b^4)*cos(d*x + c)^5 + 385*(78*C*a^2*b^2 + 52*B*a*b^3 + (13*A + 11*C)*b^4)*cos(d*x + c)^ 4 + 585*(44*C*a^3*b + 66*B*a^2*b^2 + 4*(11*A + 9*C)*a*b^3 + 9*B*b^4)*cos(d *x + c)^3 + 77*(117*C*a^4 + 468*B*a^3*b + 78*(9*A + 7*C)*a^2*b^2 + 364*B*a *b^3 + 7*(13*A + 11*C)*b^4)*cos(d*x + c)^2 + 195*(77*B*a^4 + 44*(7*A + 5*C )*a^3*b + 330*B*a^2*b^2 + 20*(11*A + 9*C)*a*b^3 + 45*B*b^4)*cos(d*x + c))* sin(d*x + c)/sqrt(cos(d*x + c)))/d
Timed out. \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\text {Timed out} \]
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \]
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)
\[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\sec \left (d x + c\right )}} \,d x } \]
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. \[ \int \frac {(a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]
int(((a + b*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(1/co s(c + d*x))^(1/2),x)