Integrand size = 43, antiderivative size = 463 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 \left (117 a^3 B+273 a b^2 B+39 a^2 b (9 A+7 C)+7 b^3 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {2 \left (165 a^2 b B+45 b^3 B+11 a^3 (7 A+5 C)+15 a b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 b \left (143 A b^2+195 a b B+24 a^2 C+121 b^2 C\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (338 a^2 b B+117 b^3 B+24 a^3 C+39 a b^2 (11 A+9 C)\right ) \sin (c+d x)}{1001 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (13 b B+6 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{143 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (117 a^3 B+273 a b^2 B+39 a^2 b (9 A+7 C)+7 b^3 (13 A+11 C)\right ) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (165 a^2 b B+45 b^3 B+11 a^3 (7 A+5 C)+15 a b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}} \] Output:
2/195*(117*B*a^3+273*B*a*b^2+39*a^2*b*(9*A+7*C)+7*b^3*(13*A+11*C))*cos(d*x +c)^(1/2)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*sec(d*x+c)^(1/2)/d+2/231*( 165*B*a^2*b+45*B*b^3+11*a^3*(7*A+5*C)+15*a*b^2*(11*A+9*C))*cos(d*x+c)^(1/2 )*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*sec(d*x+c)^(1/2)/d+2/1287*b*(143* A*b^2+195*B*a*b+24*C*a^2+121*C*b^2)*sin(d*x+c)/d/sec(d*x+c)^(7/2)+2/1001*( 338*B*a^2*b+117*B*b^3+24*a^3*C+39*a*b^2*(11*A+9*C))*sin(d*x+c)/d/sec(d*x+c )^(5/2)+2/143*(13*B*b+6*C*a)*(a+b*cos(d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(5 /2)+2/13*C*(a+b*cos(d*x+c))^3*sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/585*(117*B*a ^3+273*B*a*b^2+39*a^2*b*(9*A+7*C)+7*b^3*(13*A+11*C))*sin(d*x+c)/d/sec(d*x+ c)^(3/2)+2/231*(165*B*a^2*b+45*B*b^3+11*a^3*(7*A+5*C)+15*a*b^2*(11*A+9*C)) *sin(d*x+c)/d/sec(d*x+c)^(1/2)
Time = 5.79 (sec) , antiderivative size = 355, normalized size of antiderivative = 0.77 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {\sec (c+d x)} \left (7392 \left (117 a^3 B+273 a b^2 B+39 a^2 b (9 A+7 C)+7 b^3 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+6240 \left (165 a^2 b B+45 b^3 B+11 a^3 (7 A+5 C)+15 a b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\left (154 \left (936 a^3 B+3354 a b^2 B+78 a^2 b (36 A+43 C)+b^3 (1118 A+1171 C)\right ) \cos (c+d x)+5 \left (78 \left (1716 a^2 b B+531 b^3 B+44 a^3 (14 A+13 C)+3 a b^2 (572 A+531 C)\right )+936 \left (33 a^2 b B+16 b^3 B+11 a^3 C+3 a b^2 (11 A+16 C)\right ) \cos (2 (c+d x))+77 b \left (52 A b^2+156 a b B+156 a^2 C+89 b^2 C\right ) \cos (3 (c+d x))+1638 b^2 (b B+3 a C) \cos (4 (c+d x))+693 b^3 C \cos (5 (c+d x))\right )\right ) \sin (2 (c+d x))\right )}{720720 d} \] Input:
Integrate[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)) /Sec[c + d*x]^(3/2),x]
Output:
(Sqrt[Sec[c + d*x]]*(7392*(117*a^3*B + 273*a*b^2*B + 39*a^2*b*(9*A + 7*C) + 7*b^3*(13*A + 11*C))*Sqrt[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 6240 *(165*a^2*b*B + 45*b^3*B + 11*a^3*(7*A + 5*C) + 15*a*b^2*(11*A + 9*C))*Sqr t[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (154*(936*a^3*B + 3354*a*b^2*B + 78*a^2*b*(36*A + 43*C) + b^3*(1118*A + 1171*C))*Cos[c + d*x] + 5*(78*(1 716*a^2*b*B + 531*b^3*B + 44*a^3*(14*A + 13*C) + 3*a*b^2*(572*A + 531*C)) + 936*(33*a^2*b*B + 16*b^3*B + 11*a^3*C + 3*a*b^2*(11*A + 16*C))*Cos[2*(c + d*x)] + 77*b*(52*A*b^2 + 156*a*b*B + 156*a^2*C + 89*b^2*C)*Cos[3*(c + d* x)] + 1638*b^2*(b*B + 3*a*C)*Cos[4*(c + d*x)] + 693*b^3*C*Cos[5*(c + d*x)] ))*Sin[2*(c + d*x)]))/(720720*d)
Time = 2.27 (sec) , antiderivative size = 392, normalized size of antiderivative = 0.85, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.488, Rules used = {3042, 4709, 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.
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos (c+d x)^2\right )}{\sec (c+d x)^{3/2}}dx\) |
| \(\Big \downarrow \) 4709 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \left (C \cos ^2(c+d x)+B \cos (c+d x)+A\right )dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+B \sin \left (c+d x+\frac {\pi }{2}\right )+A\right )dx\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{13} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left ((13 b B+6 a C) \cos ^2(c+d x)+(13 A b+11 C b+13 a B) \cos (c+d x)+a (13 A+5 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left ((13 b B+6 a C) \cos ^2(c+d x)+(13 A b+11 C b+13 a B) \cos (c+d x)+a (13 A+5 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left ((13 b B+6 a C) \sin \left (c+d x+\frac {\pi }{2}\right )^2+(13 A b+11 C b+13 a B) \sin \left (c+d x+\frac {\pi }{2}\right )+a (13 A+5 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\) |
| \(\Big \downarrow \) 3528 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {2}{11} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x)) \left (\left (24 C a^2+195 b B a+143 A b^2+121 b^2 C\right ) \cos ^2(c+d x)+\left (143 B a^2+286 A b a+230 b C a+117 b^2 B\right ) \cos (c+d x)+a (143 a A+65 b B+85 a C)\right )dx+\frac {2 (6 a C+13 b B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x)) \left (\left (24 C a^2+195 b B a+143 A b^2+121 b^2 C\right ) \cos ^2(c+d x)+\left (143 B a^2+286 A b a+230 b C a+117 b^2 B\right ) \cos (c+d x)+a (143 a A+65 b B+85 a C)\right )dx+\frac {2 (6 a C+13 b B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (\left (24 C a^2+195 b B a+143 A b^2+121 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (143 B a^2+286 A b a+230 b C a+117 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )+a (143 a A+65 b B+85 a C)\right )dx+\frac {2 (6 a C+13 b B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\) |
| \(\Big \downarrow \) 3512 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {2}{9} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) \left (9 (143 a A+65 b B+85 a C) a^2+9 \left (24 C a^3+338 b B a^2+39 b^2 (11 A+9 C) a+117 b^3 B\right ) \cos ^2(c+d x)+11 \left (117 B a^3+39 b (9 A+7 C) a^2+273 b^2 B a+7 b^3 (13 A+11 C)\right ) \cos (c+d x)\right )dx+\frac {2 b \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (24 a^2 C+195 a b B+143 A b^2+121 b^2 C\right )}{9 d}\right )+\frac {2 (6 a C+13 b B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \int \cos ^{\frac {3}{2}}(c+d x) \left (9 (143 a A+65 b B+85 a C) a^2+9 \left (24 C a^3+338 b B a^2+39 b^2 (11 A+9 C) a+117 b^3 B\right ) \cos ^2(c+d x)+11 \left (117 B a^3+39 b (9 A+7 C) a^2+273 b^2 B a+7 b^3 (13 A+11 C)\right ) \cos (c+d x)\right )dx+\frac {2 b \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (24 a^2 C+195 a b B+143 A b^2+121 b^2 C\right )}{9 d}\right )+\frac {2 (6 a C+13 b B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (9 (143 a A+65 b B+85 a C) a^2+9 \left (24 C a^3+338 b B a^2+39 b^2 (11 A+9 C) a+117 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+11 \left (117 B a^3+39 b (9 A+7 C) a^2+273 b^2 B a+7 b^3 (13 A+11 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 b \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (24 a^2 C+195 a b B+143 A b^2+121 b^2 C\right )}{9 d}\right )+\frac {2 (6 a C+13 b B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\) |
| \(\Big \downarrow \) 3502 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) \left (117 \left (11 (7 A+5 C) a^3+165 b B a^2+15 b^2 (11 A+9 C) a+45 b^3 B\right )+77 \left (117 B a^3+39 b (9 A+7 C) a^2+273 b^2 B a+7 b^3 (13 A+11 C)\right ) \cos (c+d x)\right )dx+\frac {18 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^3 C+338 a^2 b B+39 a b^2 (11 A+9 C)+117 b^3 B\right )}{7 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (24 a^2 C+195 a b B+143 A b^2+121 b^2 C\right )}{9 d}\right )+\frac {2 (6 a C+13 b B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \cos ^{\frac {3}{2}}(c+d x) \left (117 \left (11 (7 A+5 C) a^3+165 b B a^2+15 b^2 (11 A+9 C) a+45 b^3 B\right )+77 \left (117 B a^3+39 b (9 A+7 C) a^2+273 b^2 B a+7 b^3 (13 A+11 C)\right ) \cos (c+d x)\right )dx+\frac {18 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^3 C+338 a^2 b B+39 a b^2 (11 A+9 C)+117 b^3 B\right )}{7 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (24 a^2 C+195 a b B+143 A b^2+121 b^2 C\right )}{9 d}\right )+\frac {2 (6 a C+13 b B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (117 \left (11 (7 A+5 C) a^3+165 b B a^2+15 b^2 (11 A+9 C) a+45 b^3 B\right )+77 \left (117 B a^3+39 b (9 A+7 C) a^2+273 b^2 B a+7 b^3 (13 A+11 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {18 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^3 C+338 a^2 b B+39 a b^2 (11 A+9 C)+117 b^3 B\right )}{7 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (24 a^2 C+195 a b B+143 A b^2+121 b^2 C\right )}{9 d}\right )+\frac {2 (6 a C+13 b B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\) |
| \(\Big \downarrow \) 3227 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (117 \left (11 a^3 (7 A+5 C)+165 a^2 b B+15 a b^2 (11 A+9 C)+45 b^3 B\right ) \int \cos ^{\frac {3}{2}}(c+d x)dx+77 \left (117 a^3 B+39 a^2 b (9 A+7 C)+273 a b^2 B+7 b^3 (13 A+11 C)\right ) \int \cos ^{\frac {5}{2}}(c+d x)dx\right )+\frac {18 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^3 C+338 a^2 b B+39 a b^2 (11 A+9 C)+117 b^3 B\right )}{7 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (24 a^2 C+195 a b B+143 A b^2+121 b^2 C\right )}{9 d}\right )+\frac {2 (6 a C+13 b B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (117 \left (11 a^3 (7 A+5 C)+165 a^2 b B+15 a b^2 (11 A+9 C)+45 b^3 B\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+77 \left (117 a^3 B+39 a^2 b (9 A+7 C)+273 a b^2 B+7 b^3 (13 A+11 C)\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}dx\right )+\frac {18 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^3 C+338 a^2 b B+39 a b^2 (11 A+9 C)+117 b^3 B\right )}{7 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (24 a^2 C+195 a b B+143 A b^2+121 b^2 C\right )}{9 d}\right )+\frac {2 (6 a C+13 b B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (77 \left (117 a^3 B+39 a^2 b (9 A+7 C)+273 a b^2 B+7 b^3 (13 A+11 C)\right ) \left (\frac {3}{5} \int \sqrt {\cos (c+d x)}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+117 \left (11 a^3 (7 A+5 C)+165 a^2 b B+15 a b^2 (11 A+9 C)+45 b^3 B\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {18 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^3 C+338 a^2 b B+39 a b^2 (11 A+9 C)+117 b^3 B\right )}{7 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (24 a^2 C+195 a b B+143 A b^2+121 b^2 C\right )}{9 d}\right )+\frac {2 (6 a C+13 b B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (77 \left (117 a^3 B+39 a^2 b (9 A+7 C)+273 a b^2 B+7 b^3 (13 A+11 C)\right ) \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+117 \left (11 a^3 (7 A+5 C)+165 a^2 b B+15 a b^2 (11 A+9 C)+45 b^3 B\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {18 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^3 C+338 a^2 b B+39 a b^2 (11 A+9 C)+117 b^3 B\right )}{7 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (24 a^2 C+195 a b B+143 A b^2+121 b^2 C\right )}{9 d}\right )+\frac {2 (6 a C+13 b B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\) |
| \(\Big \downarrow \) 3119 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (117 \left (11 a^3 (7 A+5 C)+165 a^2 b B+15 a b^2 (11 A+9 C)+45 b^3 B\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+77 \left (117 a^3 B+39 a^2 b (9 A+7 C)+273 a b^2 B+7 b^3 (13 A+11 C)\right ) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {18 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^3 C+338 a^2 b B+39 a b^2 (11 A+9 C)+117 b^3 B\right )}{7 d}\right )+\frac {2 b \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (24 a^2 C+195 a b B+143 A b^2+121 b^2 C\right )}{9 d}\right )+\frac {2 (6 a C+13 b B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\) |
| \(\Big \downarrow \) 3120 |
| \(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {2 b \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) \left (24 a^2 C+195 a b B+143 A b^2+121 b^2 C\right )}{9 d}+\frac {1}{9} \left (\frac {18 \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (24 a^3 C+338 a^2 b B+39 a b^2 (11 A+9 C)+117 b^3 B\right )}{7 d}+\frac {1}{7} \left (77 \left (117 a^3 B+39 a^2 b (9 A+7 C)+273 a b^2 B+7 b^3 (13 A+11 C)\right ) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+117 \left (11 a^3 (7 A+5 C)+165 a^2 b B+15 a b^2 (11 A+9 C)+45 b^3 B\right ) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )\right )\right )+\frac {2 (6 a C+13 b B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\) |
Input:
Int[((a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/2),x]
Output:
Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*C*Cos[c + d*x]^(5/2)*(a + b*Cos[ c + d*x])^3*Sin[c + d*x])/(13*d) + ((2*(13*b*B + 6*a*C)*Cos[c + d*x]^(5/2) *(a + b*Cos[c + d*x])^2*Sin[c + d*x])/(11*d) + ((2*b*(143*A*b^2 + 195*a*b* B + 24*a^2*C + 121*b^2*C)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d) + ((18*(3 38*a^2*b*B + 117*b^3*B + 24*a^3*C + 39*a*b^2*(11*A + 9*C))*Cos[c + d*x]^(5 /2)*Sin[c + d*x])/(7*d) + (117*(165*a^2*b*B + 45*b^3*B + 11*a^3*(7*A + 5*C ) + 15*a*b^2*(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)) + 77*(117*a^3*B + 273*a*b^2*B + 39*a^2* b*(9*A + 7*C) + 7*b^3*(13*A + 11*C))*((6*EllipticE[(c + d*x)/2, 2])/(5*d) + (2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d)))/7)/9)/11)/13)
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. \(1187\) vs. \(2(430)=860\).
Time = 46.70 (sec) , antiderivative size = 1188, normalized size of antiderivative = 2.57
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1188\) |
| parts | \(\text {Expression too large to display}\) | \(1325\) |
Input:
int((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x, method=_RETURNVERBOSE)
Output:
-2/45045*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(-443520* C*b^3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^14+(262080*B*b^3+786240*C*a*b^ 2+1330560*C*b^3)*sin(1/2*d*x+1/2*c)^12*cos(1/2*d*x+1/2*c)+(-160160*A*b^3-4 80480*B*a*b^2-655200*B*b^3-480480*C*a^2*b-1965600*C*a*b^2-1798720*C*b^3)*s in(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(308880*A*a*b^2+320320*A*b^3+30888 0*B*a^2*b+960960*B*a*b^2+739440*B*b^3+102960*C*a^3+960960*C*a^2*b+2218320* C*a*b^2+1379840*C*b^3)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-216216*A* a^2*b-463320*A*a*b^2-296296*A*b^3-72072*B*a^3-463320*B*a^2*b-888888*B*a*b^ 2-453960*B*b^3-154440*C*a^3-888888*C*a^2*b-1361880*C*a*b^2-666512*C*b^3)*s in(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(60060*A*a^3+216216*A*a^2*b+360360* A*a*b^2+136136*A*b^3+72072*B*a^3+360360*B*a^2*b+408408*B*a*b^2+180180*B*b^ 3+120120*C*a^3+408408*C*a^2*b+540540*C*a*b^2+198352*C*b^3)*sin(1/2*d*x+1/2 *c)^4*cos(1/2*d*x+1/2*c)+(-30030*A*a^3-54054*A*a^2*b-102960*A*a*b^2-24024* A*b^3-18018*B*a^3-102960*B*a^2*b-72072*B*a*b^2-36270*B*b^3-34320*C*a^3-720 72*C*a^2*b-108810*C*a*b^2-27258*C*b^3)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/ 2*c)+15015*A*a^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))+32175*a*A*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))-81081*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^2*b-21021*A*(sin(1/2*d*x+1...
Result contains complex when optimal does not.
Time = 0.16 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx =\text {Too large to display} \] Input:
integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(3 /2),x, algorithm="fricas")
Output:
-1/45045*(195*sqrt(2)*(11*I*(7*A + 5*C)*a^3 + 165*I*B*a^2*b + 15*I*(11*A + 9*C)*a*b^2 + 45*I*B*b^3)*weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin( d*x + c)) + 195*sqrt(2)*(-11*I*(7*A + 5*C)*a^3 - 165*I*B*a^2*b - 15*I*(11* A + 9*C)*a*b^2 - 45*I*B*b^3)*weierstrassPInverse(-4, 0, cos(d*x + c) - I*s in(d*x + c)) + 231*sqrt(2)*(-117*I*B*a^3 - 39*I*(9*A + 7*C)*a^2*b - 273*I* B*a*b^2 - 7*I*(13*A + 11*C)*b^3)*weierstrassZeta(-4, 0, weierstrassPInvers e(-4, 0, cos(d*x + c) + I*sin(d*x + c))) + 231*sqrt(2)*(117*I*B*a^3 + 39*I *(9*A + 7*C)*a^2*b + 273*I*B*a*b^2 + 7*I*(13*A + 11*C)*b^3)*weierstrassZet a(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - 2*(3 465*C*b^3*cos(d*x + c)^6 + 4095*(3*C*a*b^2 + B*b^3)*cos(d*x + c)^5 + 385*( 39*C*a^2*b + 39*B*a*b^2 + (13*A + 11*C)*b^3)*cos(d*x + c)^4 + 585*(11*C*a^ 3 + 33*B*a^2*b + 3*(11*A + 9*C)*a*b^2 + 9*B*b^3)*cos(d*x + c)^3 + 77*(117* B*a^3 + 39*(9*A + 7*C)*a^2*b + 273*B*a*b^2 + 7*(13*A + 11*C)*b^3)*cos(d*x + c)^2 + 195*(11*(7*A + 5*C)*a^3 + 165*B*a^2*b + 15*(11*A + 9*C)*a*b^2 + 4 5*B*b^3)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (a + b \cos {\left (c + d x \right )}\right )^{3} \left (A + B \cos {\left (c + d x \right )} + C \cos ^{2}{\left (c + d x \right )}\right )}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \] Input:
integrate((a+b*cos(d*x+c))**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)/sec(d*x+c)* *(3/2),x)
Output:
Integral((a + b*cos(c + d*x))**3*(A + B*cos(c + d*x) + C*cos(c + d*x)**2)/ sec(c + d*x)**(3/2), x)
\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(3 /2),x, algorithm="maxima")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^3/s ec(d*x + c)^(3/2), x)
\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(3 /2),x, algorithm="giac")
Output:
integrate((C*cos(d*x + c)^2 + B*cos(d*x + c) + A)*(b*cos(d*x + c) + a)^3/s ec(d*x + c)^(3/2), x)
Timed out. \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \] Input:
int(((a + b*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(1/co s(c + d*x))^(3/2),x)
Output:
int(((a + b*cos(c + d*x))^3*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/(1/co s(c + d*x))^(3/2), x)
\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{2}}d x \right ) a^{4}+4 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )}{\sec \left (d x +c \right )^{2}}d x \right ) a^{3} b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}}{\sec \left (d x +c \right )^{2}}d x \right ) b^{3} c +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}}{\sec \left (d x +c \right )^{2}}d x \right ) a \,b^{2} c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}}{\sec \left (d x +c \right )^{2}}d x \right ) b^{4}+3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\sec \left (d x +c \right )^{2}}d x \right ) a^{2} b c +4 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\sec \left (d x +c \right )^{2}}d x \right ) a \,b^{3}+\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\sec \left (d x +c \right )^{2}}d x \right ) a^{3} c +6 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\sec \left (d x +c \right )^{2}}d x \right ) a^{2} b^{2} \] Input:
int((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x)
Output:
int(sqrt(sec(c + d*x))/sec(c + d*x)**2,x)*a**4 + 4*int((sqrt(sec(c + d*x)) *cos(c + d*x))/sec(c + d*x)**2,x)*a**3*b + int((sqrt(sec(c + d*x))*cos(c + d*x)**5)/sec(c + d*x)**2,x)*b**3*c + 3*int((sqrt(sec(c + d*x))*cos(c + d* x)**4)/sec(c + d*x)**2,x)*a*b**2*c + int((sqrt(sec(c + d*x))*cos(c + d*x)* *4)/sec(c + d*x)**2,x)*b**4 + 3*int((sqrt(sec(c + d*x))*cos(c + d*x)**3)/s ec(c + d*x)**2,x)*a**2*b*c + 4*int((sqrt(sec(c + d*x))*cos(c + d*x)**3)/se c(c + d*x)**2,x)*a*b**3 + int((sqrt(sec(c + d*x))*cos(c + d*x)**2)/sec(c + d*x)**2,x)*a**3*c + 6*int((sqrt(sec(c + d*x))*cos(c + d*x)**2)/sec(c + d* x)**2,x)*a**2*b**2