\(\int \frac {(a+b \cos (c+d x))^3 (A+C \cos ^2(c+d x))}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\) [1382]

Optimal result

Integrand size = 35, antiderivative size = 386 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {2 b \left (39 a^2 (9 A+7 C)+7 b^2 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {2 a \left (11 a^2 (7 A+5 C)+15 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 b \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x)}{1287 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {6 a \left (143 A b^2+8 a^2 C+117 b^2 C\right ) \sin (c+d x)}{1001 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {12 a C (a+b \cos (c+d x))^2 \sin (c+d x)}{143 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{13 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (39 a^2 (9 A+7 C)+7 b^2 (13 A+11 C)\right ) \sin (c+d x)}{585 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 a \left (11 a^2 (7 A+5 C)+15 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}} \] Output:

2/195*b*(39*a^2*(9*A+7*C)+7*b^2*(13*A+11*C))*cos(d*x+c)^(1/2)*EllipticE(si 
n(1/2*d*x+1/2*c),2^(1/2))*sec(d*x+c)^(1/2)/d+2/231*a*(11*a^2*(7*A+5*C)+15* 
b^2*(11*A+9*C))*cos(d*x+c)^(1/2)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))*se 
c(d*x+c)^(1/2)/d+2/1287*b*(24*a^2*C+11*b^2*(13*A+11*C))*sin(d*x+c)/d/sec(d 
*x+c)^(7/2)+6/1001*a*(143*A*b^2+8*C*a^2+117*C*b^2)*sin(d*x+c)/d/sec(d*x+c) 
^(5/2)+12/143*a*C*(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*b*(39*a^2*(9*A+7*C) 
+7*b^2*(13*A+11*C))*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/231*a*(11*a^2*(7*A+5*C 
)+15*b^2*(11*A+9*C))*sin(d*x+c)/d/sec(d*x+c)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 6.57 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.72 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt {\sec (c+d x)} \left (7392 b \left (39 a^2 (9 A+7 C)+7 b^2 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+6240 a \left (11 a^2 (7 A+5 C)+15 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\left (154 b \left (78 a^2 (36 A+43 C)+b^2 (1118 A+1171 C)\right ) \cos (c+d x)+5 \left (3432 a^3 (14 A+13 C)+234 a b^2 (572 A+531 C)+936 a \left (33 A b^2+11 a^2 C+48 b^2 C\right ) \cos (2 (c+d x))+77 \left (52 A b^3+156 a^2 b C+89 b^3 C\right ) \cos (3 (c+d x))+4914 a b^2 C \cos (4 (c+d x))+693 b^3 C \cos (5 (c+d x))\right )\right ) \sin (2 (c+d x))\right )}{720720 d} \] Input:

Integrate[((a + b*Cos[c + d*x])^3*(A + C*Cos[c + d*x]^2))/Sec[c + d*x]^(3/ 
2),x]
 

Output:

(Sqrt[Sec[c + d*x]]*(7392*b*(39*a^2*(9*A + 7*C) + 7*b^2*(13*A + 11*C))*Sqr 
t[Cos[c + d*x]]*EllipticE[(c + d*x)/2, 2] + 6240*a*(11*a^2*(7*A + 5*C) + 1 
5*b^2*(11*A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (154*b* 
(78*a^2*(36*A + 43*C) + b^2*(1118*A + 1171*C))*Cos[c + d*x] + 5*(3432*a^3* 
(14*A + 13*C) + 234*a*b^2*(572*A + 531*C) + 936*a*(33*A*b^2 + 11*a^2*C + 4 
8*b^2*C)*Cos[2*(c + d*x)] + 77*(52*A*b^3 + 156*a^2*b*C + 89*b^3*C)*Cos[3*( 
c + d*x)] + 4914*a*b^2*C*Cos[4*(c + d*x)] + 693*b^3*C*Cos[5*(c + d*x)]))*S 
in[2*(c + d*x)]))/(720720*d)
 

Rubi [A] (verified)

Time = 2.00 (sec) , antiderivative size = 341, normalized size of antiderivative = 0.88, number of steps used = 21, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4709, 3042, 3529, 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.

\(\displaystyle \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos (c+d x)^2\right )}{\sec (c+d x)^{3/2}}dx\)

\(\Big \downarrow \) 4709

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3 \left (C \cos ^2(c+d x)+A\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (C \sin \left (c+d x+\frac {\pi }{2}\right )^2+A\right )dx\)

\(\Big \downarrow \) 3529

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{13} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left (6 a C \cos ^2(c+d x)+b (13 A+11 C) \cos (c+d x)+a (13 A+5 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2 \left (6 a C \cos ^2(c+d x)+b (13 A+11 C) \cos (c+d x)+a (13 A+5 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (6 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (13 A+11 C) \sin \left (c+d x+\frac {\pi }{2}\right )+a (13 A+5 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\)

\(\Big \downarrow \) 3528

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {2}{11} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x)) \left ((143 A+85 C) a^2+2 b (143 A+115 C) \cos (c+d x) a+\left (24 C a^2+11 b^2 (13 A+11 C)\right ) \cos ^2(c+d x)\right )dx+\frac {12 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \int \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x)) \left ((143 A+85 C) a^2+2 b (143 A+115 C) \cos (c+d x) a+\left (24 C a^2+11 b^2 (13 A+11 C)\right ) \cos ^2(c+d x)\right )dx+\frac {12 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left ((143 A+85 C) a^2+2 b (143 A+115 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+\left (24 C a^2+11 b^2 (13 A+11 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+\frac {12 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\)

\(\Big \downarrow \) 3512

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {2}{9} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) \left (9 (143 A+85 C) a^3+27 \left (8 C a^2+143 A b^2+117 b^2 C\right ) \cos ^2(c+d x) a+11 b \left (39 (9 A+7 C) a^2+7 b^2 (13 A+11 C)\right ) \cos (c+d x)\right )dx+\frac {2 b \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {12 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \int \cos ^{\frac {3}{2}}(c+d x) \left (9 (143 A+85 C) a^3+27 \left (8 C a^2+143 A b^2+117 b^2 C\right ) \cos ^2(c+d x) a+11 b \left (39 (9 A+7 C) a^2+7 b^2 (13 A+11 C)\right ) \cos (c+d x)\right )dx+\frac {2 b \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {12 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (9 (143 A+85 C) a^3+27 \left (8 C a^2+143 A b^2+117 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a+11 b \left (39 (9 A+7 C) a^2+7 b^2 (13 A+11 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 b \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {12 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\)

\(\Big \downarrow \) 3502

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) \left (117 a \left (11 (7 A+5 C) a^2+15 b^2 (11 A+9 C)\right )+77 b \left (39 (9 A+7 C) a^2+7 b^2 (13 A+11 C)\right ) \cos (c+d x)\right )dx+\frac {54 a \left (8 a^2 C+143 A b^2+117 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {12 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \cos ^{\frac {3}{2}}(c+d x) \left (117 a \left (11 (7 A+5 C) a^2+15 b^2 (11 A+9 C)\right )+77 b \left (39 (9 A+7 C) a^2+7 b^2 (13 A+11 C)\right ) \cos (c+d x)\right )dx+\frac {54 a \left (8 a^2 C+143 A b^2+117 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {12 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (117 a \left (11 (7 A+5 C) a^2+15 b^2 (11 A+9 C)\right )+77 b \left (39 (9 A+7 C) a^2+7 b^2 (13 A+11 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {54 a \left (8 a^2 C+143 A b^2+117 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {12 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\)

\(\Big \downarrow \) 3227

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (117 a \left (11 a^2 (7 A+5 C)+15 b^2 (11 A+9 C)\right ) \int \cos ^{\frac {3}{2}}(c+d x)dx+77 b \left (39 a^2 (9 A+7 C)+7 b^2 (13 A+11 C)\right ) \int \cos ^{\frac {5}{2}}(c+d x)dx\right )+\frac {54 a \left (8 a^2 C+143 A b^2+117 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {12 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (117 a \left (11 a^2 (7 A+5 C)+15 b^2 (11 A+9 C)\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+77 b \left (39 a^2 (9 A+7 C)+7 b^2 (13 A+11 C)\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}dx\right )+\frac {54 a \left (8 a^2 C+143 A b^2+117 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {12 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (77 b \left (39 a^2 (9 A+7 C)+7 b^2 (13 A+11 C)\right ) \left (\frac {3}{5} \int \sqrt {\cos (c+d x)}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+117 a \left (11 a^2 (7 A+5 C)+15 b^2 (11 A+9 C)\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {54 a \left (8 a^2 C+143 A b^2+117 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {12 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (77 b \left (39 a^2 (9 A+7 C)+7 b^2 (13 A+11 C)\right ) \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+117 a \left (11 a^2 (7 A+5 C)+15 b^2 (11 A+9 C)\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {54 a \left (8 a^2 C+143 A b^2+117 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {12 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\)

\(\Big \downarrow \) 3119

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {1}{9} \left (\frac {1}{7} \left (117 a \left (11 a^2 (7 A+5 C)+15 b^2 (11 A+9 C)\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+77 b \left (39 a^2 (9 A+7 C)+7 b^2 (13 A+11 C)\right ) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {54 a \left (8 a^2 C+143 A b^2+117 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {2 b \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}\right )+\frac {12 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\)

\(\Big \downarrow \) 3120

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{13} \left (\frac {1}{11} \left (\frac {2 b \left (24 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{9 d}+\frac {1}{9} \left (\frac {54 a \left (8 a^2 C+143 A b^2+117 b^2 C\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}+\frac {1}{7} \left (77 b \left (39 a^2 (9 A+7 C)+7 b^2 (13 A+11 C)\right ) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+117 a \left (11 a^2 (7 A+5 C)+15 b^2 (11 A+9 C)\right ) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )\right )\right )+\frac {12 a C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^2}{11 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{13 d}\right )\)

Input:

Int[((a + b*Cos[c + d*x])^3*(A + 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) + ((12*a*C*Cos[c + d*x]^(5/2)*(a + b*Cos[ 
c + d*x])^2*Sin[c + d*x])/(11*d) + ((2*b*(24*a^2*C + 11*b^2*(13*A + 11*C)) 
*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(9*d) + ((54*a*(143*A*b^2 + 8*a^2*C + 11 
7*b^2*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (117*a*(11*a^2*(7*A + 5* 
C) + 15*b^2*(11*A + 9*C))*((2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*Sqrt[C 
os[c + d*x]]*Sin[c + d*x])/(3*d)) + 77*b*(39*a^2*(9*A + 7*C) + 7*b^2*(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)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3115

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]
 

rule 3119

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]
 

rule 3120

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]
 

rule 3227

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]
 

rule 3502

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]
 

rule 3512

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]
 

rule 3528

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])))
 

rule 3529

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) 
+ (f_.)*(x_)])^(n_.)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] : 
> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^(n + 
1)/(d*f*(m + n + 2))), x] + Simp[1/(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)) + (A*b*d*(m + n + 2) - C*(a*c - b*d*(m + n + 1)))*Sin[e + f*x] + C* 
(a*d*m - b*c*(m + 1))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f 
, A, 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])))
 

rule 4709

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(872\) vs. \(2(353)=706\).

Time = 55.92 (sec) , antiderivative size = 873, normalized size of antiderivative = 2.26

method	result	size
default	\(\text {Expression too large to display}\)	\(873\)
parts	\(\text {Expression too large to display}\)	\(1295\)

Input:

int((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x,method=_RETUR 
NVERBOSE)
 

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+(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^3-480480*C*a^2*b 
-1965600*C*a*b^2-1798720*C*b^3)*sin(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+102960*C*a^3+960960*C*a^2*b+2218320*C*a*b^2+13 
79840*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-4633 
20*A*a*b^2-296296*A*b^3-154440*C*a^3-888888*C*a^2*b-1361880*C*a*b^2-666512 
*C*b^3)*sin(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+120120*C*a^3+408408*C*a^2*b+540540*C*a*b^2+1 
98352*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^3-34320*C*a^3-72072*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+1/2*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))*b^3+10725*a^3*C* 
(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(co 
s(1/2*d*x+1/2*c),2^(1/2))+26325*C*a*b^2*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.14 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=-\frac {195 \, \sqrt {2} {\left (11 i \, {\left (7 \, A + 5 \, C\right )} a^{3} + 15 i \, {\left (11 \, A + 9 \, C\right )} a b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 195 \, \sqrt {2} {\left (-11 i \, {\left (7 \, A + 5 \, C\right )} a^{3} - 15 i \, {\left (11 \, A + 9 \, C\right )} a b^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 \, \sqrt {2} {\left (-39 i \, {\left (9 \, A + 7 \, C\right )} a^{2} b - 7 i \, {\left (13 \, A + 11 \, C\right )} b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 \, \sqrt {2} {\left (39 i \, {\left (9 \, A + 7 \, C\right )} a^{2} b + 7 i \, {\left (13 \, A + 11 \, C\right )} b^{3}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (3465 \, C b^{3} \cos \left (d x + c\right )^{6} + 12285 \, C a b^{2} \cos \left (d x + c\right )^{5} + 385 \, {\left (39 \, C a^{2} b + {\left (13 \, A + 11 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{4} + 585 \, {\left (11 \, C a^{3} + 3 \, {\left (11 \, A + 9 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{3} + 77 \, {\left (39 \, {\left (9 \, A + 7 \, C\right )} a^{2} b + 7 \, {\left (13 \, A + 11 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} + 195 \, {\left (11 \, {\left (7 \, A + 5 \, C\right )} a^{3} + 15 \, {\left (11 \, A + 9 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{45045 \, d} \] Input:

integrate((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x, algori 
thm="fricas")
 

Output:

-1/45045*(195*sqrt(2)*(11*I*(7*A + 5*C)*a^3 + 15*I*(11*A + 9*C)*a*b^2)*wei 
erstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 195*sqrt(2)*(-11* 
I*(7*A + 5*C)*a^3 - 15*I*(11*A + 9*C)*a*b^2)*weierstrassPInverse(-4, 0, co 
s(d*x + c) - I*sin(d*x + c)) + 231*sqrt(2)*(-39*I*(9*A + 7*C)*a^2*b - 7*I* 
(13*A + 11*C)*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d 
*x + c) + I*sin(d*x + c))) + 231*sqrt(2)*(39*I*(9*A + 7*C)*a^2*b + 7*I*(13 
*A + 11*C)*b^3)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x 
+ c) - I*sin(d*x + c))) - 2*(3465*C*b^3*cos(d*x + c)^6 + 12285*C*a*b^2*cos 
(d*x + c)^5 + 385*(39*C*a^2*b + (13*A + 11*C)*b^3)*cos(d*x + c)^4 + 585*(1 
1*C*a^3 + 3*(11*A + 9*C)*a*b^2)*cos(d*x + c)^3 + 77*(39*(9*A + 7*C)*a^2*b 
+ 7*(13*A + 11*C)*b^3)*cos(d*x + c)^2 + 195*(11*(7*A + 5*C)*a^3 + 15*(11*A 
 + 9*C)*a*b^2)*cos(d*x + c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
 

Sympy [F]

\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \left (a + b \cos {\left (c + d x \right )}\right )^{3}}{\sec ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \] Input:

integrate((a+b*cos(d*x+c))**3*(A+C*cos(d*x+c)**2)/sec(d*x+c)**(3/2),x)
 

Output:

Integral((A + C*cos(c + d*x)**2)*(a + b*cos(c + d*x))**3/sec(c + d*x)**(3/ 
2), x)
 

Maxima [F]

\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x, algori 
thm="maxima")
 

Output:

integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^3/sec(d*x + c)^(3/2) 
, x)
 

Giac [F]

\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int { \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{3}}{\sec \left (d x + c\right )^{\frac {3}{2}}} \,d x } \] Input:

integrate((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(3/2),x, algori 
thm="giac")
 

Output:

integrate((C*cos(d*x + c)^2 + A)*(b*cos(d*x + c) + a)^3/sec(d*x + c)^(3/2) 
, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^3}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \] Input:

int(((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3)/(1/cos(c + d*x))^(3/2) 
,x)
 

Output:

int(((A + C*cos(c + d*x)^2)*(a + b*cos(c + d*x))^3)/(1/cos(c + d*x))^(3/2) 
, x)
 

Reduce [F]

\[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sec ^{\frac {3}{2}}(c+d x)} \, dx=\left (\int \frac {\sqrt {\sec \left (d x +c \right )}}{\sec \left (d x +c \right )^{2}}d x \right ) a^{4}+3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )}{\sec \left (d x +c \right )^{2}}d x \right ) a^{3} b +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}}{\sec \left (d x +c \right )^{2}}d x \right ) b^{3} c +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}}{\sec \left (d x +c \right )^{2}}d x \right ) a \,b^{2} c +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\sec \left (d x +c \right )^{2}}d x \right ) a^{2} b c +\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{3}}{\sec \left (d x +c \right )^{2}}d x \right ) a \,b^{3}+\left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\sec \left (d x +c \right )^{2}}d x \right ) a^{3} c +3 \left (\int \frac {\sqrt {\sec \left (d x +c \right )}\, \cos \left (d x +c \right )^{2}}{\sec \left (d x +c \right )^{2}}d x \right ) a^{2} b^{2} \] Input:

int((a+b*cos(d*x+c))^3*(A+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 + 3*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 + 3*int((sqrt(sec(c + d*x))*cos(c + d*x 
)**3)/sec(c + d*x)**2,x)*a**2*b*c + int((sqrt(sec(c + d*x))*cos(c + d*x)** 
3)/sec(c + d*x)**2,x)*a*b**3 + int((sqrt(sec(c + d*x))*cos(c + d*x)**2)/se 
c(c + d*x)**2,x)*a**3*c + 3*int((sqrt(sec(c + d*x))*cos(c + d*x)**2)/sec(c 
 + d*x)**2,x)*a**2*b**2