\(\int \frac {(a+b \cos (c+d x))^3 (A+C \cos ^2(c+d x))}{\sqrt {\sec (c+d x)}} \, dx\) [1381]

Optimal result

Integrand size = 35, antiderivative size = 335 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {2 a \left (a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a \left (99 A b^2+8 a^2 C+77 b^2 C\right ) \sin (c+d x)}{165 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a C (a+b \cos (c+d x))^2 \sin (c+d x)}{33 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^3 \sin (c+d x)}{11 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}} \] Output:

2/5*a*(a^2*(5*A+3*C)+b^2*(9*A+7*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*b*(33*a^2*(7*A+5*C)+5*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+2/231*b*(8*a^2*C+3*b^2*(11*A+9*C))*sin(d*x+c)/d/sec(d*x+c)^(5/2)+2/ 
165*a*(99*A*b^2+8*C*a^2+77*C*b^2)*sin(d*x+c)/d/sec(d*x+c)^(3/2)+4/33*a*C*( 
a+b*cos(d*x+c))^2*sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/11*C*(a+b*cos(d*x+c))^3* 
sin(d*x+c)/d/sec(d*x+c)^(3/2)+2/231*b*(33*a^2*(7*A+5*C)+5*b^2*(11*A+9*C))* 
sin(d*x+c)/d/sec(d*x+c)^(1/2)
 

Mathematica [A] (verified)

Time = 5.83 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.70 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=\frac {\sqrt {\sec (c+d x)} \left (3696 a \left (a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+80 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )+\left (154 a \left (36 A b^2+12 a^2 C+43 b^2 C\right ) \cos (c+d x)+5 b \left (1848 a^2 A+572 A b^2+1716 a^2 C+531 b^2 C+12 \left (11 A b^2+33 a^2 C+16 b^2 C\right ) \cos (2 (c+d x))+154 a b C \cos (3 (c+d x))+21 b^2 C \cos (4 (c+d x))\right )\right ) \sin (2 (c+d x))\right )}{9240 d} \] Input:

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

Output:

(Sqrt[Sec[c + d*x]]*(3696*a*(a^2*(5*A + 3*C) + b^2*(9*A + 7*C))*Sqrt[Cos[c 
 + d*x]]*EllipticE[(c + d*x)/2, 2] + 80*b*(33*a^2*(7*A + 5*C) + 5*b^2*(11* 
A + 9*C))*Sqrt[Cos[c + d*x]]*EllipticF[(c + d*x)/2, 2] + (154*a*(36*A*b^2 
+ 12*a^2*C + 43*b^2*C)*Cos[c + d*x] + 5*b*(1848*a^2*A + 572*A*b^2 + 1716*a 
^2*C + 531*b^2*C + 12*(11*A*b^2 + 33*a^2*C + 16*b^2*C)*Cos[2*(c + d*x)] + 
154*a*b*C*Cos[3*(c + d*x)] + 21*b^2*C*Cos[4*(c + d*x)]))*Sin[2*(c + d*x)]) 
)/(9240*d)
 

Rubi [A] (verified)

Time = 1.94 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.93, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4709

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3529

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {2}{11} \int \frac {1}{2} \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \left (6 a C \cos ^2(c+d x)+b (11 A+9 C) \cos (c+d x)+a (11 A+3 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \left (6 a C \cos ^2(c+d x)+b (11 A+9 C) \cos (c+d x)+a (11 A+3 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (6 a C \sin \left (c+d x+\frac {\pi }{2}\right )^2+b (11 A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right )+a (11 A+3 C)\right )dx+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )\)

\(\Big \downarrow \) 3528

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {2}{9} \int \frac {3}{2} \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (3 (11 A+5 C) a^2+2 b (33 A+25 C) \cos (c+d x) a+\left (8 C a^2+3 b^2 (11 A+9 C)\right ) \cos ^2(c+d x)\right )dx+\frac {4 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{3} \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (3 (11 A+5 C) a^2+2 b (33 A+25 C) \cos (c+d x) a+\left (8 C a^2+3 b^2 (11 A+9 C)\right ) \cos ^2(c+d x)\right )dx+\frac {4 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{3} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (3 (11 A+5 C) a^2+2 b (33 A+25 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a+\left (8 C a^2+3 b^2 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+\frac {4 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )\)

\(\Big \downarrow \) 3512

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{3} \left (\frac {2}{7} \int \frac {1}{2} \sqrt {\cos (c+d x)} \left (21 (11 A+5 C) a^3+7 \left (8 C a^2+99 A b^2+77 b^2 C\right ) \cos ^2(c+d x) a+3 b \left (33 (7 A+5 C) a^2+5 b^2 (11 A+9 C)\right ) \cos (c+d x)\right )dx+\frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {4 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \int \sqrt {\cos (c+d x)} \left (21 (11 A+5 C) a^3+7 \left (8 C a^2+99 A b^2+77 b^2 C\right ) \cos ^2(c+d x) a+3 b \left (33 (7 A+5 C) a^2+5 b^2 (11 A+9 C)\right ) \cos (c+d x)\right )dx+\frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {4 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (21 (11 A+5 C) a^3+7 \left (8 C a^2+99 A b^2+77 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a+3 b \left (33 (7 A+5 C) a^2+5 b^2 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {4 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )\)

\(\Big \downarrow \) 3502

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {2}{5} \int \frac {3}{2} \sqrt {\cos (c+d x)} \left (77 a \left ((5 A+3 C) a^2+b^2 (9 A+7 C)\right )+5 b \left (33 (7 A+5 C) a^2+5 b^2 (11 A+9 C)\right ) \cos (c+d x)\right )dx+\frac {14 a \left (8 a^2 C+99 A b^2+77 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {4 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {3}{5} \int \sqrt {\cos (c+d x)} \left (77 a \left ((5 A+3 C) a^2+b^2 (9 A+7 C)\right )+5 b \left (33 (7 A+5 C) a^2+5 b^2 (11 A+9 C)\right ) \cos (c+d x)\right )dx+\frac {14 a \left (8 a^2 C+99 A b^2+77 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {4 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (77 a \left ((5 A+3 C) a^2+b^2 (9 A+7 C)\right )+5 b \left (33 (7 A+5 C) a^2+5 b^2 (11 A+9 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {14 a \left (8 a^2 C+99 A b^2+77 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {4 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )\)

\(\Big \downarrow \) 3227

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {3}{5} \left (5 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \int \cos ^{\frac {3}{2}}(c+d x)dx+77 a \left (a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \int \sqrt {\cos (c+d x)}dx\right )+\frac {14 a \left (8 a^2 C+99 A b^2+77 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {4 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {3}{5} \left (77 a \left (a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+5 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx\right )+\frac {14 a \left (8 a^2 C+99 A b^2+77 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {4 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )\)

\(\Big \downarrow \) 3115

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {3}{5} \left (77 a \left (a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+5 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {14 a \left (8 a^2 C+99 A b^2+77 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {4 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {3}{5} \left (77 a \left (a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+5 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {14 a \left (8 a^2 C+99 A b^2+77 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {4 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )\)

\(\Big \downarrow \) 3119

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{3} \left (\frac {1}{7} \left (\frac {3}{5} \left (5 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {154 a \left (a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {14 a \left (8 a^2 C+99 A b^2+77 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {4 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )\)

\(\Big \downarrow \) 3120

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{11} \left (\frac {1}{3} \left (\frac {2 b \left (8 a^2 C+3 b^2 (11 A+9 C)\right ) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}+\frac {1}{7} \left (\frac {14 a \left (8 a^2 C+99 A b^2+77 b^2 C\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {3}{5} \left (\frac {154 a \left (a^2 (5 A+3 C)+b^2 (9 A+7 C)\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+5 b \left (33 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )\right )\right )+\frac {4 a C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {2 C \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^3}{11 d}\right )\)

Input:

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

Output:

Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*C*Cos[c + d*x]^(3/2)*(a + b*Cos[ 
c + d*x])^3*Sin[c + d*x])/(11*d) + ((4*a*C*Cos[c + d*x]^(3/2)*(a + b*Cos[c 
 + d*x])^2*Sin[c + d*x])/(3*d) + ((2*b*(8*a^2*C + 3*b^2*(11*A + 9*C))*Cos[ 
c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + ((14*a*(99*A*b^2 + 8*a^2*C + 77*b^2*C 
)*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5*d) + (3*((154*a*(a^2*(5*A + 3*C) + b 
^2*(9*A + 7*C))*EllipticE[(c + d*x)/2, 2])/d + 5*b*(33*a^2*(7*A + 5*C) + 5 
*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))))/5)/7)/3)/11)
 

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. \(792\) vs. \(2(306)=612\).

Time = 40.69 (sec) , antiderivative size = 793, normalized size of antiderivative = 2.37

method	result	size
default	\(\text {Expression too large to display}\)	\(793\)
parts	\(\text {Expression too large to display}\)	\(1190\)

Input:

int((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x,method=_RETUR 
NVERBOSE)
 

Output:

-2/1155*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(6720*C*b^ 
3*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+(-12320*C*a*b^2-16800*C*b^3)*si 
n(1/2*d*x+1/2*c)^10*cos(1/2*d*x+1/2*c)+(2640*A*b^3+7920*C*a^2*b+24640*C*a* 
b^2+18960*C*b^3)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c)+(-5544*A*a*b^2-39 
60*A*b^3-1848*C*a^3-11880*C*a^2*b-22792*C*a*b^2-11640*C*b^3)*sin(1/2*d*x+1 
/2*c)^6*cos(1/2*d*x+1/2*c)+(4620*A*a^2*b+5544*A*a*b^2+3080*A*b^3+1848*C*a^ 
3+9240*C*a^2*b+10472*C*a*b^2+4620*C*b^3)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+ 
1/2*c)+(-2310*A*a^2*b-1386*A*a*b^2-880*A*b^3-462*C*a^3-2640*C*a^2*b-1848*C 
*a*b^2-930*C*b^3)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+1155*A*a^2*b*(si 
n(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))+275*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))-1155*A*(sin(1/2 
*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d* 
x+1/2*c),2^(1/2))*a^3-2079*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d*x+1 
/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))*a*b^2+825*a^2*b*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(cos 
(1/2*d*x+1/2*c),2^(1/2))+225*C*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))-693*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)*EllipticE(cos(1/2*d 
*x+1/2*c),2^(1/2))*a^3-1617*C*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(2*sin(1/2*d...
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.03 \[ \int \frac {(a+b \cos (c+d x))^3 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx=-\frac {5 \, \sqrt {2} {\left (33 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b + 5 i \, {\left (11 \, A + 9 \, C\right )} b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, \sqrt {2} {\left (-33 i \, {\left (7 \, A + 5 \, C\right )} a^{2} b - 5 i \, {\left (11 \, A + 9 \, C\right )} b^{3}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 231 \, \sqrt {2} {\left (-i \, {\left (5 \, A + 3 \, C\right )} a^{3} - i \, {\left (9 \, A + 7 \, C\right )} a b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 \, \sqrt {2} {\left (i \, {\left (5 \, A + 3 \, C\right )} a^{3} + i \, {\left (9 \, A + 7 \, C\right )} a b^{2}\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - \frac {2 \, {\left (105 \, C b^{3} \cos \left (d x + c\right )^{5} + 385 \, C a b^{2} \cos \left (d x + c\right )^{4} + 15 \, {\left (33 \, C a^{2} b + {\left (11 \, A + 9 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )^{3} + 77 \, {\left (3 \, C a^{3} + {\left (9 \, A + 7 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left (33 \, {\left (7 \, A + 5 \, C\right )} a^{2} b + 5 \, {\left (11 \, A + 9 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{1155 \, d} \] Input:

integrate((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x, algori 
thm="fricas")
 

Output:

-1/1155*(5*sqrt(2)*(33*I*(7*A + 5*C)*a^2*b + 5*I*(11*A + 9*C)*b^3)*weierst 
rassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c)) + 5*sqrt(2)*(-33*I*(7*A 
 + 5*C)*a^2*b - 5*I*(11*A + 9*C)*b^3)*weierstrassPInverse(-4, 0, cos(d*x + 
 c) - I*sin(d*x + c)) + 231*sqrt(2)*(-I*(5*A + 3*C)*a^3 - I*(9*A + 7*C)*a* 
b^2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*si 
n(d*x + c))) + 231*sqrt(2)*(I*(5*A + 3*C)*a^3 + I*(9*A + 7*C)*a*b^2)*weier 
strassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c) 
)) - 2*(105*C*b^3*cos(d*x + c)^5 + 385*C*a*b^2*cos(d*x + c)^4 + 15*(33*C*a 
^2*b + (11*A + 9*C)*b^3)*cos(d*x + c)^3 + 77*(3*C*a^3 + (9*A + 7*C)*a*b^2) 
*cos(d*x + c)^2 + 5*(33*(7*A + 5*C)*a^2*b + 5*(11*A + 9*C)*b^3)*cos(d*x + 
c))*sin(d*x + c)/sqrt(cos(d*x + c)))/d
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x, algori 
thm="maxima")
 

Output:

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

Giac [F]

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

integrate((a+b*cos(d*x+c))^3*(A+C*cos(d*x+c)^2)/sec(d*x+c)^(1/2),x, algori 
thm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(sec(c + d*x))/sec(c + d*x),x)*a**4 + 3*int((sqrt(sec(c + d*x))*co 
s(c + d*x))/sec(c + d*x),x)*a**3*b + int((sqrt(sec(c + d*x))*cos(c + d*x)* 
*5)/sec(c + d*x),x)*b**3*c + 3*int((sqrt(sec(c + d*x))*cos(c + d*x)**4)/se 
c(c + d*x),x)*a*b**2*c + 3*int((sqrt(sec(c + d*x))*cos(c + d*x)**3)/sec(c 
+ d*x),x)*a**2*b*c + int((sqrt(sec(c + d*x))*cos(c + d*x)**3)/sec(c + d*x) 
,x)*a*b**3 + int((sqrt(sec(c + d*x))*cos(c + d*x)**2)/sec(c + d*x),x)*a**3 
*c + 3*int((sqrt(sec(c + d*x))*cos(c + d*x)**2)/sec(c + d*x),x)*a**2*b**2