\(\int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)+C \sec ^2(c+d x)) \, dx\) [1202]

Optimal result

Integrand size = 43, antiderivative size = 267 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {4 a^3 (15 A+17 B+21 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^3 (105 A+121 B+143 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {4 a^3 (105 A+121 B+143 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {4 a^3 (210 A+253 B+264 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{1155 d}+\frac {2 A \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3 \sin (c+d x)}{11 d}+\frac {2 (6 A+11 B) \cos ^{\frac {3}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{99 a d}+\frac {2 (105 A+143 B+99 C) \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right ) \sin (c+d x)}{693 d} \] Output:

4/15*a^3*(15*A+17*B+21*C)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+4/231*a^ 
3*(105*A+121*B+143*C)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+4/231*a^3*( 
105*A+121*B+143*C)*cos(d*x+c)^(1/2)*sin(d*x+c)/d+4/1155*a^3*(210*A+253*B+2 
64*C)*cos(d*x+c)^(3/2)*sin(d*x+c)/d+2/11*A*cos(d*x+c)^(3/2)*(a+a*cos(d*x+c 
))^3*sin(d*x+c)/d+2/99*(6*A+11*B)*cos(d*x+c)^(3/2)*(a^2+a^2*cos(d*x+c))^2* 
sin(d*x+c)/a/d+2/693*(105*A+143*B+99*C)*cos(d*x+c)^(3/2)*(a^3+a^3*cos(d*x+ 
c))*sin(d*x+c)/d
                                                                                    
                                                                                    
 

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 6.89 (sec) , antiderivative size = 1364, normalized size of antiderivative = 5.11 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:

Integrate[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + 
 C*Sec[c + d*x]^2),x]
 

Output:

a^3*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])^3*Sec[c/2 + (d*x)/2]^6*(-1/30*( 
(15*A + 17*B + 21*C)*Cot[c])/d + ((1953*A + 2134*B + 2354*C)*Cos[d*x]*Sin[ 
c])/(7392*d) + ((75*A + 73*B + 54*C)*Cos[2*d*x]*Sin[2*c])/(720*d) + ((189* 
A + 132*B + 44*C)*Cos[3*d*x]*Sin[3*c])/(4928*d) + ((3*A + B)*Cos[4*d*x]*Si 
n[4*c])/(288*d) + (A*Cos[5*d*x]*Sin[5*c])/(704*d) + ((1953*A + 2134*B + 23 
54*C)*Cos[c]*Sin[d*x])/(7392*d) + ((75*A + 73*B + 54*C)*Cos[2*c]*Sin[2*d*x 
])/(720*d) + ((189*A + 132*B + 44*C)*Cos[3*c]*Sin[3*d*x])/(4928*d) + ((3*A 
 + B)*Cos[4*c]*Sin[4*d*x])/(288*d) + (A*Cos[5*c]*Sin[5*d*x])/(704*d)) - (5 
*A*(1 + Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d* 
x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[Cot[c]]]*Sqrt 
[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin[c]*Sin[d*x - 
 ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(22*d*Sqrt[1 + Cot 
[c]^2]) - (11*B*(1 + Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, 
{5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[d*x - ArcTan[ 
Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + Cot[c]^2]*Sin 
[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(42*d 
*Sqrt[1 + Cot[c]^2]) - (13*C*(1 + Cos[c + d*x])^3*Csc[c]*HypergeometricPFQ 
[{1/4, 1/2}, {5/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^6*Sec[ 
d*x - ArcTan[Cot[c]]]*Sqrt[1 - Sin[d*x - ArcTan[Cot[c]]]]*Sqrt[-(Sqrt[1 + 
Cot[c]^2]*Sin[c]*Sin[d*x - ArcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[...
 

Rubi [A] (verified)

Time = 1.91 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.03, number of steps used = 23, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.535, Rules used = {3042, 4600, 3042, 3524, 27, 3042, 3455, 27, 3042, 3455, 27, 3042, 3447, 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.

\(\displaystyle \int \cos ^{\frac {11}{2}}(c+d x) (a \sec (c+d x)+a)^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^{11/2} (a \sec (c+d x)+a)^3 \left (A+B \sec (c+d x)+C \sec (c+d x)^2\right )dx\)

\(\Big \downarrow \) 4600

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3524

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

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^3 (a (3 A+11 C)+a (6 A+11 B) \cos (c+d x))dx}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3455

\(\displaystyle \frac {\frac {2}{9} \int \frac {1}{2} \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^2 \left (3 (15 A+11 B+33 C) a^2+(105 A+143 B+99 C) \cos (c+d x) a^2\right )dx+\frac {2 (6 A+11 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{9} \int \sqrt {\cos (c+d x)} (\cos (c+d x) a+a)^2 \left (3 (15 A+11 B+33 C) a^2+(105 A+143 B+99 C) \cos (c+d x) a^2\right )dx+\frac {2 (6 A+11 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{9} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (3 (15 A+11 B+33 C) a^2+(105 A+143 B+99 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {2 (6 A+11 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\)

\(\Big \downarrow \) 3455

\(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \int 3 \sqrt {\cos (c+d x)} (\cos (c+d x) a+a) \left (5 (21 A+22 B+33 C) a^3+(210 A+253 B+264 C) \cos (c+d x) a^3\right )dx+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \int \sqrt {\cos (c+d x)} (\cos (c+d x) a+a) \left (5 (21 A+22 B+33 C) a^3+(210 A+253 B+264 C) \cos (c+d x) a^3\right )dx+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (5 (21 A+22 B+33 C) a^3+(210 A+253 B+264 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\)

\(\Big \downarrow \) 3447

\(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \int \sqrt {\cos (c+d x)} \left ((210 A+253 B+264 C) \cos ^2(c+d x) a^4+5 (21 A+22 B+33 C) a^4+\left (5 (21 A+22 B+33 C) a^4+(210 A+253 B+264 C) a^4\right ) \cos (c+d x)\right )dx+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left ((210 A+253 B+264 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^4+5 (21 A+22 B+33 C) a^4+\left (5 (21 A+22 B+33 C) a^4+(210 A+253 B+264 C) a^4\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {2}{5} \int \frac {1}{2} \sqrt {\cos (c+d x)} \left (77 (15 A+17 B+21 C) a^4+15 (105 A+121 B+143 C) \cos (c+d x) a^4\right )dx+\frac {2 a^4 (210 A+253 B+264 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \int \sqrt {\cos (c+d x)} \left (77 (15 A+17 B+21 C) a^4+15 (105 A+121 B+143 C) \cos (c+d x) a^4\right )dx+\frac {2 a^4 (210 A+253 B+264 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )} \left (77 (15 A+17 B+21 C) a^4+15 (105 A+121 B+143 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^4\right )dx+\frac {2 a^4 (210 A+253 B+264 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\)

\(\Big \downarrow \) 3227

\(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (15 a^4 (105 A+121 B+143 C) \int \cos ^{\frac {3}{2}}(c+d x)dx+77 a^4 (15 A+17 B+21 C) \int \sqrt {\cos (c+d x)}dx\right )+\frac {2 a^4 (210 A+253 B+264 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (77 a^4 (15 A+17 B+21 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+15 a^4 (105 A+121 B+143 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx\right )+\frac {2 a^4 (210 A+253 B+264 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (77 a^4 (15 A+17 B+21 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+15 a^4 (105 A+121 B+143 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {2 a^4 (210 A+253 B+264 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (77 a^4 (15 A+17 B+21 C) \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+15 a^4 (105 A+121 B+143 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {2 a^4 (210 A+253 B+264 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\frac {1}{9} \left (\frac {6}{7} \left (\frac {1}{5} \left (15 a^4 (105 A+121 B+143 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+\frac {154 a^4 (15 A+17 B+21 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\right )+\frac {2 a^4 (210 A+253 B+264 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+\frac {2 (105 A+143 B+99 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}\right )+\frac {2 (6 A+11 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {\frac {1}{9} \left (\frac {2 (105 A+143 B+99 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^4 \cos (c+d x)+a^4\right )}{7 d}+\frac {6}{7} \left (\frac {2 a^4 (210 A+253 B+264 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {1}{5} \left (\frac {154 a^4 (15 A+17 B+21 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+15 a^4 (105 A+121 B+143 C) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )\right )\right )+\frac {2 (6 A+11 B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \left (a^2 \cos (c+d x)+a^2\right )^2}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3}{11 d}\)

Input:

Int[Cos[c + d*x]^(11/2)*(a + a*Sec[c + d*x])^3*(A + B*Sec[c + d*x] + C*Sec 
[c + d*x]^2),x]
 

Output:

(2*A*Cos[c + d*x]^(3/2)*(a + a*Cos[c + d*x])^3*Sin[c + d*x])/(11*d) + ((2* 
(6*A + 11*B)*Cos[c + d*x]^(3/2)*(a^2 + a^2*Cos[c + d*x])^2*Sin[c + d*x])/( 
9*d) + ((2*(105*A + 143*B + 99*C)*Cos[c + d*x]^(3/2)*(a^4 + a^4*Cos[c + d* 
x])*Sin[c + d*x])/(7*d) + (6*((2*a^4*(210*A + 253*B + 264*C)*Cos[c + d*x]^ 
(3/2)*Sin[c + d*x])/(5*d) + ((154*a^4*(15*A + 17*B + 21*C)*EllipticE[(c + 
d*x)/2, 2])/d + 15*a^4*(105*A + 121*B + 143*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*a)
 

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 3447

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a 
 + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x]^2), 
 x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]
 

rule 3455

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
(f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim 
p[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x])^(n 
 + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1))   Int[(a + b*Sin[e + 
f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1 
) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))*Sin[e + 
f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 
 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1 
] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])
 

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 3524

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/(b*d*(m + 
 n + 2))   Int[(a + b*Sin[e + f*x])^m*(c + d*Sin[e + f*x])^n*Simp[A*b*d*(m 
+ n + 2) + C*(a*c*m + b*d*(n + 1)) + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + 
n + 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, m, n} 
, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !Lt 
Q[m, -2^(-1)] && NeQ[m + n + 2, 0]
 

rule 4600

Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x 
_)])^(m_.)*((A_.) + (B_.)*sec[(e_.) + (f_.)*(x_)] + (C_.)*sec[(e_.) + (f_.) 
*(x_)]^2), x_Symbol] :> Simp[d^(m + 2)   Int[(b + a*Cos[e + f*x])^m*(d*Cos[ 
e + f*x])^(n - m - 2)*(C + B*Cos[e + f*x] + A*Cos[e + f*x]^2), x], x] /; Fr 
eeQ[{a, b, d, e, f, A, B, C, n}, x] &&  !IntegerQ[n] && IntegerQ[m]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(544\) vs. \(2(246)=492\).

Time = 162.31 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.04

method	result	size
default	\(-\frac {4 \sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, a^{3} \left (10080 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (-43680 A -6160 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (77280 A +24200 B +3960 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-72240 A -37532 B -14256 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (39270 A +29722 B +19866 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-8820 A -8118 B -6864 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+1575 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3465 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+1815 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3927 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+2145 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-4851 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{3465 \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\)	\(545\)

Input:

int(cos(d*x+c)^(11/2)*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x 
,method=_RETURNVERBOSE)
 

Output:

-4/3465*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*a^3*(10080 
*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12+(-43680*A-6160*B)*sin(1/2*d*x+ 
1/2*c)^10*cos(1/2*d*x+1/2*c)+(77280*A+24200*B+3960*C)*sin(1/2*d*x+1/2*c)^8 
*cos(1/2*d*x+1/2*c)+(-72240*A-37532*B-14256*C)*sin(1/2*d*x+1/2*c)^6*cos(1/ 
2*d*x+1/2*c)+(39270*A+29722*B+19866*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/ 
2*c)+(-8820*A-8118*B-6864*C)*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)+1575* 
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)*EllipticF( 
cos(1/2*d*x+1/2*c),2^(1/2))-3465*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))+1815*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))-3927*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)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))+2145*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))-4851*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)))/(-2*sin(1/2*d*x+1/2*c)^4+sin 
(1/2*d*x+1/2*c)^2)^(1/2)/sin(1/2*d*x+1/2*c)/(2*cos(1/2*d*x+1/2*c)^2-1)^(1/ 
2)/d
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 264, normalized size of antiderivative = 0.99 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (15 i \, \sqrt {2} {\left (105 \, A + 121 \, B + 143 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 15 i \, \sqrt {2} {\left (105 \, A + 121 \, B + 143 \, C\right )} a^{3} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 i \, \sqrt {2} {\left (15 \, A + 17 \, B + 21 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 i \, \sqrt {2} {\left (15 \, A + 17 \, B + 21 \, C\right )} a^{3} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (315 \, A a^{3} \cos \left (d x + c\right )^{4} + 385 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{3} + 45 \, {\left (42 \, A + 33 \, B + 11 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 77 \, {\left (30 \, A + 34 \, B + 27 \, C\right )} a^{3} \cos \left (d x + c\right ) + 30 \, {\left (105 \, A + 121 \, B + 143 \, C\right )} a^{3}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{3465 \, d} \] Input:

integrate(cos(d*x+c)^(11/2)*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c 
)^2),x, algorithm="fricas")
 

Output:

-2/3465*(15*I*sqrt(2)*(105*A + 121*B + 143*C)*a^3*weierstrassPInverse(-4, 
0, cos(d*x + c) + I*sin(d*x + c)) - 15*I*sqrt(2)*(105*A + 121*B + 143*C)*a 
^3*weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 231*I*sqrt( 
2)*(15*A + 17*B + 21*C)*a^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 
 0, cos(d*x + c) + I*sin(d*x + c))) + 231*I*sqrt(2)*(15*A + 17*B + 21*C)*a 
^3*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin( 
d*x + c))) - (315*A*a^3*cos(d*x + c)^4 + 385*(3*A + B)*a^3*cos(d*x + c)^3 
+ 45*(42*A + 33*B + 11*C)*a^3*cos(d*x + c)^2 + 77*(30*A + 34*B + 27*C)*a^3 
*cos(d*x + c) + 30*(105*A + 121*B + 143*C)*a^3)*sqrt(cos(d*x + c))*sin(d*x 
 + c))/d
 

Sympy [F(-1)]

Timed out. \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**(11/2)*(a+a*sec(d*x+c))**3*(A+B*sec(d*x+c)+C*sec(d*x 
+c)**2),x)
 

Output:

Timed out
 

Maxima [F(-1)]

Timed out. \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)^(11/2)*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c 
)^2),x, algorithm="maxima")
 

Output:

Timed out
 

Giac [F]

\[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {11}{2}} \,d x } \] Input:

integrate(cos(d*x+c)^(11/2)*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c 
)^2),x, algorithm="giac")
 

Output:

integrate((C*sec(d*x + c)^2 + B*sec(d*x + c) + A)*(a*sec(d*x + c) + a)^3*c 
os(d*x + c)^(11/2), x)
 

Mupad [B] (verification not implemented)

Time = 14.37 (sec) , antiderivative size = 507, normalized size of antiderivative = 1.90 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:

int(cos(c + d*x)^(11/2)*(a + a/cos(c + d*x))^3*(A + B/cos(c + d*x) + C/cos 
(c + d*x)^2),x)
 

Output:

(2*(C*a^3*ellipticE(c/2 + (d*x)/2, 2) + C*a^3*ellipticF(c/2 + (d*x)/2, 2) 
+ C*a^3*cos(c + d*x)^(1/2)*sin(c + d*x)))/d + (B*a^3*((2*cos(c + d*x)^(1/2 
)*sin(c + d*x))/3 + (2*ellipticF(c/2 + (d*x)/2, 2))/3))/d - (2*A*a^3*cos(c 
 + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c + d*x)^2))/(7 
*d*(sin(c + d*x)^2)^(1/2)) - (2*A*a^3*cos(c + d*x)^(9/2)*sin(c + d*x)*hype 
rgeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(3*d*(sin(c + d*x)^2)^(1/2)) - (6 
*A*a^3*cos(c + d*x)^(11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15/4, cos(c 
 + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2)) - (2*A*a^3*cos(c + d*x)^(13/2)*s 
in(c + d*x)*hypergeom([1/2, 13/4], 17/4, cos(c + d*x)^2))/(13*d*(sin(c + d 
*x)^2)^(1/2)) - (6*B*a^3*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7 
/4], 11/4, cos(c + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2)) - (2*B*a^3*cos(c 
+ d*x)^(9/2)*sin(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(3* 
d*(sin(c + d*x)^2)^(1/2)) - (2*B*a^3*cos(c + d*x)^(11/2)*sin(c + d*x)*hype 
rgeom([1/2, 11/4], 15/4, cos(c + d*x)^2))/(11*d*(sin(c + d*x)^2)^(1/2)) - 
(6*C*a^3*cos(c + d*x)^(7/2)*sin(c + d*x)*hypergeom([1/2, 7/4], 11/4, cos(c 
 + d*x)^2))/(7*d*(sin(c + d*x)^2)^(1/2)) - (2*C*a^3*cos(c + d*x)^(9/2)*sin 
(c + d*x)*hypergeom([1/2, 9/4], 13/4, cos(c + d*x)^2))/(9*d*(sin(c + d*x)^ 
2)^(1/2))
 

Reduce [F]

\[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=a^{3} \left (\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{5}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{4}d x \right ) b +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{4}d x \right ) c +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{3}d x \right ) a +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{3}d x \right ) b +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{3}d x \right ) c +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{2}d x \right ) a +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{2}d x \right ) b +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )^{2}d x \right ) c +3 \left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )d x \right ) a +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5} \sec \left (d x +c \right )d x \right ) b +\left (\int \sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}d x \right ) a \right ) \] Input:

int(cos(d*x+c)^(11/2)*(a+a*sec(d*x+c))^3*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x 
)
 

Output:

a**3*(int(sqrt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d*x)**5,x)*c + int(sq 
rt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d*x)**4,x)*b + 3*int(sqrt(cos(c + 
 d*x))*cos(c + d*x)**5*sec(c + d*x)**4,x)*c + int(sqrt(cos(c + d*x))*cos(c 
 + d*x)**5*sec(c + d*x)**3,x)*a + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)**5 
*sec(c + d*x)**3,x)*b + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d 
*x)**3,x)*c + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d*x)**2,x)* 
a + 3*int(sqrt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d*x)**2,x)*b + int(sq 
rt(cos(c + d*x))*cos(c + d*x)**5*sec(c + d*x)**2,x)*c + 3*int(sqrt(cos(c + 
 d*x))*cos(c + d*x)**5*sec(c + d*x),x)*a + int(sqrt(cos(c + d*x))*cos(c + 
d*x)**5*sec(c + d*x),x)*b + int(sqrt(cos(c + d*x))*cos(c + d*x)**5,x)*a)