\(\int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^2 (A+C \sec ^2(c+d x)) \, dx\) [1090]

Optimal result

Integrand size = 35, antiderivative size = 230 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {4 a^2 (7 A+9 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {8 a^2 (25 A+33 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{231 d}+\frac {8 a^2 (25 A+33 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{231 d}+\frac {4 a^2 (7 A+9 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{45 d}+\frac {2 a^2 (89 A+99 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{693 d}+\frac {2 A \cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^2 \sin (c+d x)}{11 d}+\frac {8 A \cos ^{\frac {5}{2}}(c+d x) \left (a^2+a^2 \cos (c+d x)\right ) \sin (c+d x)}{99 d} \] Output:

4/15*a^2*(7*A+9*C)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/d+8/231*a^2*(25*A 
+33*C)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/d+8/231*a^2*(25*A+33*C)*cos( 
d*x+c)^(1/2)*sin(d*x+c)/d+4/45*a^2*(7*A+9*C)*cos(d*x+c)^(3/2)*sin(d*x+c)/d 
+2/693*a^2*(89*A+99*C)*cos(d*x+c)^(5/2)*sin(d*x+c)/d+2/11*A*cos(d*x+c)^(5/ 
2)*(a+a*cos(d*x+c))^2*sin(d*x+c)/d+8/99*A*cos(d*x+c)^(5/2)*(a^2+a^2*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 = 6.60 (sec) , antiderivative size = 976, normalized size of antiderivative = 4.24 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx =\text {Too large to display} \] Input:

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

Output:

a^2*(Sqrt[Cos[c + d*x]]*(1 + Cos[c + d*x])^2*Sec[c/2 + (d*x)/2]^4*(-1/15*( 
(7*A + 9*C)*Cot[c])/d + ((941*A + 1122*C)*Cos[d*x]*Sin[c])/(3696*d) + ((19 
*A + 18*C)*Cos[2*d*x]*Sin[2*c])/(180*d) + ((101*A + 44*C)*Cos[3*d*x]*Sin[3 
*c])/(2464*d) + (A*Cos[4*d*x]*Sin[4*c])/(72*d) + (A*Cos[5*d*x]*Sin[5*c])/( 
352*d) + ((941*A + 1122*C)*Cos[c]*Sin[d*x])/(3696*d) + ((19*A + 18*C)*Cos[ 
2*c]*Sin[2*d*x])/(180*d) + ((101*A + 44*C)*Cos[3*c]*Sin[3*d*x])/(2464*d) + 
 (A*Cos[4*c]*Sin[4*d*x])/(72*d) + (A*Cos[5*c]*Sin[5*d*x])/(352*d)) - (50*A 
*(1 + Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5/4}, Sin[d*x 
- ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*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 - A 
rcTan[Cot[c]]])]*Sqrt[1 + Sin[d*x - ArcTan[Cot[c]]]])/(231*d*Sqrt[1 + Cot[ 
c]^2]) - (2*C*(1 + Cos[c + d*x])^2*Csc[c]*HypergeometricPFQ[{1/4, 1/2}, {5 
/4}, Sin[d*x - ArcTan[Cot[c]]]^2]*Sec[c/2 + (d*x)/2]^4*Sec[d*x - ArcTan[Co 
t[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]]]])/(7*d*Sq 
rt[1 + Cot[c]^2]) - (7*A*(1 + Cos[c + d*x])^2*Csc[c]*Sec[c/2 + (d*x)/2]^4* 
((HypergeometricPFQ[{-1/2, -1/4}, {3/4}, Cos[d*x + ArcTan[Tan[c]]]^2]*Sin[ 
d*x + ArcTan[Tan[c]]]*Tan[c])/(Sqrt[1 - Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[1 
+ Cos[d*x + ArcTan[Tan[c]]]]*Sqrt[Cos[c]*Cos[d*x + ArcTan[Tan[c]]]*Sqrt[1 
+ Tan[c]^2]]*Sqrt[1 + Tan[c]^2]) - ((Sin[d*x + ArcTan[Tan[c]]]*Tan[c])/...
 

Rubi [A] (verified)

Time = 1.37 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.02, number of steps used = 19, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.543, Rules used = {3042, 4602, 3042, 3525, 27, 3042, 3455, 27, 3042, 3447, 3042, 3502, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4602

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3525

\(\displaystyle \frac {2 \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a)^2 (a (5 A+11 C)+4 a A \cos (c+d x))dx}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a)^2 (a (5 A+11 C)+4 a A \cos (c+d x))dx}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (a (5 A+11 C)+4 a A \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}\)

\(\Big \downarrow \) 3455

\(\displaystyle \frac {\frac {2}{9} \int \frac {1}{2} \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a) \left ((65 A+99 C) a^2+(89 A+99 C) \cos (c+d x) a^2\right )dx+\frac {8 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{9} \int \cos ^{\frac {3}{2}}(c+d x) (\cos (c+d x) a+a) \left ((65 A+99 C) a^2+(89 A+99 C) \cos (c+d x) a^2\right )dx+\frac {8 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{9} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((65 A+99 C) a^2+(89 A+99 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^2\right )dx+\frac {8 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}\)

\(\Big \downarrow \) 3447

\(\displaystyle \frac {\frac {1}{9} \int \cos ^{\frac {3}{2}}(c+d x) \left ((89 A+99 C) \cos ^2(c+d x) a^3+(65 A+99 C) a^3+\left ((65 A+99 C) a^3+(89 A+99 C) a^3\right ) \cos (c+d x)\right )dx+\frac {8 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{9} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left ((89 A+99 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^3+(65 A+99 C) a^3+\left ((65 A+99 C) a^3+(89 A+99 C) a^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx+\frac {8 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \int \cos ^{\frac {3}{2}}(c+d x) \left (18 (25 A+33 C) a^3+77 (7 A+9 C) \cos (c+d x) a^3\right )dx+\frac {2 a^3 (89 A+99 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {8 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (18 (25 A+33 C) a^3+77 (7 A+9 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx+\frac {2 a^3 (89 A+99 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {8 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}\)

\(\Big \downarrow \) 3227

\(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (18 a^3 (25 A+33 C) \int \cos ^{\frac {3}{2}}(c+d x)dx+77 a^3 (7 A+9 C) \int \cos ^{\frac {5}{2}}(c+d x)dx\right )+\frac {2 a^3 (89 A+99 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {8 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (18 a^3 (25 A+33 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx+77 a^3 (7 A+9 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}dx\right )+\frac {2 a^3 (89 A+99 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {8 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (77 a^3 (7 A+9 C) \left (\frac {3}{5} \int \sqrt {\cos (c+d x)}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+18 a^3 (25 A+33 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {2 a^3 (89 A+99 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {8 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (77 a^3 (7 A+9 C) \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+18 a^3 (25 A+33 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )+\frac {2 a^3 (89 A+99 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {8 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\frac {1}{9} \left (\frac {2}{7} \left (18 a^3 (25 A+33 C) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )+77 a^3 (7 A+9 C) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )+\frac {2 a^3 (89 A+99 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}\right )+\frac {8 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {\frac {1}{9} \left (\frac {2 a^3 (89 A+99 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 d}+\frac {2}{7} \left (77 a^3 (7 A+9 C) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )+18 a^3 (25 A+33 C) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )\right )\right )+\frac {8 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}{9 d}}{11 a}+\frac {2 A \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) (a \cos (c+d x)+a)^2}{11 d}\)

Input:

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

Output:

(2*A*Cos[c + d*x]^(5/2)*(a + a*Cos[c + d*x])^2*Sin[c + d*x])/(11*d) + ((8* 
A*Cos[c + d*x]^(5/2)*(a^3 + a^3*Cos[c + d*x])*Sin[c + d*x])/(9*d) + ((2*a^ 
3*(89*A + 99*C)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(7*d) + (2*(18*a^3*(25*A 
+ 33*C)*((2*EllipticF[(c + d*x)/2, 2])/(3*d) + (2*Sqrt[Cos[c + d*x]]*Sin[c 
 + d*x])/(3*d)) + 77*a^3*(7*A + 9*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*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 3525

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

rule 4602

Int[(cos[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x 
_)])^(m_.)*((A_.) + (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 + A*Cos 
[e + f*x]^2), x], x] /; FreeQ[{a, b, d, e, f, A, C, n}, x] &&  !IntegerQ[n] 
 && IntegerQ[m]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(435\) vs. \(2(209)=418\).

Time = 58.58 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.90

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^{2} \left (10080 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}-37520 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (57040 A +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 (-46192 A -11484 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (22022 A +12474 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-4563 A -3861 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+750 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 )-1617 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 )+990 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 )-2079 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}\)	\(436\)

Input:

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

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^2*(10080 
*A*cos(1/2*d*x+1/2*c)*sin(1/2*d*x+1/2*c)^12-37520*A*cos(1/2*d*x+1/2*c)*sin 
(1/2*d*x+1/2*c)^10+(57040*A+3960*C)*sin(1/2*d*x+1/2*c)^8*cos(1/2*d*x+1/2*c 
)+(-46192*A-11484*C)*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/2*c)+(22022*A+1247 
4*C)*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)+(-4563*A-3861*C)*sin(1/2*d*x+ 
1/2*c)^2*cos(1/2*d*x+1/2*c)+750*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))-1617*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))+990*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))-2079*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 = 0.10 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.04 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (30 i \, \sqrt {2} {\left (25 \, A + 33 \, C\right )} a^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 30 i \, \sqrt {2} {\left (25 \, A + 33 \, C\right )} a^{2} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 231 i \, \sqrt {2} {\left (7 \, A + 9 \, C\right )} a^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 231 i \, \sqrt {2} {\left (7 \, A + 9 \, C\right )} a^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (315 \, A a^{2} \cos \left (d x + c\right )^{4} + 770 \, A a^{2} \cos \left (d x + c\right )^{3} + 45 \, {\left (20 \, A + 11 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 154 \, {\left (7 \, A + 9 \, C\right )} a^{2} \cos \left (d x + c\right ) + 60 \, {\left (25 \, A + 33 \, C\right )} a^{2}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{3465 \, d} \] Input:

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

Output:

-2/3465*(30*I*sqrt(2)*(25*A + 33*C)*a^2*weierstrassPInverse(-4, 0, cos(d*x 
 + c) + I*sin(d*x + c)) - 30*I*sqrt(2)*(25*A + 33*C)*a^2*weierstrassPInver 
se(-4, 0, cos(d*x + c) - I*sin(d*x + c)) - 231*I*sqrt(2)*(7*A + 9*C)*a^2*w 
eierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x 
+ c))) + 231*I*sqrt(2)*(7*A + 9*C)*a^2*weierstrassZeta(-4, 0, weierstrassP 
Inverse(-4, 0, cos(d*x + c) - I*sin(d*x + c))) - (315*A*a^2*cos(d*x + c)^4 
 + 770*A*a^2*cos(d*x + c)^3 + 45*(20*A + 11*C)*a^2*cos(d*x + c)^2 + 154*(7 
*A + 9*C)*a^2*cos(d*x + c) + 60*(25*A + 33*C)*a^2)*sqrt(cos(d*x + c))*sin( 
d*x + c))/d
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-1)]

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

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

Output:

Timed out
 

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 14.26 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.16 \[ \int \cos ^{\frac {11}{2}}(c+d x) (a+a \sec (c+d x))^2 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {2\,C\,a^2\,\left (\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )+\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )\right )}{3\,d}-\frac {2\,A\,a^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,A\,a^2\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{11\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,A\,a^2\,{\cos \left (c+d\,x\right )}^{13/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {13}{4};\ \frac {17}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{13\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {4\,C\,a^2\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a^2\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \] Input:

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

Output:

(2*C*a^2*(cos(c + d*x)^(1/2)*sin(c + d*x) + ellipticF(c/2 + (d*x)/2, 2)))/ 
(3*d) - (2*A*a^2*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)) - (4*A*a^2*cos(c + d*x)^( 
11/2)*sin(c + d*x)*hypergeom([1/2, 11/4], 15/4, cos(c + d*x)^2))/(11*d*(si 
n(c + d*x)^2)^(1/2)) - (2*A*a^2*cos(c + d*x)^(13/2)*sin(c + d*x)*hypergeom 
([1/2, 13/4], 17/4, cos(c + d*x)^2))/(13*d*(sin(c + d*x)^2)^(1/2)) - (4*C* 
a^2*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^2*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]

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

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

Output:

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