\(\int \sqrt {a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac {11}{2}}(c+d x) \, dx\) [493]

Optimal result

Integrand size = 35, antiderivative size = 220 \[ \int \sqrt {a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {32 a (8 A+9 B) \sqrt {\sec (c+d x)} \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {16 a (8 A+9 B) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{315 d \sqrt {a+a \cos (c+d x)}}+\frac {4 a (8 A+9 B) \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{105 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a (8 A+9 B) \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{63 d \sqrt {a+a \cos (c+d x)}}+\frac {2 a A \sec ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{9 d \sqrt {a+a \cos (c+d x)}} \] Output:

32/315*a*(8*A+9*B)*sec(d*x+c)^(1/2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+16 
/315*a*(8*A+9*B)*sec(d*x+c)^(3/2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+4/10 
5*a*(8*A+9*B)*sec(d*x+c)^(5/2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+2/63*a* 
(8*A+9*B)*sec(d*x+c)^(7/2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)+2/9*a*A*sec 
(d*x+c)^(9/2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^(1/2)
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.56 \[ \int \sqrt {a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {2 \sqrt {a (1+\cos (c+d x))} (107 A+81 B+11 (8 A+9 B) \cos (c+d x)+11 (8 A+9 B) \cos (2 (c+d x))+16 A \cos (3 (c+d x))+18 B \cos (3 (c+d x))+16 A \cos (4 (c+d x))+18 B \cos (4 (c+d x))) \sec ^{\frac {9}{2}}(c+d x) \tan \left (\frac {1}{2} (c+d x)\right )}{315 d} \] Input:

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

Output:

(2*Sqrt[a*(1 + Cos[c + d*x])]*(107*A + 81*B + 11*(8*A + 9*B)*Cos[c + d*x] 
+ 11*(8*A + 9*B)*Cos[2*(c + d*x)] + 16*A*Cos[3*(c + d*x)] + 18*B*Cos[3*(c 
+ d*x)] + 16*A*Cos[4*(c + d*x)] + 18*B*Cos[4*(c + d*x)])*Sec[c + d*x]^(9/2 
)*Tan[(c + d*x)/2])/(315*d)
 

Rubi [A] (verified)

Time = 1.16 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.343, Rules used = {3042, 3440, 3042, 3459, 3042, 3251, 3042, 3251, 3042, 3251, 3042, 3250}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3440

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3459

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} (8 A+9 B) \int \frac {\sqrt {\cos (c+d x) a+a}}{\cos ^{\frac {9}{2}}(c+d x)}dx+\frac {2 a A \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} (8 A+9 B) \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2}}dx+\frac {2 a A \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )\)

\(\Big \downarrow \) 3251

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} (8 A+9 B) \left (\frac {6}{7} \int \frac {\sqrt {\cos (c+d x) a+a}}{\cos ^{\frac {7}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a A \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} (8 A+9 B) \left (\frac {6}{7} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2}}dx+\frac {2 a \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a A \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )\)

\(\Big \downarrow \) 3251

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} (8 A+9 B) \left (\frac {6}{7} \left (\frac {4}{5} \int \frac {\sqrt {\cos (c+d x) a+a}}{\cos ^{\frac {5}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a A \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} (8 A+9 B) \left (\frac {6}{7} \left (\frac {4}{5} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}}dx+\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a A \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )\)

\(\Big \downarrow \) 3251

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} (8 A+9 B) \left (\frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\sqrt {\cos (c+d x) a+a}}{\cos ^{\frac {3}{2}}(c+d x)}dx+\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a A \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} (8 A+9 B) \left (\frac {6}{7} \left (\frac {4}{5} \left (\frac {2}{3} \int \frac {\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}}{\sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}}dx+\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )+\frac {2 a A \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )\)

\(\Big \downarrow \) 3250

\(\displaystyle \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \left (\frac {1}{9} (8 A+9 B) \left (\frac {2 a \sin (c+d x)}{7 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {6}{7} \left (\frac {2 a \sin (c+d x)}{5 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {4}{5} \left (\frac {2 a \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}+\frac {4 a \sin (c+d x)}{3 d \sqrt {\cos (c+d x)} \sqrt {a \cos (c+d x)+a}}\right )\right )\right )+\frac {2 a A \sin (c+d x)}{9 d \cos ^{\frac {9}{2}}(c+d x) \sqrt {a \cos (c+d x)+a}}\right )\)

Input:

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

Output:

Sqrt[Cos[c + d*x]]*Sqrt[Sec[c + d*x]]*((2*a*A*Sin[c + d*x])/(9*d*Cos[c + d 
*x]^(9/2)*Sqrt[a + a*Cos[c + d*x]]) + ((8*A + 9*B)*((2*a*Sin[c + d*x])/(7* 
d*Cos[c + d*x]^(7/2)*Sqrt[a + a*Cos[c + d*x]]) + (6*((2*a*Sin[c + d*x])/(5 
*d*Cos[c + d*x]^(5/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*((2*a*Sin[c + d*x])/( 
3*d*Cos[c + d*x]^(3/2)*Sqrt[a + a*Cos[c + d*x]]) + (4*a*Sin[c + d*x])/(3*d 
*Sqrt[Cos[c + d*x]]*Sqrt[a + a*Cos[c + d*x]])))/5))/7))/9)
 

Defintions of rubi rules used

rule 3042

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

rule 3250

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + ( 
f_.)*(x_)])^(3/2), x_Symbol] :> Simp[-2*b^2*(Cos[e + f*x]/(f*(b*c + a*d)*Sq 
rt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]])), x] /; FreeQ[{a, b, c, d, 
 e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
 

rule 3251

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((c_.) + (d_.)*sin[(e_.) + ( 
f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*c - a*d)*Cos[e + f*x]*((c + d*Sin[e 
+ f*x])^(n + 1)/(f*(n + 1)*(c^2 - d^2)*Sqrt[a + b*Sin[e + f*x]])), x] + Sim 
p[(2*n + 3)*((b*c - a*d)/(2*b*(n + 1)*(c^2 - d^2)))   Int[Sqrt[a + b*Sin[e 
+ f*x]]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x 
] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, 
-1] && NeQ[2*n + 3, 0] && IntegerQ[2*n]
 

rule 3440

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*((a_.) + (b_.)*sin[(e_.) + (f_.)* 
(x_)])^(m_.)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Sim 
p[(g*Csc[e + f*x])^p*(g*Sin[e + f*x])^p   Int[(a + b*Sin[e + f*x])^m*((c + 
d*Sin[e + f*x])^n/(g*Sin[e + f*x])^p), x], x] /; FreeQ[{a, b, c, d, e, f, g 
, m, n, p}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[p] &&  !(IntegerQ[m] && I 
ntegerQ[n])
 

rule 3459

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + ( 
f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp 
[(-b^2)*(B*c - A*d)*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(n + 1) 
*(b*c + a*d)*Sqrt[a + b*Sin[e + f*x]])), x] + Simp[(A*b*d*(2*n + 3) - B*(b* 
c - 2*a*d*(n + 1)))/(2*d*(n + 1)*(b*c + a*d))   Int[Sqrt[a + b*Sin[e + f*x] 
]*(c + d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x 
] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[n, 
-1]
 

Maple [A] (verified)

Time = 21.13 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.66

method	result	size
default	\(\sqrt {2}\, \left (\frac {2 A \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (2048 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-3584 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+2496 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-752 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+107\right ) \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )}}}{315 d \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}-\frac {2 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (2048 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-3584 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+2496 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-752 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+107\right ) \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )}}}{315 d \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}+\frac {4 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (2176 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-3808 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+2652 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-799 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+94\right ) \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )}}}{315 d \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}\right )\)	\(366\)
parts	\(\frac {2 A \sqrt {2}\, \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (2048 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-3584 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+2496 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-752 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+107\right ) \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )}}}{315 d \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}+B \sqrt {2}\, \left (-\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (2048 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-3584 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+2496 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-752 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+107\right ) \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )}}}{315 d \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}+\frac {4 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (2176 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-3808 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+2652 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-799 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+94\right ) \sqrt {a \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )}}}{315 d \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{4}}\right )\)	\(369\)

Input:

int((a+a*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c))*sec(d*x+c)^(11/2),x,method=_RE 
TURNVERBOSE)
 

Output:

2^(1/2)*(2/315*A/d*tan(1/2*d*x+1/2*c)*(2048*cos(1/2*d*x+1/2*c)^8-3584*cos( 
1/2*d*x+1/2*c)^6+2496*cos(1/2*d*x+1/2*c)^4-752*cos(1/2*d*x+1/2*c)^2+107)*( 
a*cos(1/2*d*x+1/2*c)^2)^(1/2)*(1/(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)/(2*cos( 
1/2*d*x+1/2*c)^2-1)^4-2/315*B/d*tan(1/2*d*x+1/2*c)*(2048*cos(1/2*d*x+1/2*c 
)^8-3584*cos(1/2*d*x+1/2*c)^6+2496*cos(1/2*d*x+1/2*c)^4-752*cos(1/2*d*x+1/ 
2*c)^2+107)*(a*cos(1/2*d*x+1/2*c)^2)^(1/2)*(1/(2*cos(1/2*d*x+1/2*c)^2-1))^ 
(1/2)/(2*cos(1/2*d*x+1/2*c)^2-1)^4+4/315*B/d*tan(1/2*d*x+1/2*c)*(2176*cos( 
1/2*d*x+1/2*c)^8-3808*cos(1/2*d*x+1/2*c)^6+2652*cos(1/2*d*x+1/2*c)^4-799*c 
os(1/2*d*x+1/2*c)^2+94)*(a*cos(1/2*d*x+1/2*c)^2)^(1/2)*(1/(2*cos(1/2*d*x+1 
/2*c)^2-1))^(1/2)/(2*cos(1/2*d*x+1/2*c)^2-1)^4)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.55 \[ \int \sqrt {a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac {11}{2}}(c+d x) \, dx=\frac {2 \, {\left (16 \, {\left (8 \, A + 9 \, B\right )} \cos \left (d x + c\right )^{4} + 8 \, {\left (8 \, A + 9 \, B\right )} \cos \left (d x + c\right )^{3} + 6 \, {\left (8 \, A + 9 \, B\right )} \cos \left (d x + c\right )^{2} + 5 \, {\left (8 \, A + 9 \, B\right )} \cos \left (d x + c\right ) + 35 \, A\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{315 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\right )} \sqrt {\cos \left (d x + c\right )}} \] Input:

integrate((a+a*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c))*sec(d*x+c)^(11/2),x, alg 
orithm="fricas")
 

Output:

2/315*(16*(8*A + 9*B)*cos(d*x + c)^4 + 8*(8*A + 9*B)*cos(d*x + c)^3 + 6*(8 
*A + 9*B)*cos(d*x + c)^2 + 5*(8*A + 9*B)*cos(d*x + c) + 35*A)*sqrt(a*cos(d 
*x + c) + a)*sin(d*x + c)/((d*cos(d*x + c)^5 + d*cos(d*x + c)^4)*sqrt(cos( 
d*x + c)))
 

Sympy [F(-1)]

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

integrate((a+a*cos(d*x+c))**(1/2)*(A+B*cos(d*x+c))*sec(d*x+c)**(11/2),x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 659 vs. \(2 (190) = 380\).

Time = 0.24 (sec) , antiderivative size = 659, normalized size of antiderivative = 3.00 \[ \int \sqrt {a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac {11}{2}}(c+d x) \, dx =\text {Too large to display} \] Input:

integrate((a+a*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c))*sec(d*x+c)^(11/2),x, alg 
orithm="maxima")
 

Output:

2/315*(A*(315*sqrt(2)*sqrt(a)*sin(d*x + c)/(cos(d*x + c) + 1) - 735*sqrt(2 
)*sqrt(a)*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 + 1302*sqrt(2)*sqrt(a)*sin(d 
*x + c)^5/(cos(d*x + c) + 1)^5 - 1206*sqrt(2)*sqrt(a)*sin(d*x + c)^7/(cos( 
d*x + c) + 1)^7 + 431*sqrt(2)*sqrt(a)*sin(d*x + c)^9/(cos(d*x + c) + 1)^9 
- 107*sqrt(2)*sqrt(a)*sin(d*x + c)^11/(cos(d*x + c) + 1)^11)*(sin(d*x + c) 
^2/(cos(d*x + c) + 1)^2 + 1)^5/((sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(11/ 
2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(11/2)*(5*sin(d*x + c)^2/(cos(d* 
x + c) + 1)^2 + 10*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 + 10*sin(d*x + c)^6 
/(cos(d*x + c) + 1)^6 + 5*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + sin(d*x + 
c)^10/(cos(d*x + c) + 1)^10 + 1)) + 9*B*(35*sqrt(2)*sqrt(a)*sin(d*x + c)/( 
cos(d*x + c) + 1) - 105*sqrt(2)*sqrt(a)*sin(d*x + c)^3/(cos(d*x + c) + 1)^ 
3 + 154*sqrt(2)*sqrt(a)*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 - 142*sqrt(2)* 
sqrt(a)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 67*sqrt(2)*sqrt(a)*sin(d*x + 
 c)^9/(cos(d*x + c) + 1)^9 - 9*sqrt(2)*sqrt(a)*sin(d*x + c)^11/(cos(d*x + 
c) + 1)^11)*(sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 1)^5/((sin(d*x + c)/(co 
s(d*x + c) + 1) + 1)^(11/2)*(-sin(d*x + c)/(cos(d*x + c) + 1) + 1)^(11/2)* 
(5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 10*sin(d*x + c)^4/(cos(d*x + c) + 
 1)^4 + 10*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 5*sin(d*x + c)^8/(cos(d*x 
 + c) + 1)^8 + sin(d*x + c)^10/(cos(d*x + c) + 1)^10 + 1)))/d
 

Giac [F(-1)]

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

integrate((a+a*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c))*sec(d*x+c)^(11/2),x, alg 
orithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 32.97 (sec) , antiderivative size = 479, normalized size of antiderivative = 2.18 \[ \int \sqrt {a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac {11}{2}}(c+d x) \, dx =\text {Too large to display} \] Input:

int((A + B*cos(c + d*x))*(1/cos(c + d*x))^(11/2)*(a + a*cos(c + d*x))^(1/2 
),x)
 

Output:

((1/(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(((a + a*(exp(- 
 c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(256*A + 288*B)*1i)/(315* 
d) - (exp(c*9i + d*x*9i)*(a + a*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1 
i)/2))^(1/2)*(256*A + 288*B)*1i)/(315*d) + (exp(c*2i + d*x*2i)*(a + a*(exp 
(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(1152*A + 1296*B)*1i)/( 
315*d) - (exp(c*7i + d*x*7i)*(a + a*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d 
*x*1i)/2))^(1/2)*(1152*A + 1296*B)*1i)/(315*d) + (exp(c*4i + d*x*4i)*(a + 
a*(exp(- c*1i - d*x*1i)/2 + exp(c*1i + d*x*1i)/2))^(1/2)*(2016*A + 1008*B) 
*1i)/(315*d) - (exp(c*5i + d*x*5i)*(a + a*(exp(- c*1i - d*x*1i)/2 + exp(c* 
1i + d*x*1i)/2))^(1/2)*(2016*A + 1008*B)*1i)/(315*d)))/(exp(c*1i + d*x*1i) 
 + 4*exp(c*2i + d*x*2i) + 4*exp(c*3i + d*x*3i) + 6*exp(c*4i + d*x*4i) + 6* 
exp(c*5i + d*x*5i) + 4*exp(c*6i + d*x*6i) + 4*exp(c*7i + d*x*7i) + exp(c*8 
i + d*x*8i) + exp(c*9i + d*x*9i) + 1)
 

Reduce [F]

\[ \int \sqrt {a+a \cos (c+d x)} (A+B \cos (c+d x)) \sec ^{\frac {11}{2}}(c+d x) \, dx=\sqrt {a}\, \left (\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )+1}\, \cos \left (d x +c \right ) \sec \left (d x +c \right )^{5}d x \right ) b +\left (\int \sqrt {\sec \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )+1}\, \sec \left (d x +c \right )^{5}d x \right ) a \right ) \] Input:

int((a+a*cos(d*x+c))^(1/2)*(A+B*cos(d*x+c))*sec(d*x+c)^(11/2),x)
 

Output:

sqrt(a)*(int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x) + 1)*cos(c + d*x)*sec(c 
+ d*x)**5,x)*b + int(sqrt(sec(c + d*x))*sqrt(cos(c + d*x) + 1)*sec(c + d*x 
)**5,x)*a)