\(\int \frac {\cos ^{\frac {9}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx\) [158]

Optimal result

Integrand size = 33, antiderivative size = 254 \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx=-\frac {7 (17 A-33 B) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {(11 A-21 B) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{2 a^3 d}+\frac {(11 A-21 B) \sqrt {\cos (c+d x)} \sin (c+d x)}{2 a^3 d}-\frac {7 (17 A-33 B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{30 a^3 d}+\frac {(A-B) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{5 d (a+a \cos (c+d x))^3}+\frac {(7 A-12 B) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{15 a d (a+a \cos (c+d x))^2}+\frac {3 (11 A-21 B) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{10 d \left (a^3+a^3 \cos (c+d x)\right )} \] Output:

-7/10*(17*A-33*B)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))/a^3/d+1/2*(11*A-21 
*B)*InverseJacobiAM(1/2*d*x+1/2*c,2^(1/2))/a^3/d+1/2*(11*A-21*B)*cos(d*x+c 
)^(1/2)*sin(d*x+c)/a^3/d-7/30*(17*A-33*B)*cos(d*x+c)^(3/2)*sin(d*x+c)/a^3/ 
d+1/5*(A-B)*cos(d*x+c)^(9/2)*sin(d*x+c)/d/(a+a*cos(d*x+c))^3+1/15*(7*A-12* 
B)*cos(d*x+c)^(7/2)*sin(d*x+c)/a/d/(a+a*cos(d*x+c))^2+3/10*(11*A-21*B)*cos 
(d*x+c)^(5/2)*sin(d*x+c)/d/(a^3+a^3*cos(d*x+c))
 

Mathematica [C] (verified)

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

Time = 8.97 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.90 \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx=-\frac {\sqrt {\cos (c+d x)} \csc ^5(c+d x) \left (-396 A+396 B-680 A \cos (c+d x)+680 B \cos (c+d x)+792 A \cos ^2(c+d x)-792 B \cos ^2(c+d x)+680 A \cos ^3(c+d x)-680 B \cos ^3(c+d x)-440 A \cos ^4(c+d x)+440 B \cos ^4(c+d x)-180 A \cos ^5(c+d x)+180 B \cos ^5(c+d x)+20 A \cos ^6(c+d x)-20 B \cos ^6(c+d x)+66 A \sin ^2(c+d x)+234 B \sin ^2(c+d x)+448 B \cos (c+d x) \sin ^2(c+d x)+40 B \cos ^4(c+d x) \sin ^2(c+d x)-12 B \cos ^5(c+d x) \sin ^2(c+d x)+165 A \sin ^4(c+d x)-315 B \sin ^4(c+d x)+15 (11 A-21 B) \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\cos ^2(c+d x)\right ) \sin ^4(c+d x) \sqrt {\sin ^2(c+d x)}-448 B \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {5}{2},\frac {7}{4},\cos ^2(c+d x)\right ) \sin ^4(c+d x) \sqrt {\sin ^2(c+d x)}+680 A \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {7}{2},\frac {7}{4},\cos ^2(c+d x)\right ) \sin ^4(c+d x) \sqrt {\sin ^2(c+d x)}-680 B \cos (c+d x) \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {7}{2},\frac {7}{4},\cos ^2(c+d x)\right ) \sin ^4(c+d x) \sqrt {\sin ^2(c+d x)}-90 B \sin ^2(2 (c+d x))-24 B \csc (c+d x) \sin ^3(2 (c+d x))\right )}{30 a^3 d} \] Input:

Integrate[(Cos[c + d*x]^(9/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^3 
,x]
 

Output:

-1/30*(Sqrt[Cos[c + d*x]]*Csc[c + d*x]^5*(-396*A + 396*B - 680*A*Cos[c + d 
*x] + 680*B*Cos[c + d*x] + 792*A*Cos[c + d*x]^2 - 792*B*Cos[c + d*x]^2 + 6 
80*A*Cos[c + d*x]^3 - 680*B*Cos[c + d*x]^3 - 440*A*Cos[c + d*x]^4 + 440*B* 
Cos[c + d*x]^4 - 180*A*Cos[c + d*x]^5 + 180*B*Cos[c + d*x]^5 + 20*A*Cos[c 
+ d*x]^6 - 20*B*Cos[c + d*x]^6 + 66*A*Sin[c + d*x]^2 + 234*B*Sin[c + d*x]^ 
2 + 448*B*Cos[c + d*x]*Sin[c + d*x]^2 + 40*B*Cos[c + d*x]^4*Sin[c + d*x]^2 
 - 12*B*Cos[c + d*x]^5*Sin[c + d*x]^2 + 165*A*Sin[c + d*x]^4 - 315*B*Sin[c 
 + d*x]^4 + 15*(11*A - 21*B)*Hypergeometric2F1[1/4, 1/2, 5/4, Cos[c + d*x] 
^2]*Sin[c + d*x]^4*Sqrt[Sin[c + d*x]^2] - 448*B*Cos[c + d*x]*Hypergeometri 
c2F1[3/4, 5/2, 7/4, Cos[c + d*x]^2]*Sin[c + d*x]^4*Sqrt[Sin[c + d*x]^2] + 
680*A*Cos[c + d*x]*Hypergeometric2F1[3/4, 7/2, 7/4, Cos[c + d*x]^2]*Sin[c 
+ d*x]^4*Sqrt[Sin[c + d*x]^2] - 680*B*Cos[c + d*x]*Hypergeometric2F1[3/4, 
7/2, 7/4, Cos[c + d*x]^2]*Sin[c + d*x]^4*Sqrt[Sin[c + d*x]^2] - 90*B*Sin[2 
*(c + d*x)]^2 - 24*B*Csc[c + d*x]*Sin[2*(c + d*x)]^3))/(a^3*d)
 

Rubi [A] (verified)

Time = 1.37 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.02, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {3042, 3456, 27, 3042, 3456, 3042, 3456, 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.

\(\displaystyle \int \frac {\cos ^{\frac {9}{2}}(c+d x) (A+B \cos (c+d x))}{(a \cos (c+d x)+a)^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{9/2} \left (A+B \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\)

\(\Big \downarrow \) 3456

\(\displaystyle \frac {\int \frac {\cos ^{\frac {7}{2}}(c+d x) (9 a (A-B)-5 a (A-3 B) \cos (c+d x))}{2 (\cos (c+d x) a+a)^2}dx}{5 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\cos ^{\frac {7}{2}}(c+d x) (9 a (A-B)-5 a (A-3 B) \cos (c+d x))}{(\cos (c+d x) a+a)^2}dx}{10 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{7/2} \left (9 a (A-B)-5 a (A-3 B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2}dx}{10 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\)

\(\Big \downarrow \) 3456

\(\displaystyle \frac {\frac {\int \frac {\cos ^{\frac {5}{2}}(c+d x) \left (7 a^2 (7 A-12 B)-5 a^2 (10 A-21 B) \cos (c+d x)\right )}{\cos (c+d x) a+a}dx}{3 a^2}+\frac {2 a (7 A-12 B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\int \frac {\sin \left (c+d x+\frac {\pi }{2}\right )^{5/2} \left (7 a^2 (7 A-12 B)-5 a^2 (10 A-21 B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right ) a+a}dx}{3 a^2}+\frac {2 a (7 A-12 B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\)

\(\Big \downarrow \) 3456

\(\displaystyle \frac {\frac {\frac {\int \frac {5}{2} \cos ^{\frac {3}{2}}(c+d x) \left (9 a^3 (11 A-21 B)-7 a^3 (17 A-33 B) \cos (c+d x)\right )dx}{a^2}+\frac {9 a^2 (11 A-21 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {2 a (7 A-12 B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {5 \int \cos ^{\frac {3}{2}}(c+d x) \left (9 a^3 (11 A-21 B)-7 a^3 (17 A-33 B) \cos (c+d x)\right )dx}{2 a^2}+\frac {9 a^2 (11 A-21 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {2 a (7 A-12 B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {5 \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2} \left (9 a^3 (11 A-21 B)-7 a^3 (17 A-33 B) \sin \left (c+d x+\frac {\pi }{2}\right )\right )dx}{2 a^2}+\frac {9 a^2 (11 A-21 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {2 a (7 A-12 B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\)

\(\Big \downarrow \) 3227

\(\displaystyle \frac {\frac {\frac {5 \left (9 a^3 (11 A-21 B) \int \cos ^{\frac {3}{2}}(c+d x)dx-7 a^3 (17 A-33 B) \int \cos ^{\frac {5}{2}}(c+d x)dx\right )}{2 a^2}+\frac {9 a^2 (11 A-21 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {2 a (7 A-12 B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {5 \left (9 a^3 (11 A-21 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{3/2}dx-7 a^3 (17 A-33 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )^{5/2}dx\right )}{2 a^2}+\frac {9 a^2 (11 A-21 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {2 a (7 A-12 B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {\frac {\frac {5 \left (9 a^3 (11 A-21 B) \left (\frac {1}{3} \int \frac {1}{\sqrt {\cos (c+d x)}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )-7 a^3 (17 A-33 B) \left (\frac {3}{5} \int \sqrt {\cos (c+d x)}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )}{2 a^2}+\frac {9 a^2 (11 A-21 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {2 a (7 A-12 B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\frac {5 \left (9 a^3 (11 A-21 B) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )-7 a^3 (17 A-33 B) \left (\frac {3}{5} \int \sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )}{2 a^2}+\frac {9 a^2 (11 A-21 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {2 a (7 A-12 B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\)

\(\Big \downarrow \) 3119

\(\displaystyle \frac {\frac {\frac {5 \left (9 a^3 (11 A-21 B) \left (\frac {1}{3} \int \frac {1}{\sqrt {\sin \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )-7 a^3 (17 A-33 B) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )}{2 a^2}+\frac {9 a^2 (11 A-21 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}}{3 a^2}+\frac {2 a (7 A-12 B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\)

\(\Big \downarrow \) 3120

\(\displaystyle \frac {\frac {\frac {9 a^2 (11 A-21 B) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac {5 \left (9 a^3 (11 A-21 B) \left (\frac {2 \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{3 d}+\frac {2 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}\right )-7 a^3 (17 A-33 B) \left (\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}\right )\right )}{2 a^2}}{3 a^2}+\frac {2 a (7 A-12 B) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2}}{10 a^2}+\frac {(A-B) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{5 d (a \cos (c+d x)+a)^3}\)

Input:

Int[(Cos[c + d*x]^(9/2)*(A + B*Cos[c + d*x]))/(a + a*Cos[c + d*x])^3,x]
 

Output:

((A - B)*Cos[c + d*x]^(9/2)*Sin[c + d*x])/(5*d*(a + a*Cos[c + d*x])^3) + ( 
(2*a*(7*A - 12*B)*Cos[c + d*x]^(7/2)*Sin[c + d*x])/(3*d*(a + a*Cos[c + d*x 
])^2) + ((9*a^2*(11*A - 21*B)*Cos[c + d*x]^(5/2)*Sin[c + d*x])/(d*(a + a*C 
os[c + d*x])) + (5*(9*a^3*(11*A - 21*B)*((2*EllipticF[(c + d*x)/2, 2])/(3* 
d) + (2*Sqrt[Cos[c + d*x]]*Sin[c + d*x])/(3*d)) - 7*a^3*(17*A - 33*B)*((6* 
EllipticE[(c + d*x)/2, 2])/(5*d) + (2*Cos[c + d*x]^(3/2)*Sin[c + d*x])/(5* 
d))))/(2*a^2))/(3*a^2))/(10*a^2)
 

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 3456

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[(A*b - a*B)*Cos[e + f*x]*(a + b*Sin[e + f*x])^m*((c + d*Sin[e + f*x])^n/( 
a*f*(2*m + 1))), x] - Simp[1/(a*b*(2*m + 1))   Int[(a + b*Sin[e + f*x])^(m 
+ 1)*(c + d*Sin[e + f*x])^(n - 1)*Simp[A*(a*d*n - b*c*(m + 1)) - B*(a*c*m + 
 b*d*n) - d*(a*B*(m - n) + A*b*(m + n + 1))*Sin[e + f*x], x], x], x] /; Fre 
eQ[{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[m, -2^(-1)] && GtQ[n, 0] && IntegerQ[2*m] && (In 
tegerQ[2*n] || EqQ[c, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(492\) vs. \(2(233)=466\).

Time = 18.81 (sec) , antiderivative size = 493, normalized size of antiderivative = 1.94

method	result	size
default	\(-\frac {\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}}\, \left (192 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+160 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-864 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+468 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+330 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \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 ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+714 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \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 )-228 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}-630 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \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 ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-1386 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \cos \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 )-1058 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+1590 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+474 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-744 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}-47 A \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+57 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+3 A -3 B \right )}{60 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} \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}\)	\(493\)

Input:

int(cos(d*x+c)^(9/2)*(A+B*cos(d*x+c))/(a+a*cos(d*x+c))^3,x,method=_RETURNV 
ERBOSE)
 

Output:

-1/60*((2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*(192*B*cos(1 
/2*d*x+1/2*c)^12+160*A*cos(1/2*d*x+1/2*c)^10-864*B*cos(1/2*d*x+1/2*c)^10+4 
68*A*cos(1/2*d*x+1/2*c)^8+330*A*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d 
*x+1/2*c)^2+1)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*cos(1/2*d*x+1/2 
*c)^5+714*A*cos(1/2*d*x+1/2*c)^5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2* 
d*x+1/2*c)^2+1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-228*B*cos(1/2* 
d*x+1/2*c)^8-630*B*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2+1 
)^(1/2)*EllipticF(cos(1/2*d*x+1/2*c),2^(1/2))*cos(1/2*d*x+1/2*c)^5-1386*B* 
cos(1/2*d*x+1/2*c)^5*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2*c)^2 
+1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-1058*A*cos(1/2*d*x+1/2*c)^ 
6+1590*B*cos(1/2*d*x+1/2*c)^6+474*A*cos(1/2*d*x+1/2*c)^4-744*B*cos(1/2*d*x 
+1/2*c)^4-47*A*cos(1/2*d*x+1/2*c)^2+57*B*cos(1/2*d*x+1/2*c)^2+3*A-3*B)/a^3 
/cos(1/2*d*x+1/2*c)^5/(-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 = 0.12 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.95 \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx =\text {Too large to display} \] Input:

integrate(cos(d*x+c)^(9/2)*(A+B*cos(d*x+c))/(a+a*cos(d*x+c))^3,x, algorith 
m="fricas")
 

Output:

1/60*(2*(12*B*cos(d*x + c)^4 + 4*(5*A - 6*B)*cos(d*x + c)^3 + 3*(79*A - 14 
7*B)*cos(d*x + c)^2 + 2*(188*A - 357*B)*cos(d*x + c) + 165*A - 315*B)*sqrt 
(cos(d*x + c))*sin(d*x + c) - 15*(sqrt(2)*(11*I*A - 21*I*B)*cos(d*x + c)^3 
 + 3*sqrt(2)*(11*I*A - 21*I*B)*cos(d*x + c)^2 + 3*sqrt(2)*(11*I*A - 21*I*B 
)*cos(d*x + c) + sqrt(2)*(11*I*A - 21*I*B))*weierstrassPInverse(-4, 0, cos 
(d*x + c) + I*sin(d*x + c)) - 15*(sqrt(2)*(-11*I*A + 21*I*B)*cos(d*x + c)^ 
3 + 3*sqrt(2)*(-11*I*A + 21*I*B)*cos(d*x + c)^2 + 3*sqrt(2)*(-11*I*A + 21* 
I*B)*cos(d*x + c) + sqrt(2)*(-11*I*A + 21*I*B))*weierstrassPInverse(-4, 0, 
 cos(d*x + c) - I*sin(d*x + c)) - 21*(sqrt(2)*(17*I*A - 33*I*B)*cos(d*x + 
c)^3 + 3*sqrt(2)*(17*I*A - 33*I*B)*cos(d*x + c)^2 + 3*sqrt(2)*(17*I*A - 33 
*I*B)*cos(d*x + c) + sqrt(2)*(17*I*A - 33*I*B))*weierstrassZeta(-4, 0, wei 
erstrassPInverse(-4, 0, cos(d*x + c) + I*sin(d*x + c))) - 21*(sqrt(2)*(-17 
*I*A + 33*I*B)*cos(d*x + c)^3 + 3*sqrt(2)*(-17*I*A + 33*I*B)*cos(d*x + c)^ 
2 + 3*sqrt(2)*(-17*I*A + 33*I*B)*cos(d*x + c) + sqrt(2)*(-17*I*A + 33*I*B) 
)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c) - I*sin(d 
*x + c))))/(a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d* 
x + c) + a^3*d)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**(9/2)*(A+B*cos(d*x+c))/(a+a*cos(d*x+c))**3,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {9}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:

integrate(cos(d*x+c)^(9/2)*(A+B*cos(d*x+c))/(a+a*cos(d*x+c))^3,x, algorith 
m="maxima")
 

Output:

integrate((B*cos(d*x + c) + A)*cos(d*x + c)^(9/2)/(a*cos(d*x + c) + a)^3, 
x)
 

Giac [F]

\[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {9}{2}}}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3}} \,d x } \] Input:

integrate(cos(d*x+c)^(9/2)*(A+B*cos(d*x+c))/(a+a*cos(d*x+c))^3,x, algorith 
m="giac")
 

Output:

integrate((B*cos(d*x + c) + A)*cos(d*x + c)^(9/2)/(a*cos(d*x + c) + a)^3, 
x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{9/2}\,\left (A+B\,\cos \left (c+d\,x\right )\right )}{{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3} \,d x \] Input:

int((cos(c + d*x)^(9/2)*(A + B*cos(c + d*x)))/(a + a*cos(c + d*x))^3,x)
                                                                                    
                                                                                    
 

Output:

int((cos(c + d*x)^(9/2)*(A + B*cos(c + d*x)))/(a + a*cos(c + d*x))^3, x)
 

Reduce [F]

\[ \int \frac {\cos ^{\frac {9}{2}}(c+d x) (A+B \cos (c+d x))}{(a+a \cos (c+d x))^3} \, dx=\frac {\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{5}}{\cos \left (d x +c \right )^{3}+3 \cos \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )+1}d x \right ) b +\left (\int \frac {\sqrt {\cos \left (d x +c \right )}\, \cos \left (d x +c \right )^{4}}{\cos \left (d x +c \right )^{3}+3 \cos \left (d x +c \right )^{2}+3 \cos \left (d x +c \right )+1}d x \right ) a}{a^{3}} \] Input:

int(cos(d*x+c)^(9/2)*(A+B*cos(d*x+c))/(a+a*cos(d*x+c))^3,x)
 

Output:

(int((sqrt(cos(c + d*x))*cos(c + d*x)**5)/(cos(c + d*x)**3 + 3*cos(c + d*x 
)**2 + 3*cos(c + d*x) + 1),x)*b + int((sqrt(cos(c + d*x))*cos(c + d*x)**4) 
/(cos(c + d*x)**3 + 3*cos(c + d*x)**2 + 3*cos(c + d*x) + 1),x)*a)/a**3