\(\int (a+a \cos (c+d x))^4 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^5(c+d x) \, dx\) [335]

Optimal result

Integrand size = 41, antiderivative size = 217 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=a^4 (B+4 C) x+\frac {a^4 (35 A+48 B+52 C) \text {arctanh}(\sin (c+d x))}{8 d}-\frac {5 a^4 (7 A+8 B+4 C) \sin (c+d x)}{8 d}+\frac {(35 A+44 B+36 C) \left (a^4+a^4 \cos (c+d x)\right ) \tan (c+d x)}{12 d}+\frac {(7 A+8 B+4 C) \left (a^2+a^2 \cos (c+d x)\right )^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a (A+B) (a+a \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+a \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d} \] Output:

a^4*(B+4*C)*x+1/8*a^4*(35*A+48*B+52*C)*arctanh(sin(d*x+c))/d-5/8*a^4*(7*A+ 
8*B+4*C)*sin(d*x+c)/d+1/12*(35*A+44*B+36*C)*(a^4+a^4*cos(d*x+c))*tan(d*x+c 
)/d+1/8*(7*A+8*B+4*C)*(a^2+a^2*cos(d*x+c))^2*sec(d*x+c)*tan(d*x+c)/d+1/3*a 
*(A+B)*(a+a*cos(d*x+c))^3*sec(d*x+c)^2*tan(d*x+c)/d+1/4*A*(a+a*cos(d*x+c)) 
^4*sec(d*x+c)^3*tan(d*x+c)/d
                                                                                    
                                                                                    
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(838\) vs. \(2(217)=434\).

Time = 13.17 (sec) , antiderivative size = 838, normalized size of antiderivative = 3.86 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx =\text {Too large to display} \] Input:

Integrate[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*S 
ec[c + d*x]^5,x]
 

Output:

((B + 4*C)*(c + d*x)*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8)/(16*d) + 
 ((-35*A - 48*B - 52*C)*(a + a*Cos[c + d*x])^4*Log[Cos[(c + d*x)/2] - Sin[ 
(c + d*x)/2]]*Sec[c/2 + (d*x)/2]^8)/(128*d) + ((35*A + 48*B + 52*C)*(a + a 
*Cos[c + d*x])^4*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sec[c/2 + (d*x)/ 
2]^8)/(128*d) + (A*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8)/(256*d*(Co 
s[(c + d*x)/2] - Sin[(c + d*x)/2])^4) + ((97*A + 52*B + 12*C)*(a + a*Cos[c 
 + d*x])^4*Sec[c/2 + (d*x)/2]^8)/(768*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/ 
2])^2) - (A*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8)/(256*d*(Cos[(c + 
d*x)/2] + Sin[(c + d*x)/2])^4) + ((-97*A - 52*B - 12*C)*(a + a*Cos[c + d*x 
])^4*Sec[c/2 + (d*x)/2]^8)/(768*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) 
 + ((a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(4*A*Sin[(c + d*x)/2] + B* 
Sin[(c + d*x)/2]))/(96*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + ((a + 
a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(4*A*Sin[(c + d*x)/2] + B*Sin[(c + 
d*x)/2]))/(96*d*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) + ((a + a*Cos[c + 
 d*x])^4*Sec[c/2 + (d*x)/2]^8*(5*A*Sin[(c + d*x)/2] + 5*B*Sin[(c + d*x)/2] 
 + 3*C*Sin[(c + d*x)/2]))/(12*d*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + ( 
(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*(5*A*Sin[(c + d*x)/2] + 5*B*Si 
n[(c + d*x)/2] + 3*C*Sin[(c + d*x)/2]))/(12*d*(Cos[(c + d*x)/2] + Sin[(c + 
 d*x)/2])) + (C*(a + a*Cos[c + d*x])^4*Sec[c/2 + (d*x)/2]^8*Sin[c + d*x])/ 
(16*d)
 

Rubi [A] (verified)

Time = 1.86 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.07, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.415, Rules used = {3042, 3522, 3042, 3454, 3042, 3454, 3042, 3454, 27, 3042, 3447, 3042, 3502, 3042, 3214, 3042, 4257}

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.

\(\displaystyle \int \sec ^5(c+d x) (a \cos (c+d x)+a)^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3522

\(\displaystyle \frac {\int (\cos (c+d x) a+a)^4 (4 a (A+B)-a (A-4 C) \cos (c+d x)) \sec ^4(c+d x)dx}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^4 \left (4 a (A+B)-a (A-4 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\)

\(\Big \downarrow \) 3454

\(\displaystyle \frac {\frac {1}{3} \int (\cos (c+d x) a+a)^3 \left (3 a^2 (7 A+8 B+4 C)-a^2 (7 A+4 B-12 C) \cos (c+d x)\right ) \sec ^3(c+d x)dx+\frac {4 a^2 (A+B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^3 \left (3 a^2 (7 A+8 B+4 C)-a^2 (7 A+4 B-12 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {4 a^2 (A+B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\)

\(\Big \downarrow \) 3454

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \int (\cos (c+d x) a+a)^2 \left (2 a^3 (35 A+44 B+36 C)-a^3 (35 A+32 B-12 C) \cos (c+d x)\right ) \sec ^2(c+d x)dx+\frac {3 a^3 (7 A+8 B+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 (A+B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (2 a^3 (35 A+44 B+36 C)-a^3 (35 A+32 B-12 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {3 a^3 (7 A+8 B+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 (A+B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\)

\(\Big \downarrow \) 3454

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (\int 3 (\cos (c+d x) a+a) \left (a^4 (35 A+48 B+52 C)-5 a^4 (7 A+8 B+4 C) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {2 (35 A+44 B+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+8 B+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 (A+B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \int (\cos (c+d x) a+a) \left (a^4 (35 A+48 B+52 C)-5 a^4 (7 A+8 B+4 C) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {2 (35 A+44 B+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+8 B+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 (A+B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \int \frac {\left (\sin \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left (a^4 (35 A+48 B+52 C)-5 a^4 (7 A+8 B+4 C) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 (35 A+44 B+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+8 B+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 (A+B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\)

\(\Big \downarrow \) 3447

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \int \left (-5 (7 A+8 B+4 C) \cos ^2(c+d x) a^5+(35 A+48 B+52 C) a^5+\left (a^5 (35 A+48 B+52 C)-5 a^5 (7 A+8 B+4 C)\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {2 (35 A+44 B+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+8 B+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 (A+B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \int \frac {-5 (7 A+8 B+4 C) \sin \left (c+d x+\frac {\pi }{2}\right )^2 a^5+(35 A+48 B+52 C) a^5+\left (a^5 (35 A+48 B+52 C)-5 a^5 (7 A+8 B+4 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {2 (35 A+44 B+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+8 B+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 (A+B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \left (\int \left ((35 A+48 B+52 C) a^5+8 (B+4 C) \cos (c+d x) a^5\right ) \sec (c+d x)dx-\frac {5 a^5 (7 A+8 B+4 C) \sin (c+d x)}{d}\right )+\frac {2 (35 A+44 B+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+8 B+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 (A+B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \left (\int \frac {(35 A+48 B+52 C) a^5+8 (B+4 C) \sin \left (c+d x+\frac {\pi }{2}\right ) a^5}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {5 a^5 (7 A+8 B+4 C) \sin (c+d x)}{d}\right )+\frac {2 (35 A+44 B+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+8 B+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 (A+B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\)

\(\Big \downarrow \) 3214

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \left (a^5 (35 A+48 B+52 C) \int \sec (c+d x)dx-\frac {5 a^5 (7 A+8 B+4 C) \sin (c+d x)}{d}+8 a^5 x (B+4 C)\right )+\frac {2 (35 A+44 B+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+8 B+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 (A+B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {1}{3} \left (\frac {1}{2} \left (3 \left (a^5 (35 A+48 B+52 C) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx-\frac {5 a^5 (7 A+8 B+4 C) \sin (c+d x)}{d}+8 a^5 x (B+4 C)\right )+\frac {2 (35 A+44 B+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+8 B+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )+\frac {4 a^2 (A+B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {\frac {4 a^2 (A+B) \tan (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+a)^3}{3 d}+\frac {1}{3} \left (\frac {1}{2} \left (3 \left (\frac {a^5 (35 A+48 B+52 C) \text {arctanh}(\sin (c+d x))}{d}-\frac {5 a^5 (7 A+8 B+4 C) \sin (c+d x)}{d}+8 a^5 x (B+4 C)\right )+\frac {2 (35 A+44 B+36 C) \tan (c+d x) \left (a^5 \cos (c+d x)+a^5\right )}{d}\right )+\frac {3 a^3 (7 A+8 B+4 C) \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^2}{2 d}\right )}{4 a}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a \cos (c+d x)+a)^4}{4 d}\)

Input:

Int[(a + a*Cos[c + d*x])^4*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + 
 d*x]^5,x]
 

Output:

(A*(a + a*Cos[c + d*x])^4*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((4*a^2*(A 
+ B)*(a + a*Cos[c + d*x])^3*Sec[c + d*x]^2*Tan[c + d*x])/(3*d) + ((3*a^3*( 
7*A + 8*B + 4*C)*(a + a*Cos[c + d*x])^2*Sec[c + d*x]*Tan[c + d*x])/(2*d) + 
 (3*(8*a^5*(B + 4*C)*x + (a^5*(35*A + 48*B + 52*C)*ArcTanh[Sin[c + d*x]])/ 
d - (5*a^5*(7*A + 8*B + 4*C)*Sin[c + d*x])/d) + (2*(35*A + 44*B + 36*C)*(a 
^5 + a^5*Cos[c + d*x])*Tan[c + d*x])/d)/2)/3)/(4*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 3214

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. 
)*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d   Int[1/(c + d 
*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
 

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 3454

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^2)*(B*c - A*d)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[ 
e + f*x])^(n + 1)/(d*f*(n + 1)*(b*c + a*d))), x] - Simp[b/(d*(n + 1)*(b*c + 
 a*d))   Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp 
[a*A*d*(m - n - 2) - B*(a*c*(m - 1) + b*d*(n + 1)) - (A*b*d*(m + n + 1) - B 
*(b*c*m - a*d*(n + 1)))*Sin[e + f*x], x], 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] 
&& 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 3522

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

rule 4257

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] 
 /; FreeQ[{c, d}, x]
 

Maple [A] (verified)

Time = 14.92 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.12

method	result	size
parts	\(\frac {a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (4 a^{4} A +a^{4} B \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (a^{4} B +4 a^{4} C \right ) \left (d x +c \right )}{d}+\frac {\left (a^{4} A +4 a^{4} B +6 a^{4} C \right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (4 a^{4} A +6 a^{4} B +4 a^{4} C \right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 a^{4} A +4 a^{4} B +a^{4} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\sin \left (d x +c \right ) a^{4} C}{d}\)	\(242\)
parallelrisch	\(\frac {35 a^{4} \left (-2 \left (A +\frac {48 B}{35}+\frac {52 C}{35}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+2 \left (A +\frac {48 B}{35}+\frac {52 C}{35}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\frac {16 d x \left (B +4 C \right ) \cos \left (2 d x +2 c \right )}{35}+\frac {4 d x \left (B +4 C \right ) \cos \left (4 d x +4 c \right )}{35}+\left (\frac {32 A}{15}+\frac {176 B}{105}+\frac {32 C}{35}\right ) \sin \left (2 d x +2 c \right )+\left (\frac {16 B}{35}+\frac {2 C}{7}+\frac {27 A}{35}\right ) \sin \left (3 d x +3 c \right )+\left (\frac {16 A}{21}+\frac {16 C}{35}+\frac {16 B}{21}\right ) \sin \left (4 d x +4 c \right )+\frac {2 C \sin \left (5 d x +5 c \right )}{35}+\left (A +\frac {16 B}{35}+\frac {8 C}{35}\right ) \sin \left (d x +c \right )+\frac {12 d x \left (B +4 C \right )}{35}\right )}{4 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\)	\(254\)
derivativedivides	\(\frac {a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} B \left (d x +c \right )+a^{4} C \sin \left (d x +c \right )+4 a^{4} A \tan \left (d x +c \right )+4 a^{4} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{4} C \left (d x +c \right )+6 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} B \tan \left (d x +c \right )+6 a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} B \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{4} C \tan \left (d x +c \right )+a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-a^{4} B \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\)	\(340\)
default	\(\frac {a^{4} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} B \left (d x +c \right )+a^{4} C \sin \left (d x +c \right )+4 a^{4} A \tan \left (d x +c \right )+4 a^{4} B \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a^{4} C \left (d x +c \right )+6 a^{4} A \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 a^{4} B \tan \left (d x +c \right )+6 a^{4} C \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )-4 a^{4} A \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+4 a^{4} B \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{4} C \tan \left (d x +c \right )+a^{4} A \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-a^{4} B \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\)	\(340\)
risch	\(a^{4} x B +4 a^{4} C x -\frac {i {\mathrm e}^{i \left (d x +c \right )} a^{4} C}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} a^{4} C}{2 d}-\frac {i a^{4} \left (81 A \,{\mathrm e}^{7 i \left (d x +c \right )}+48 B \,{\mathrm e}^{7 i \left (d x +c \right )}+12 C \,{\mathrm e}^{7 i \left (d x +c \right )}-96 A \,{\mathrm e}^{6 i \left (d x +c \right )}-144 B \,{\mathrm e}^{6 i \left (d x +c \right )}-96 C \,{\mathrm e}^{6 i \left (d x +c \right )}+105 A \,{\mathrm e}^{5 i \left (d x +c \right )}+48 B \,{\mathrm e}^{5 i \left (d x +c \right )}+12 C \,{\mathrm e}^{5 i \left (d x +c \right )}-480 A \,{\mathrm e}^{4 i \left (d x +c \right )}-480 B \,{\mathrm e}^{4 i \left (d x +c \right )}-288 C \,{\mathrm e}^{4 i \left (d x +c \right )}-105 A \,{\mathrm e}^{3 i \left (d x +c \right )}-48 B \,{\mathrm e}^{3 i \left (d x +c \right )}-12 C \,{\mathrm e}^{3 i \left (d x +c \right )}-544 A \,{\mathrm e}^{2 i \left (d x +c \right )}-496 B \,{\mathrm e}^{2 i \left (d x +c \right )}-288 C \,{\mathrm e}^{2 i \left (d x +c \right )}-81 A \,{\mathrm e}^{i \left (d x +c \right )}-48 B \,{\mathrm e}^{i \left (d x +c \right )}-12 C \,{\mathrm e}^{i \left (d x +c \right )}-160 A -160 B -96 C \right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {35 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{d}+\frac {13 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 d}-\frac {35 a^{4} A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}-\frac {6 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}-\frac {13 a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 d}\)	\(469\)

Input:

int((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5,x,meth 
od=_RETURNVERBOSE)
 

Output:

a^4*A/d*(-(-1/4*sec(d*x+c)^3-3/8*sec(d*x+c))*tan(d*x+c)+3/8*ln(sec(d*x+c)+ 
tan(d*x+c)))-(4*A*a^4+B*a^4)/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+(B*a^4+4 
*C*a^4)/d*(d*x+c)+(A*a^4+4*B*a^4+6*C*a^4)/d*ln(sec(d*x+c)+tan(d*x+c))+(4*A 
*a^4+6*B*a^4+4*C*a^4)/d*tan(d*x+c)+(6*A*a^4+4*B*a^4+C*a^4)/d*(1/2*sec(d*x+ 
c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))+1/d*sin(d*x+c)*a^4*C
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.88 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {48 \, {\left (B + 4 \, C\right )} a^{4} d x \cos \left (d x + c\right )^{4} + 3 \, {\left (35 \, A + 48 \, B + 52 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (35 \, A + 48 \, B + 52 \, C\right )} a^{4} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (24 \, C a^{4} \cos \left (d x + c\right )^{4} + 32 \, {\left (5 \, A + 5 \, B + 3 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 3 \, {\left (27 \, A + 16 \, B + 4 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 8 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right ) + 6 \, A a^{4}\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \] Input:

integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5, 
x, algorithm="fricas")
 

Output:

1/48*(48*(B + 4*C)*a^4*d*x*cos(d*x + c)^4 + 3*(35*A + 48*B + 52*C)*a^4*cos 
(d*x + c)^4*log(sin(d*x + c) + 1) - 3*(35*A + 48*B + 52*C)*a^4*cos(d*x + c 
)^4*log(-sin(d*x + c) + 1) + 2*(24*C*a^4*cos(d*x + c)^4 + 32*(5*A + 5*B + 
3*C)*a^4*cos(d*x + c)^3 + 3*(27*A + 16*B + 4*C)*a^4*cos(d*x + c)^2 + 8*(4* 
A + B)*a^4*cos(d*x + c) + 6*A*a^4)*sin(d*x + c))/(d*cos(d*x + c)^4)
 

Sympy [F(-1)]

Timed out. \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \] Input:

integrate((a+a*cos(d*x+c))**4*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)* 
*5,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 416 vs. \(2 (205) = 410\).

Time = 0.07 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.92 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {64 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} + 48 \, {\left (d x + c\right )} B a^{4} + 192 \, {\left (d x + c\right )} C a^{4} - 3 \, A a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, A a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 48 \, B a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, A a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 96 \, B a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 144 \, C a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, C a^{4} \sin \left (d x + c\right ) + 192 \, A a^{4} \tan \left (d x + c\right ) + 288 \, B a^{4} \tan \left (d x + c\right ) + 192 \, C a^{4} \tan \left (d x + c\right )}{48 \, d} \] Input:

integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5, 
x, algorithm="maxima")
 

Output:

1/48*(64*(tan(d*x + c)^3 + 3*tan(d*x + c))*A*a^4 + 16*(tan(d*x + c)^3 + 3* 
tan(d*x + c))*B*a^4 + 48*(d*x + c)*B*a^4 + 192*(d*x + c)*C*a^4 - 3*A*a^4*( 
2*(3*sin(d*x + c)^3 - 5*sin(d*x + c))/(sin(d*x + c)^4 - 2*sin(d*x + c)^2 + 
 1) - 3*log(sin(d*x + c) + 1) + 3*log(sin(d*x + c) - 1)) - 72*A*a^4*(2*sin 
(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) 
- 1)) - 48*B*a^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 
 1) + log(sin(d*x + c) - 1)) - 12*C*a^4*(2*sin(d*x + c)/(sin(d*x + c)^2 - 
1) - log(sin(d*x + c) + 1) + log(sin(d*x + c) - 1)) + 24*A*a^4*(log(sin(d* 
x + c) + 1) - log(sin(d*x + c) - 1)) + 96*B*a^4*(log(sin(d*x + c) + 1) - l 
og(sin(d*x + c) - 1)) + 144*C*a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c 
) - 1)) + 48*C*a^4*sin(d*x + c) + 192*A*a^4*tan(d*x + c) + 288*B*a^4*tan(d 
*x + c) + 192*C*a^4*tan(d*x + c))/d
 

Giac [A] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 339, normalized size of antiderivative = 1.56 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {\frac {48 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1} + 24 \, {\left (B a^{4} + 4 \, C a^{4}\right )} {\left (d x + c\right )} + 3 \, {\left (35 \, A a^{4} + 48 \, B a^{4} + 52 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (35 \, A a^{4} + 48 \, B a^{4} + 52 \, C a^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (105 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 120 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 84 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 385 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 424 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 276 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 511 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 520 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 300 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 279 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 216 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 108 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \] Input:

integrate((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5, 
x, algorithm="giac")
 

Output:

1/24*(48*C*a^4*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 + 1) + 24*(B*a 
^4 + 4*C*a^4)*(d*x + c) + 3*(35*A*a^4 + 48*B*a^4 + 52*C*a^4)*log(abs(tan(1 
/2*d*x + 1/2*c) + 1)) - 3*(35*A*a^4 + 48*B*a^4 + 52*C*a^4)*log(abs(tan(1/2 
*d*x + 1/2*c) - 1)) - 2*(105*A*a^4*tan(1/2*d*x + 1/2*c)^7 + 120*B*a^4*tan( 
1/2*d*x + 1/2*c)^7 + 84*C*a^4*tan(1/2*d*x + 1/2*c)^7 - 385*A*a^4*tan(1/2*d 
*x + 1/2*c)^5 - 424*B*a^4*tan(1/2*d*x + 1/2*c)^5 - 276*C*a^4*tan(1/2*d*x + 
 1/2*c)^5 + 511*A*a^4*tan(1/2*d*x + 1/2*c)^3 + 520*B*a^4*tan(1/2*d*x + 1/2 
*c)^3 + 300*C*a^4*tan(1/2*d*x + 1/2*c)^3 - 279*A*a^4*tan(1/2*d*x + 1/2*c) 
- 216*B*a^4*tan(1/2*d*x + 1/2*c) - 108*C*a^4*tan(1/2*d*x + 1/2*c))/(tan(1/ 
2*d*x + 1/2*c)^2 - 1)^4)/d
 

Mupad [B] (verification not implemented)

Time = 1.35 (sec) , antiderivative size = 1342, normalized size of antiderivative = 6.18 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Too large to display} \] Input:

int(((a + a*cos(c + d*x))^4*(A + B*cos(c + d*x) + C*cos(c + d*x)^2))/cos(c 
 + d*x)^5,x)
 

Output:

((105*A*a^4*atanh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/32 + (9*B*a^4*at 
anh(sin(c/2 + (d*x)/2)/cos(c/2 + (d*x)/2)))/2 + (39*C*a^4*atanh(sin(c/2 + 
(d*x)/2)/cos(c/2 + (d*x)/2)))/8 + (7*A*a^4*sin(2*c + 2*d*x))/3 + (27*A*a^4 
*sin(3*c + 3*d*x))/32 + (5*A*a^4*sin(4*c + 4*d*x))/6 + (11*B*a^4*sin(2*c + 
 2*d*x))/6 + (B*a^4*sin(3*c + 3*d*x))/2 + (5*B*a^4*sin(4*c + 4*d*x))/6 + C 
*a^4*sin(2*c + 2*d*x) + (5*C*a^4*sin(3*c + 3*d*x))/16 + (C*a^4*sin(4*c + 4 
*d*x))/2 + (C*a^4*sin(5*c + 5*d*x))/16 + (3*B*a^4*atan((1225*A^2*sin(c/2 + 
 (d*x)/2) + 2368*B^2*sin(c/2 + (d*x)/2) + 3728*C^2*sin(c/2 + (d*x)/2) + 33 
60*A*B*sin(c/2 + (d*x)/2) + 3640*A*C*sin(c/2 + (d*x)/2) + 5504*B*C*sin(c/2 
 + (d*x)/2))/(cos(c/2 + (d*x)/2)*(1225*A^2 + 2368*B^2 + 3728*C^2 + 3360*A* 
B + 3640*A*C + 5504*B*C))))/4 + 3*C*a^4*atan((1225*A^2*sin(c/2 + (d*x)/2) 
+ 2368*B^2*sin(c/2 + (d*x)/2) + 3728*C^2*sin(c/2 + (d*x)/2) + 3360*A*B*sin 
(c/2 + (d*x)/2) + 3640*A*C*sin(c/2 + (d*x)/2) + 5504*B*C*sin(c/2 + (d*x)/2 
))/(cos(c/2 + (d*x)/2)*(1225*A^2 + 2368*B^2 + 3728*C^2 + 3360*A*B + 3640*A 
*C + 5504*B*C))) + (35*A*a^4*sin(c + d*x))/32 + (B*a^4*sin(c + d*x))/2 + ( 
C*a^4*sin(c + d*x))/4 + B*a^4*atan((1225*A^2*sin(c/2 + (d*x)/2) + 2368*B^2 
*sin(c/2 + (d*x)/2) + 3728*C^2*sin(c/2 + (d*x)/2) + 3360*A*B*sin(c/2 + (d* 
x)/2) + 3640*A*C*sin(c/2 + (d*x)/2) + 5504*B*C*sin(c/2 + (d*x)/2))/(cos(c/ 
2 + (d*x)/2)*(1225*A^2 + 2368*B^2 + 3728*C^2 + 3360*A*B + 3640*A*C + 5504* 
B*C)))*cos(2*c + 2*d*x) + (B*a^4*atan((1225*A^2*sin(c/2 + (d*x)/2) + 23...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 817, normalized size of antiderivative = 3.76 \[ \int (a+a \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx =\text {Too large to display} \] Input:

int((a+a*cos(d*x+c))^4*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^5,x)
 

Output:

(a**4*( - 105*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a - 1 
44*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*b - 156*cos(c + 
d*x)*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*c + 210*cos(c + d*x)*log(ta 
n((c + d*x)/2) - 1)*sin(c + d*x)**2*a + 288*cos(c + d*x)*log(tan((c + d*x) 
/2) - 1)*sin(c + d*x)**2*b + 312*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*si 
n(c + d*x)**2*c - 105*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a - 144*cos(c 
 + d*x)*log(tan((c + d*x)/2) - 1)*b - 156*cos(c + d*x)*log(tan((c + d*x)/2 
) - 1)*c + 105*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*a + 
144*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*b + 156*cos(c + 
 d*x)*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4*c - 210*cos(c + d*x)*log(t 
an((c + d*x)/2) + 1)*sin(c + d*x)**2*a - 288*cos(c + d*x)*log(tan((c + d*x 
)/2) + 1)*sin(c + d*x)**2*b - 312*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*s 
in(c + d*x)**2*c + 105*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*a + 144*cos( 
c + d*x)*log(tan((c + d*x)/2) + 1)*b + 156*cos(c + d*x)*log(tan((c + d*x)/ 
2) + 1)*c + 24*cos(c + d*x)*sin(c + d*x)**5*c + 24*cos(c + d*x)*sin(c + d* 
x)**4*b*d*x + 96*cos(c + d*x)*sin(c + d*x)**4*c*d*x - 81*cos(c + d*x)*sin( 
c + d*x)**3*a - 48*cos(c + d*x)*sin(c + d*x)**3*b - 60*cos(c + d*x)*sin(c 
+ d*x)**3*c - 48*cos(c + d*x)*sin(c + d*x)**2*b*d*x - 192*cos(c + d*x)*sin 
(c + d*x)**2*c*d*x + 87*cos(c + d*x)*sin(c + d*x)*a + 48*cos(c + d*x)*sin( 
c + d*x)*b + 36*cos(c + d*x)*sin(c + d*x)*c + 24*cos(c + d*x)*b*d*x + 9...