\(\int (a+b \cos (c+d x)) (B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^6(c+d x) \, dx\) [777]

Optimal result

Integrand size = 38, antiderivative size = 114 \[ \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {(3 a B+4 b C) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {(b B+a C) \tan (c+d x)}{d}+\frac {(3 a B+4 b C) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {a B \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {(b B+a C) \tan ^3(c+d x)}{3 d} \] Output:

1/8*(3*B*a+4*C*b)*arctanh(sin(d*x+c))/d+(B*b+C*a)*tan(d*x+c)/d+1/8*(3*B*a+ 
4*C*b)*sec(d*x+c)*tan(d*x+c)/d+1/4*a*B*sec(d*x+c)^3*tan(d*x+c)/d+1/3*(B*b+ 
C*a)*tan(d*x+c)^3/d
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.75 \[ \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {3 (3 a B+4 b C) \text {arctanh}(\sin (c+d x))+\sec (c+d x) \left (9 a B+12 b C+8 (b B+a C) (2+\cos (2 (c+d x))) \sec (c+d x)+6 a B \sec ^2(c+d x)\right ) \tan (c+d x)}{24 d} \] Input:

Integrate[(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + 
 d*x]^6,x]
 

Output:

(3*(3*a*B + 4*b*C)*ArcTanh[Sin[c + d*x]] + Sec[c + d*x]*(9*a*B + 12*b*C + 
8*(b*B + a*C)*(2 + Cos[2*(c + d*x)])*Sec[c + d*x] + 6*a*B*Sec[c + d*x]^2)* 
Tan[c + d*x])/(24*d)
 

Rubi [A] (verified)

Time = 0.83 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.93, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {3042, 3508, 3042, 3447, 3042, 3500, 3042, 3227, 3042, 4254, 2009, 4255, 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.

\(\displaystyle \int \sec ^6(c+d x) (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (B \sin \left (c+d x+\frac {\pi }{2}\right )+C \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^6}dx\)

\(\Big \downarrow \) 3508

\(\displaystyle \int \sec ^5(c+d x) (a+b \cos (c+d x)) (B+C \cos (c+d x))dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3447

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3500

\(\displaystyle \frac {1}{4} \int (4 (b B+a C)+(3 a B+4 b C) \cos (c+d x)) \sec ^4(c+d x)dx+\frac {a B \tan (c+d x) \sec ^3(c+d x)}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \int \frac {4 (b B+a C)+(3 a B+4 b C) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {a B \tan (c+d x) \sec ^3(c+d x)}{4 d}\)

\(\Big \downarrow \) 3227

\(\displaystyle \frac {1}{4} \left (4 (a C+b B) \int \sec ^4(c+d x)dx+(3 a B+4 b C) \int \sec ^3(c+d x)dx\right )+\frac {a B \tan (c+d x) \sec ^3(c+d x)}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left ((3 a B+4 b C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx+4 (a C+b B) \int \csc \left (c+d x+\frac {\pi }{2}\right )^4dx\right )+\frac {a B \tan (c+d x) \sec ^3(c+d x)}{4 d}\)

\(\Big \downarrow \) 4254

\(\displaystyle \frac {1}{4} \left ((3 a B+4 b C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {4 (a C+b B) \int \left (\tan ^2(c+d x)+1\right )d(-\tan (c+d x))}{d}\right )+\frac {a B \tan (c+d x) \sec ^3(c+d x)}{4 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{4} \left ((3 a B+4 b C) \int \csc \left (c+d x+\frac {\pi }{2}\right )^3dx-\frac {4 (a C+b B) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )+\frac {a B \tan (c+d x) \sec ^3(c+d x)}{4 d}\)

\(\Big \downarrow \) 4255

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

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left ((3 a B+4 b C) \left (\frac {1}{2} \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {4 (a C+b B) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )+\frac {a B \tan (c+d x) \sec ^3(c+d x)}{4 d}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {1}{4} \left ((3 a B+4 b C) \left (\frac {\text {arctanh}(\sin (c+d x))}{2 d}+\frac {\tan (c+d x) \sec (c+d x)}{2 d}\right )-\frac {4 (a C+b B) \left (-\frac {1}{3} \tan ^3(c+d x)-\tan (c+d x)\right )}{d}\right )+\frac {a B \tan (c+d x) \sec ^3(c+d x)}{4 d}\)

Input:

Int[(a + b*Cos[c + d*x])*(B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + d*x]^ 
6,x]
 

Output:

(a*B*Sec[c + d*x]^3*Tan[c + d*x])/(4*d) + ((3*a*B + 4*b*C)*(ArcTanh[Sin[c 
+ d*x]]/(2*d) + (Sec[c + d*x]*Tan[c + d*x])/(2*d)) - (4*(b*B + a*C)*(-Tan[ 
c + d*x] - Tan[c + d*x]^3/3))/d)/4
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, 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 3500

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + 
 (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 
 - a*b*B + a^2*C))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 1)* 
(a^2 - b^2))), x] + Simp[1/(b*(m + 1)*(a^2 - b^2))   Int[(a + b*Sin[e + f*x 
])^(m + 1)*Simp[b*(a*A - b*B + a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A 
*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, 
B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]
 

rule 3508

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

rule 4254

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1)   Subst[Int[Exp 
andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, 
 d}, x] && IGtQ[n/2, 0]
 

rule 4255

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* 
x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) 
  Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] 
&& IntegerQ[2*n]
 

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 = 0.89 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.04

method	result	size
parts	\(-\frac {\left (B b +C a \right ) \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {B a \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {C b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\)	\(119\)
derivativedivides	\(\frac {-C a \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B a \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+C b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B b \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\)	\(131\)
default	\(\frac {-C a \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )+B a \left (-\left (-\frac {\sec \left (d x +c \right )^{3}}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+C b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-B b \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\)	\(131\)
parallelrisch	\(\frac {-18 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (B a +\frac {4 C b}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+18 \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \left (B a +\frac {4 C b}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+32 \left (B b +C a \right ) \sin \left (2 d x +2 c \right )+3 \left (3 B a +4 C b \right ) \sin \left (3 d x +3 c \right )+8 \left (B b +C a \right ) \sin \left (4 d x +4 c \right )+3 \sin \left (d x +c \right ) \left (11 B a +4 C b \right )}{12 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\)	\(192\)
risch	\(-\frac {i \left (9 B a \,{\mathrm e}^{7 i \left (d x +c \right )}+12 C b \,{\mathrm e}^{7 i \left (d x +c \right )}+33 B a \,{\mathrm e}^{5 i \left (d x +c \right )}+12 C b \,{\mathrm e}^{5 i \left (d x +c \right )}-48 B b \,{\mathrm e}^{4 i \left (d x +c \right )}-48 C a \,{\mathrm e}^{4 i \left (d x +c \right )}-33 B a \,{\mathrm e}^{3 i \left (d x +c \right )}-12 C b \,{\mathrm e}^{3 i \left (d x +c \right )}-64 B b \,{\mathrm e}^{2 i \left (d x +c \right )}-64 C a \,{\mathrm e}^{2 i \left (d x +c \right )}-9 B a \,{\mathrm e}^{i \left (d x +c \right )}-12 C b \,{\mathrm e}^{i \left (d x +c \right )}-16 B b -16 C a \right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {3 B a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C b}{2 d}+\frac {3 B a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C b}{2 d}\)	\(266\)
norman	\(\frac {\frac {\left (3 B a -8 B b -8 C a -36 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{12 d}-\frac {\left (3 B a +8 B b +8 C a -36 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{12 d}+\frac {\left (5 B a -8 B b -8 C a +4 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{4 d}-\frac {\left (5 B a +8 B b +8 C a +4 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {\left (27 B a -40 B b -40 C a -36 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{12 d}+\frac {\left (27 B a +40 B b +40 C a -36 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{12 d}+\frac {\left (39 B a -8 B b -8 C a +12 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{12 d}-\frac {\left (39 B a +8 B b +8 C a +12 C b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{5}}-\frac {\left (3 B a +4 C b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d}+\frac {\left (3 B a +4 C b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d}\)	\(348\)

Input:

int((a+b*cos(d*x+c))*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x,method=_ 
RETURNVERBOSE)
 

Output:

-(B*b+C*a)/d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+B*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)))+C*b/d*(1/2*sec( 
d*x+c)*tan(d*x+c)+1/2*ln(sec(d*x+c)+tan(d*x+c)))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.19 \[ \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {3 \, {\left (3 \, B a + 4 \, C b\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (3 \, B a + 4 \, C b\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (16 \, {\left (C a + B b\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (3 \, B a + 4 \, C b\right )} \cos \left (d x + c\right )^{2} + 6 \, B a + 8 \, {\left (C a + B b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \] Input:

integrate((a+b*cos(d*x+c))*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, a 
lgorithm="fricas")
 

Output:

1/48*(3*(3*B*a + 4*C*b)*cos(d*x + c)^4*log(sin(d*x + c) + 1) - 3*(3*B*a + 
4*C*b)*cos(d*x + c)^4*log(-sin(d*x + c) + 1) + 2*(16*(C*a + B*b)*cos(d*x + 
 c)^3 + 3*(3*B*a + 4*C*b)*cos(d*x + c)^2 + 6*B*a + 8*(C*a + B*b)*cos(d*x + 
 c))*sin(d*x + c))/(d*cos(d*x + c)^4)
 

Sympy [F(-1)]

Timed out. \[ \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\text {Timed out} \] Input:

integrate((a+b*cos(d*x+c))*(B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c)**6,x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.43 \[ \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a + 16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B b - 3 \, B a {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \] Input:

integrate((a+b*cos(d*x+c))*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, a 
lgorithm="maxima")
 

Output:

1/48*(16*(tan(d*x + c)^3 + 3*tan(d*x + c))*C*a + 16*(tan(d*x + c)^3 + 3*ta 
n(d*x + c))*B*b - 3*B*a*(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)) - 12*C*b*(2*sin(d*x + c)/(sin(d*x + c)^2 - 1) - log(sin(d*x + c) + 
 1) + log(sin(d*x + c) - 1)))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 304 vs. \(2 (106) = 212\).

Time = 0.13 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.67 \[ \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {3 \, {\left (3 \, B a + 4 \, C b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (3 \, B a + 4 \, C b\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (15 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, B a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, C a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, B b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, C b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{4}}}{24 \, d} \] Input:

integrate((a+b*cos(d*x+c))*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x, a 
lgorithm="giac")
 

Output:

1/24*(3*(3*B*a + 4*C*b)*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 3*(3*B*a + 4* 
C*b)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) + 2*(15*B*a*tan(1/2*d*x + 1/2*c)^7 
 - 24*C*a*tan(1/2*d*x + 1/2*c)^7 - 24*B*b*tan(1/2*d*x + 1/2*c)^7 + 12*C*b* 
tan(1/2*d*x + 1/2*c)^7 + 9*B*a*tan(1/2*d*x + 1/2*c)^5 + 40*C*a*tan(1/2*d*x 
 + 1/2*c)^5 + 40*B*b*tan(1/2*d*x + 1/2*c)^5 - 12*C*b*tan(1/2*d*x + 1/2*c)^ 
5 + 9*B*a*tan(1/2*d*x + 1/2*c)^3 - 40*C*a*tan(1/2*d*x + 1/2*c)^3 - 40*B*b* 
tan(1/2*d*x + 1/2*c)^3 - 12*C*b*tan(1/2*d*x + 1/2*c)^3 + 15*B*a*tan(1/2*d* 
x + 1/2*c) + 24*C*a*tan(1/2*d*x + 1/2*c) + 24*B*b*tan(1/2*d*x + 1/2*c) + 1 
2*C*b*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 = 2.70 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.70 \[ \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {\left (\frac {5\,B\,a}{4}-2\,B\,b-2\,C\,a+C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {3\,B\,a}{4}+\frac {10\,B\,b}{3}+\frac {10\,C\,a}{3}-C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {3\,B\,a}{4}-\frac {10\,B\,b}{3}-\frac {10\,C\,a}{3}-C\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {5\,B\,a}{4}+2\,B\,b+2\,C\,a+C\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (\frac {3\,B\,a}{4}+C\,b\right )}{d} \] Input:

int(((B*cos(c + d*x) + C*cos(c + d*x)^2)*(a + b*cos(c + d*x)))/cos(c + d*x 
)^6,x)
 

Output:

(tan(c/2 + (d*x)/2)*((5*B*a)/4 + 2*B*b + 2*C*a + C*b) + tan(c/2 + (d*x)/2) 
^7*((5*B*a)/4 - 2*B*b - 2*C*a + C*b) - tan(c/2 + (d*x)/2)^3*((10*B*b)/3 - 
(3*B*a)/4 + (10*C*a)/3 + C*b) + tan(c/2 + (d*x)/2)^5*((3*B*a)/4 + (10*B*b) 
/3 + (10*C*a)/3 - C*b))/(d*(6*tan(c/2 + (d*x)/2)^4 - 4*tan(c/2 + (d*x)/2)^ 
2 - 4*tan(c/2 + (d*x)/2)^6 + tan(c/2 + (d*x)/2)^8 + 1)) + (atanh(tan(c/2 + 
 (d*x)/2))*((3*B*a)/4 + C*b))/d
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 398, normalized size of antiderivative = 3.49 \[ \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^6(c+d x) \, dx=\frac {-16 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a c -16 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{2}+24 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a c +24 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{2}-9 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{4} a b -12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{4} b c +18 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2} a b +24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) \sin \left (d x +c \right )^{2} b c -9 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a b -12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b c +9 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{4} a b +12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{4} b c -18 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2} a b -24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) \sin \left (d x +c \right )^{2} b c +9 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a b +12 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b c -9 \sin \left (d x +c \right )^{3} a b -12 \sin \left (d x +c \right )^{3} b c +15 \sin \left (d x +c \right ) a b +12 \sin \left (d x +c \right ) b c}{24 d \left (\sin \left (d x +c \right )^{4}-2 \sin \left (d x +c \right )^{2}+1\right )} \] Input:

int((a+b*cos(d*x+c))*(B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c)^6,x)
 

Output:

( - 16*cos(c + d*x)*sin(c + d*x)**3*a*c - 16*cos(c + d*x)*sin(c + d*x)**3* 
b**2 + 24*cos(c + d*x)*sin(c + d*x)*a*c + 24*cos(c + d*x)*sin(c + d*x)*b** 
2 - 9*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**4*a*b - 12*log(tan((c + d*x) 
/2) - 1)*sin(c + d*x)**4*b*c + 18*log(tan((c + d*x)/2) - 1)*sin(c + d*x)** 
2*a*b + 24*log(tan((c + d*x)/2) - 1)*sin(c + d*x)**2*b*c - 9*log(tan((c + 
d*x)/2) - 1)*a*b - 12*log(tan((c + d*x)/2) - 1)*b*c + 9*log(tan((c + d*x)/ 
2) + 1)*sin(c + d*x)**4*a*b + 12*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**4 
*b*c - 18*log(tan((c + d*x)/2) + 1)*sin(c + d*x)**2*a*b - 24*log(tan((c + 
d*x)/2) + 1)*sin(c + d*x)**2*b*c + 9*log(tan((c + d*x)/2) + 1)*a*b + 12*lo 
g(tan((c + d*x)/2) + 1)*b*c - 9*sin(c + d*x)**3*a*b - 12*sin(c + d*x)**3*b 
*c + 15*sin(c + d*x)*a*b + 12*sin(c + d*x)*b*c)/(24*d*(sin(c + d*x)**4 - 2 
*sin(c + d*x)**2 + 1))