\(\int (a+b \cos (c+d x))^3 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec (c+d x) \, dx\) [957]

Optimal result

Integrand size = 39, antiderivative size = 207 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {1}{8} \left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) x+\frac {a^3 A \text {arctanh}(\sin (c+d x))}{d}+\frac {\left (16 a^2 b B+4 b^3 B+3 a^3 C+6 a b^2 (3 A+2 C)\right ) \sin (c+d x)}{6 d}+\frac {b \left (12 A b^2+20 a b B+6 a^2 C+9 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac {(4 b B+3 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac {C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d} \] Output:

1/8*(8*B*a^3+12*B*a*b^2+12*a^2*b*(2*A+C)+b^3*(4*A+3*C))*x+a^3*A*arctanh(si 
n(d*x+c))/d+1/6*(16*B*a^2*b+4*B*b^3+3*a^3*C+6*a*b^2*(3*A+2*C))*sin(d*x+c)/ 
d+1/24*b*(12*A*b^2+20*B*a*b+6*C*a^2+9*C*b^2)*cos(d*x+c)*sin(d*x+c)/d+1/12* 
(4*B*b+3*C*a)*(a+b*cos(d*x+c))^2*sin(d*x+c)/d+1/4*C*(a+b*cos(d*x+c))^3*sin 
(d*x+c)/d
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 3.04 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.05 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {12 \left (8 a^3 B+12 a b^2 B+12 a^2 b (2 A+C)+b^3 (4 A+3 C)\right ) (c+d x)-96 a^3 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+96 a^3 A \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 \left (12 a^2 b B+3 b^3 B+4 a^3 C+3 a b^2 (4 A+3 C)\right ) \sin (c+d x)+24 b \left (A b^2+3 a b B+3 a^2 C+b^2 C\right ) \sin (2 (c+d x))+8 b^2 (b B+3 a C) \sin (3 (c+d x))+3 b^3 C \sin (4 (c+d x))}{96 d} \] Input:

Integrate[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*S 
ec[c + d*x],x]
 

Output:

(12*(8*a^3*B + 12*a*b^2*B + 12*a^2*b*(2*A + C) + b^3*(4*A + 3*C))*(c + d*x 
) - 96*a^3*A*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 96*a^3*A*Log[Cos[( 
c + d*x)/2] + Sin[(c + d*x)/2]] + 24*(12*a^2*b*B + 3*b^3*B + 4*a^3*C + 3*a 
*b^2*(4*A + 3*C))*Sin[c + d*x] + 24*b*(A*b^2 + 3*a*b*B + 3*a^2*C + b^2*C)* 
Sin[2*(c + d*x)] + 8*b^2*(b*B + 3*a*C)*Sin[3*(c + d*x)] + 3*b^3*C*Sin[4*(c 
 + d*x)])/(96*d)
 

Rubi [A] (verified)

Time = 1.43 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3042, 3528, 3042, 3528, 3042, 3512, 3042, 3502, 27, 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.

\(\displaystyle \int \sec (c+d x) (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int (a+b \cos (c+d x)) \left (12 A a^2+\left (6 C a^2+20 b B a+12 A b^2+9 b^2 C\right ) \cos ^2(c+d x)+\left (12 B a^2+24 A b a+15 b C a+8 b^2 B\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (12 A a^2+\left (6 C a^2+20 b B a+12 A b^2+9 b^2 C\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (12 B a^2+24 A b a+15 b C a+8 b^2 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\)

\(\Big \downarrow \) 3512

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \left (24 A a^3+4 \left (3 C a^3+16 b B a^2+6 b^2 (3 A+2 C) a+4 b^3 B\right ) \cos ^2(c+d x)+3 \left (8 B a^3+12 b (2 A+C) a^2+12 b^2 B a+b^3 (4 A+3 C)\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {b \sin (c+d x) \cos (c+d x) \left (6 a^2 C+20 a b B+12 A b^2+9 b^2 C\right )}{2 d}\right )+\frac {(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {24 A a^3+4 \left (3 C a^3+16 b B a^2+6 b^2 (3 A+2 C) a+4 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+3 \left (8 B a^3+12 b (2 A+C) a^2+12 b^2 B a+b^3 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {b \sin (c+d x) \cos (c+d x) \left (6 a^2 C+20 a b B+12 A b^2+9 b^2 C\right )}{2 d}\right )+\frac {(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int 3 \left (8 A a^3+\left (8 B a^3+12 b (2 A+C) a^2+12 b^2 B a+b^3 (4 A+3 C)\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {4 \sin (c+d x) \left (3 a^3 C+16 a^2 b B+6 a b^2 (3 A+2 C)+4 b^3 B\right )}{d}\right )+\frac {b \sin (c+d x) \cos (c+d x) \left (6 a^2 C+20 a b B+12 A b^2+9 b^2 C\right )}{2 d}\right )+\frac {(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \int \left (8 A a^3+\left (8 B a^3+12 b (2 A+C) a^2+12 b^2 B a+b^3 (4 A+3 C)\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {4 \sin (c+d x) \left (3 a^3 C+16 a^2 b B+6 a b^2 (3 A+2 C)+4 b^3 B\right )}{d}\right )+\frac {b \sin (c+d x) \cos (c+d x) \left (6 a^2 C+20 a b B+12 A b^2+9 b^2 C\right )}{2 d}\right )+\frac {(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \int \frac {8 A a^3+\left (8 B a^3+12 b (2 A+C) a^2+12 b^2 B a+b^3 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {4 \sin (c+d x) \left (3 a^3 C+16 a^2 b B+6 a b^2 (3 A+2 C)+4 b^3 B\right )}{d}\right )+\frac {b \sin (c+d x) \cos (c+d x) \left (6 a^2 C+20 a b B+12 A b^2+9 b^2 C\right )}{2 d}\right )+\frac {(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\)

\(\Big \downarrow \) 3214

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \left (8 a^3 A \int \sec (c+d x)dx+x \left (8 a^3 B+12 a^2 b (2 A+C)+12 a b^2 B+b^3 (4 A+3 C)\right )\right )+\frac {4 \sin (c+d x) \left (3 a^3 C+16 a^2 b B+6 a b^2 (3 A+2 C)+4 b^3 B\right )}{d}\right )+\frac {b \sin (c+d x) \cos (c+d x) \left (6 a^2 C+20 a b B+12 A b^2+9 b^2 C\right )}{2 d}\right )+\frac {(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \left (8 a^3 A \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+x \left (8 a^3 B+12 a^2 b (2 A+C)+12 a b^2 B+b^3 (4 A+3 C)\right )\right )+\frac {4 \sin (c+d x) \left (3 a^3 C+16 a^2 b B+6 a b^2 (3 A+2 C)+4 b^3 B\right )}{d}\right )+\frac {b \sin (c+d x) \cos (c+d x) \left (6 a^2 C+20 a b B+12 A b^2+9 b^2 C\right )}{2 d}\right )+\frac {(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {b \sin (c+d x) \cos (c+d x) \left (6 a^2 C+20 a b B+12 A b^2+9 b^2 C\right )}{2 d}+\frac {1}{2} \left (3 \left (\frac {8 a^3 A \text {arctanh}(\sin (c+d x))}{d}+x \left (8 a^3 B+12 a^2 b (2 A+C)+12 a b^2 B+b^3 (4 A+3 C)\right )\right )+\frac {4 \sin (c+d x) \left (3 a^3 C+16 a^2 b B+6 a b^2 (3 A+2 C)+4 b^3 B\right )}{d}\right )\right )+\frac {(3 a C+4 b B) \sin (c+d x) (a+b \cos (c+d x))^2}{3 d}\right )+\frac {C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}\)

Input:

Int[(a + b*Cos[c + d*x])^3*(A + B*Cos[c + d*x] + C*Cos[c + d*x]^2)*Sec[c + 
 d*x],x]
 

Output:

(C*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/(4*d) + (((4*b*B + 3*a*C)*(a + b*C 
os[c + d*x])^2*Sin[c + d*x])/(3*d) + ((b*(12*A*b^2 + 20*a*b*B + 6*a^2*C + 
9*b^2*C)*Cos[c + d*x]*Sin[c + d*x])/(2*d) + (3*((8*a^3*B + 12*a*b^2*B + 12 
*a^2*b*(2*A + C) + b^3*(4*A + 3*C))*x + (8*a^3*A*ArcTanh[Sin[c + d*x]])/d) 
 + (4*(16*a^2*b*B + 4*b^3*B + 3*a^3*C + 6*a*b^2*(3*A + 2*C))*Sin[c + d*x]) 
/d)/2)/3)/4
 

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 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 3512

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

rule 3528

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)*Cos[e + f*x]*(a + b*Sin[e + f*x 
])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + 
n + 2))   Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* 
d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a 
*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + 
 n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} 
, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ 
m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 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 = 2.02 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.92

method	result	size
parallelrisch	\(\frac {-96 A \,a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+96 A \,a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+24 b \left (\left (A +C \right ) b^{2}+3 B a b +3 a^{2} C \right ) \sin \left (2 d x +2 c \right )+\left (8 B \,b^{3}+24 C a \,b^{2}\right ) \sin \left (3 d x +3 c \right )+3 C \sin \left (4 d x +4 c \right ) b^{3}+\left (72 B \,b^{3}+288 a \left (A +\frac {3 C}{4}\right ) b^{2}+288 B \,a^{2} b +96 a^{3} C \right ) \sin \left (d x +c \right )+288 x d \left (\left (\frac {A}{6}+\frac {C}{8}\right ) b^{3}+\frac {B a \,b^{2}}{2}+a^{2} \left (A +\frac {C}{2}\right ) b +\frac {B \,a^{3}}{3}\right )}{96 d}\)	\(190\)
parts	\(\frac {A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) \left (d x +c \right )}{d}+\frac {\left (B \,b^{3}+3 C a \,b^{2}\right ) \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (A \,b^{3}+3 B a \,b^{2}+3 a^{2} b C \right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (3 a A \,b^{2}+3 B \,a^{2} b +a^{3} C \right ) \sin \left (d x +c \right )}{d}+\frac {C \,b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(197\)
derivativedivides	\(\frac {A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{3} \left (d x +c \right )+C \sin \left (d x +c \right ) a^{3}+3 A \,a^{2} b \left (d x +c \right )+3 B \sin \left (d x +c \right ) a^{2} b +3 a^{2} b C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 A \sin \left (d x +c \right ) a \,b^{2}+3 B a \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C a \,b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+A \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {B \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+C \,b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(251\)
default	\(\frac {A \,a^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,a^{3} \left (d x +c \right )+C \sin \left (d x +c \right ) a^{3}+3 A \,a^{2} b \left (d x +c \right )+3 B \sin \left (d x +c \right ) a^{2} b +3 a^{2} b C \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 A \sin \left (d x +c \right ) a \,b^{2}+3 B a \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C a \,b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+A \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {B \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+C \,b^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(251\)
risch	\(\frac {A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}+\frac {\sin \left (4 d x +4 c \right ) C \,b^{3}}{32 d}+\frac {\sin \left (3 d x +3 c \right ) B \,b^{3}}{12 d}+\frac {\sin \left (2 d x +2 c \right ) A \,b^{3}}{4 d}+\frac {\sin \left (2 d x +2 c \right ) C \,b^{3}}{4 d}+3 x A \,a^{2} b +\frac {3 x B a \,b^{2}}{2}+\frac {3 x \,a^{2} b C}{2}+\frac {3 b^{3} C x}{8}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a A \,b^{2}}{2 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{2} b}{2 d}+\frac {\sin \left (3 d x +3 c \right ) C a \,b^{2}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) B a \,b^{2}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) a^{2} b C}{4 d}+\frac {9 i {\mathrm e}^{-i \left (d x +c \right )} C a \,b^{2}}{8 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a A \,b^{2}}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{2} b}{2 d}-\frac {9 i {\mathrm e}^{i \left (d x +c \right )} C a \,b^{2}}{8 d}+\frac {x A \,b^{3}}{2}+x B \,a^{3}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B \,b^{3}}{8 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} a^{3} C}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} B \,b^{3}}{8 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} a^{3} C}{2 d}\)	\(414\)
norman	\(\frac {\left (\frac {3}{2} a^{2} b C +\frac {3}{2} B a \,b^{2}+\frac {1}{2} A \,b^{3}+B \,a^{3}+3 A \,a^{2} b +\frac {3}{8} C \,b^{3}\right ) x +\left (15 a^{2} b C +15 B a \,b^{2}+5 A \,b^{3}+10 B \,a^{3}+30 A \,a^{2} b +\frac {15}{4} C \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (15 a^{2} b C +15 B a \,b^{2}+5 A \,b^{3}+10 B \,a^{3}+30 A \,a^{2} b +\frac {15}{4} C \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {3}{2} a^{2} b C +\frac {3}{2} B a \,b^{2}+\frac {1}{2} A \,b^{3}+B \,a^{3}+3 A \,a^{2} b +\frac {3}{8} C \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {15}{2} a^{2} b C +\frac {15}{2} B a \,b^{2}+\frac {5}{2} A \,b^{3}+5 B \,a^{3}+15 A \,a^{2} b +\frac {15}{8} C \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {15}{2} a^{2} b C +\frac {15}{2} B a \,b^{2}+\frac {5}{2} A \,b^{3}+5 B \,a^{3}+15 A \,a^{2} b +\frac {15}{8} C \,b^{3}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\frac {4 \left (27 a A \,b^{2}+27 B \,a^{2} b +5 B \,b^{3}+9 a^{3} C +15 C a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{3 d}+\frac {\left (24 a A \,b^{2}-4 A \,b^{3}+24 B \,a^{2} b -12 B a \,b^{2}+8 B \,b^{3}+8 a^{3} C -12 a^{2} b C +24 C a \,b^{2}-5 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}+\frac {\left (24 a A \,b^{2}+4 A \,b^{3}+24 B \,a^{2} b +12 B a \,b^{2}+8 B \,b^{3}+8 a^{3} C +12 a^{2} b C +24 C a \,b^{2}+5 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (144 a A \,b^{2}-12 A \,b^{3}+144 B \,a^{2} b -36 B a \,b^{2}+32 B \,b^{3}+48 a^{3} C -36 a^{2} b C +96 C a \,b^{2}-3 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}+\frac {\left (144 a A \,b^{2}+12 A \,b^{3}+144 B \,a^{2} b +36 B a \,b^{2}+32 B \,b^{3}+48 a^{3} C +36 a^{2} b C +96 C a \,b^{2}+3 C \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}+\frac {A \,a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}-\frac {A \,a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}\)	\(717\)

Input:

int((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c),x,method 
=_RETURNVERBOSE)
 

Output:

1/96*(-96*A*a^3*ln(tan(1/2*d*x+1/2*c)-1)+96*A*a^3*ln(tan(1/2*d*x+1/2*c)+1) 
+24*b*((A+C)*b^2+3*B*a*b+3*a^2*C)*sin(2*d*x+2*c)+(8*B*b^3+24*C*a*b^2)*sin( 
3*d*x+3*c)+3*C*sin(4*d*x+4*c)*b^3+(72*B*b^3+288*a*(A+3/4*C)*b^2+288*B*a^2* 
b+96*a^3*C)*sin(d*x+c)+288*x*d*((1/6*A+1/8*C)*b^3+1/2*B*a*b^2+a^2*(A+1/2*C 
)*b+1/3*B*a^3))/d
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.91 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {12 \, A a^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, A a^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (8 \, B a^{3} + 12 \, {\left (2 \, A + C\right )} a^{2} b + 12 \, B a b^{2} + {\left (4 \, A + 3 \, C\right )} b^{3}\right )} d x + {\left (6 \, C b^{3} \cos \left (d x + c\right )^{3} + 24 \, C a^{3} + 72 \, B a^{2} b + 24 \, {\left (3 \, A + 2 \, C\right )} a b^{2} + 16 \, B b^{3} + 8 \, {\left (3 \, C a b^{2} + B b^{3}\right )} \cos \left (d x + c\right )^{2} + 3 \, {\left (12 \, C a^{2} b + 12 \, B a b^{2} + {\left (4 \, A + 3 \, C\right )} b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \] Input:

integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c),x, 
 algorithm="fricas")
 

Output:

1/24*(12*A*a^3*log(sin(d*x + c) + 1) - 12*A*a^3*log(-sin(d*x + c) + 1) + 3 
*(8*B*a^3 + 12*(2*A + C)*a^2*b + 12*B*a*b^2 + (4*A + 3*C)*b^3)*d*x + (6*C* 
b^3*cos(d*x + c)^3 + 24*C*a^3 + 72*B*a^2*b + 24*(3*A + 2*C)*a*b^2 + 16*B*b 
^3 + 8*(3*C*a*b^2 + B*b^3)*cos(d*x + c)^2 + 3*(12*C*a^2*b + 12*B*a*b^2 + ( 
4*A + 3*C)*b^3)*cos(d*x + c))*sin(d*x + c))/d
 

Sympy [F]

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

integrate((a+b*cos(d*x+c))**3*(A+B*cos(d*x+c)+C*cos(d*x+c)**2)*sec(d*x+c), 
x)
 

Output:

Integral((a + b*cos(c + d*x))**3*(A + B*cos(c + d*x) + C*cos(c + d*x)**2)* 
sec(c + d*x), x)
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.15 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {96 \, {\left (d x + c\right )} B a^{3} + 288 \, {\left (d x + c\right )} A a^{2} b + 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} b + 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b^{2} - 96 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a b^{2} + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{3} - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B b^{3} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{3} + 96 \, A a^{3} \log \left (\sec \left (d x + c\right ) + \tan \left (d x + c\right )\right ) + 96 \, C a^{3} \sin \left (d x + c\right ) + 288 \, B a^{2} b \sin \left (d x + c\right ) + 288 \, A a b^{2} \sin \left (d x + c\right )}{96 \, d} \] Input:

integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c),x, 
 algorithm="maxima")
 

Output:

1/96*(96*(d*x + c)*B*a^3 + 288*(d*x + c)*A*a^2*b + 72*(2*d*x + 2*c + sin(2 
*d*x + 2*c))*C*a^2*b + 72*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a*b^2 - 96*(s 
in(d*x + c)^3 - 3*sin(d*x + c))*C*a*b^2 + 24*(2*d*x + 2*c + sin(2*d*x + 2* 
c))*A*b^3 - 32*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*b^3 + 3*(12*d*x + 12*c 
+ sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*b^3 + 96*A*a^3*log(sec(d*x + c) 
 + tan(d*x + c)) + 96*C*a^3*sin(d*x + c) + 288*B*a^2*b*sin(d*x + c) + 288* 
A*a*b^2*sin(d*x + c))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 723 vs. \(2 (197) = 394\).

Time = 0.17 (sec) , antiderivative size = 723, normalized size of antiderivative = 3.49 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\text {Too large to display} \] Input:

integrate((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c),x, 
 algorithm="giac")
 

Output:

1/24*(24*A*a^3*log(abs(tan(1/2*d*x + 1/2*c) + 1)) - 24*A*a^3*log(abs(tan(1 
/2*d*x + 1/2*c) - 1)) + 3*(8*B*a^3 + 24*A*a^2*b + 12*C*a^2*b + 12*B*a*b^2 
+ 4*A*b^3 + 3*C*b^3)*(d*x + c) + 2*(24*C*a^3*tan(1/2*d*x + 1/2*c)^7 + 72*B 
*a^2*b*tan(1/2*d*x + 1/2*c)^7 - 36*C*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 72*A*a 
*b^2*tan(1/2*d*x + 1/2*c)^7 - 36*B*a*b^2*tan(1/2*d*x + 1/2*c)^7 + 72*C*a*b 
^2*tan(1/2*d*x + 1/2*c)^7 - 12*A*b^3*tan(1/2*d*x + 1/2*c)^7 + 24*B*b^3*tan 
(1/2*d*x + 1/2*c)^7 - 15*C*b^3*tan(1/2*d*x + 1/2*c)^7 + 72*C*a^3*tan(1/2*d 
*x + 1/2*c)^5 + 216*B*a^2*b*tan(1/2*d*x + 1/2*c)^5 - 36*C*a^2*b*tan(1/2*d* 
x + 1/2*c)^5 + 216*A*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 36*B*a*b^2*tan(1/2*d*x 
 + 1/2*c)^5 + 120*C*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 12*A*b^3*tan(1/2*d*x + 
1/2*c)^5 + 40*B*b^3*tan(1/2*d*x + 1/2*c)^5 + 9*C*b^3*tan(1/2*d*x + 1/2*c)^ 
5 + 72*C*a^3*tan(1/2*d*x + 1/2*c)^3 + 216*B*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 
 36*C*a^2*b*tan(1/2*d*x + 1/2*c)^3 + 216*A*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 
36*B*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 120*C*a*b^2*tan(1/2*d*x + 1/2*c)^3 + 1 
2*A*b^3*tan(1/2*d*x + 1/2*c)^3 + 40*B*b^3*tan(1/2*d*x + 1/2*c)^3 - 9*C*b^3 
*tan(1/2*d*x + 1/2*c)^3 + 24*C*a^3*tan(1/2*d*x + 1/2*c) + 72*B*a^2*b*tan(1 
/2*d*x + 1/2*c) + 36*C*a^2*b*tan(1/2*d*x + 1/2*c) + 72*A*a*b^2*tan(1/2*d*x 
 + 1/2*c) + 36*B*a*b^2*tan(1/2*d*x + 1/2*c) + 72*C*a*b^2*tan(1/2*d*x + 1/2 
*c) + 12*A*b^3*tan(1/2*d*x + 1/2*c) + 24*B*b^3*tan(1/2*d*x + 1/2*c) + 15*C 
*b^3*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 = 2.29 (sec) , antiderivative size = 3250, normalized size of antiderivative = 15.70 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\text {Too large to display} \] Input:

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

Output:

(tan(c/2 + (d*x)/2)^7*(2*B*b^3 - A*b^3 + 2*C*a^3 - (5*C*b^3)/4 + 6*A*a*b^2 
 - 3*B*a*b^2 + 6*B*a^2*b + 6*C*a*b^2 - 3*C*a^2*b) + tan(c/2 + (d*x)/2)^3*( 
A*b^3 + (10*B*b^3)/3 + 6*C*a^3 - (3*C*b^3)/4 + 18*A*a*b^2 + 3*B*a*b^2 + 18 
*B*a^2*b + 10*C*a*b^2 + 3*C*a^2*b) + tan(c/2 + (d*x)/2)^5*((10*B*b^3)/3 - 
A*b^3 + 6*C*a^3 + (3*C*b^3)/4 + 18*A*a*b^2 - 3*B*a*b^2 + 18*B*a^2*b + 10*C 
*a*b^2 - 3*C*a^2*b) + tan(c/2 + (d*x)/2)*(A*b^3 + 2*B*b^3 + 2*C*a^3 + (5*C 
*b^3)/4 + 6*A*a*b^2 + 3*B*a*b^2 + 6*B*a^2*b + 6*C*a*b^2 + 3*C*a^2*b))/(d*( 
4*tan(c/2 + (d*x)/2)^2 + 6*tan(c/2 + (d*x)/2)^4 + 4*tan(c/2 + (d*x)/2)^6 + 
 tan(c/2 + (d*x)/2)^8 + 1)) + (atan(((((A*b^3*1i)/2 + B*a^3*1i + (C*b^3*3i 
)/8 + A*a^2*b*3i + (B*a*b^2*3i)/2 + (C*a^2*b*3i)/2)*(32*A*a^3 + 16*A*b^3 + 
 32*B*a^3 + 12*C*b^3 + 96*A*a^2*b + 48*B*a*b^2 + 48*C*a^2*b) + tan(c/2 + ( 
d*x)/2)*(32*A^2*a^6 + 8*A^2*b^6 + 32*B^2*a^6 + (9*C^2*b^6)/2 + 96*A^2*a^2* 
b^4 + 288*A^2*a^4*b^2 + 72*B^2*a^2*b^4 + 96*B^2*a^4*b^2 + 36*C^2*a^2*b^4 + 
 72*C^2*a^4*b^2 + 12*A*C*b^6 + 48*A*B*a*b^5 + 192*A*B*a^5*b + 36*B*C*a*b^5 
 + 96*B*C*a^5*b + 320*A*B*a^3*b^3 + 120*A*C*a^2*b^4 + 288*A*C*a^4*b^2 + 16 
8*B*C*a^3*b^3))*((A*b^3*1i)/2 + B*a^3*1i + (C*b^3*3i)/8 + A*a^2*b*3i + (B* 
a*b^2*3i)/2 + (C*a^2*b*3i)/2)*1i - (((A*b^3*1i)/2 + B*a^3*1i + (C*b^3*3i)/ 
8 + A*a^2*b*3i + (B*a*b^2*3i)/2 + (C*a^2*b*3i)/2)*(32*A*a^3 + 16*A*b^3 + 3 
2*B*a^3 + 12*C*b^3 + 96*A*a^2*b + 48*B*a*b^2 + 48*C*a^2*b) - tan(c/2 + (d* 
x)/2)*(32*A^2*a^6 + 8*A^2*b^6 + 32*B^2*a^6 + (9*C^2*b^6)/2 + 96*A^2*a^2...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.09 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec (c+d x) \, dx=\frac {-6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{3} c +36 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b c +48 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a \,b^{3}+15 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{3} c -24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a^{4}+24 \,\mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a^{4}-24 \sin \left (d x +c \right )^{3} a \,b^{2} c -8 \sin \left (d x +c \right )^{3} b^{4}+24 \sin \left (d x +c \right ) a^{3} c +144 \sin \left (d x +c \right ) a^{2} b^{2}+72 \sin \left (d x +c \right ) a \,b^{2} c +24 \sin \left (d x +c \right ) b^{4}+96 a^{3} b d x +36 a^{2} b c d x +48 a \,b^{3} d x +9 b^{3} c d x}{24 d} \] Input:

int((a+b*cos(d*x+c))^3*(A+B*cos(d*x+c)+C*cos(d*x+c)^2)*sec(d*x+c),x)
 

Output:

( - 6*cos(c + d*x)*sin(c + d*x)**3*b**3*c + 36*cos(c + d*x)*sin(c + d*x)*a 
**2*b*c + 48*cos(c + d*x)*sin(c + d*x)*a*b**3 + 15*cos(c + d*x)*sin(c + d* 
x)*b**3*c - 24*log(tan((c + d*x)/2) - 1)*a**4 + 24*log(tan((c + d*x)/2) + 
1)*a**4 - 24*sin(c + d*x)**3*a*b**2*c - 8*sin(c + d*x)**3*b**4 + 24*sin(c 
+ d*x)*a**3*c + 144*sin(c + d*x)*a**2*b**2 + 72*sin(c + d*x)*a*b**2*c + 24 
*sin(c + d*x)*b**4 + 96*a**3*b*d*x + 36*a**2*b*c*d*x + 48*a*b**3*d*x + 9*b 
**3*c*d*x)/(24*d)