\(\int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)+C \cos ^2(c+d x)) \sec ^2(c+d x) \, dx\) [967]

Optimal result

Integrand size = 41, antiderivative size = 273 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {1}{8} \left (32 a^3 b B+16 a b^3 B+8 a^4 C+24 a^2 b^2 (2 A+C)+b^4 (4 A+3 C)\right ) x+\frac {a^3 (4 A b+a B) \text {arctanh}(\sin (c+d x))}{d}+\frac {b \left (34 a^2 b B+4 b^3 B-a^3 (12 A-19 C)+8 a b^2 (3 A+2 C)\right ) \sin (c+d x)}{6 d}+\frac {b^2 \left (32 a b B-a^2 (24 A-26 C)+3 b^2 (4 A+3 C)\right ) \cos (c+d x) \sin (c+d x)}{24 d}-\frac {b (12 a A-4 b B-7 a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}-\frac {b (4 A-C) (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac {A (a+b \cos (c+d x))^4 \tan (c+d x)}{d} \] Output:

1/8*(32*B*a^3*b+16*B*a*b^3+8*a^4*C+24*a^2*b^2*(2*A+C)+b^4*(4*A+3*C))*x+a^3 
*(4*A*b+B*a)*arctanh(sin(d*x+c))/d+1/6*b*(34*B*a^2*b+4*B*b^3-a^3*(12*A-19* 
C)+8*a*b^2*(3*A+2*C))*sin(d*x+c)/d+1/24*b^2*(32*B*a*b-a^2*(24*A-26*C)+3*b^ 
2*(4*A+3*C))*cos(d*x+c)*sin(d*x+c)/d-1/12*b*(12*A*a-4*B*b-7*C*a)*(a+b*cos( 
d*x+c))^2*sin(d*x+c)/d-1/4*b*(4*A-C)*(a+b*cos(d*x+c))^3*sin(d*x+c)/d+A*(a+ 
b*cos(d*x+c))^4*tan(d*x+c)/d
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 6.78 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.40 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {32 b \left (36 a^2 b B+5 b^3 B+24 a^3 C+4 a b^2 (6 A+5 C)\right ) \sin (c+d x)+b^2 \sec (c+d x) \left (3 \left (8 A b^2+32 a b B+48 a^2 C+9 b^2 C\right ) \sin (3 (c+d x))+b (8 (b B+4 a C) \sin (4 (c+d x))+3 b C \sin (5 (c+d x)))\right )+24 \left (48 a^2 A b^2 c+4 A b^4 c+32 a^3 b B c+16 a b^3 B c+8 a^4 c C+24 a^2 b^2 c C+3 b^4 c C+48 a^2 A b^2 d x+4 A b^4 d x+32 a^3 b B d x+16 a b^3 B d x+8 a^4 C d x+24 a^2 b^2 C d x+3 b^4 C d x-8 a^3 (4 A b+a B) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+32 a^3 A b \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+8 a^4 B \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\left (8 a^4 A+4 a b^3 B+6 a^2 b^2 C+b^4 (A+C)\right ) \tan (c+d x)\right )}{192 d} \] Input:

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

Output:

(32*b*(36*a^2*b*B + 5*b^3*B + 24*a^3*C + 4*a*b^2*(6*A + 5*C))*Sin[c + d*x] 
 + b^2*Sec[c + d*x]*(3*(8*A*b^2 + 32*a*b*B + 48*a^2*C + 9*b^2*C)*Sin[3*(c 
+ d*x)] + b*(8*(b*B + 4*a*C)*Sin[4*(c + d*x)] + 3*b*C*Sin[5*(c + d*x)])) + 
 24*(48*a^2*A*b^2*c + 4*A*b^4*c + 32*a^3*b*B*c + 16*a*b^3*B*c + 8*a^4*c*C 
+ 24*a^2*b^2*c*C + 3*b^4*c*C + 48*a^2*A*b^2*d*x + 4*A*b^4*d*x + 32*a^3*b*B 
*d*x + 16*a*b^3*B*d*x + 8*a^4*C*d*x + 24*a^2*b^2*C*d*x + 3*b^4*C*d*x - 8*a 
^3*(4*A*b + a*B)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 32*a^3*A*b*Log 
[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + 8*a^4*B*Log[Cos[(c + d*x)/2] + Sin 
[(c + d*x)/2]] + (8*a^4*A + 4*a*b^3*B + 6*a^2*b^2*C + b^4*(A + C))*Tan[c + 
 d*x]))/(192*d)
 

Rubi [A] (verified)

Time = 1.99 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.05, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.366, Rules used = {3042, 3526, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3526

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3528

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int (a+b \cos (c+d x)) \left (12 (4 A b+a B) a^2+b \left (-\left ((24 A-26 C) a^2\right )+32 b B a+3 b^2 (4 A+3 C)\right ) \cos ^2(c+d x)+\left (12 C a^3+36 b B a^2+b^2 (36 A+23 C) a+8 b^3 B\right ) \cos (c+d x)\right ) \sec (c+d x)dx-\frac {b \sin (c+d x) (12 a A-7 a C-4 b B) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \frac {\left (a+b \sin \left (c+d x+\frac {\pi }{2}\right )\right ) \left (12 (4 A b+a B) a^2+b \left (-\left ((24 A-26 C) a^2\right )+32 b B a+3 b^2 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+\left (12 C a^3+36 b B a^2+b^2 (36 A+23 C) a+8 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx-\frac {b \sin (c+d x) (12 a A-7 a C-4 b B) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\)

\(\Big \downarrow \) 3512

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \left (24 (4 A b+a B) a^3+4 b \left (-\left ((12 A-19 C) a^3\right )+34 b B a^2+8 b^2 (3 A+2 C) a+4 b^3 B\right ) \cos ^2(c+d x)+3 \left (8 C a^4+32 b B a^3+24 b^2 (2 A+C) a^2+16 b^3 B a+b^4 (4 A+3 C)\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {b^2 \sin (c+d x) \cos (c+d x) \left (-\left (a^2 (24 A-26 C)\right )+32 a b B+3 b^2 (4 A+3 C)\right )}{2 d}\right )-\frac {b \sin (c+d x) (12 a A-7 a C-4 b B) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \frac {24 (4 A b+a B) a^3+4 b \left (-\left ((12 A-19 C) a^3\right )+34 b B a^2+8 b^2 (3 A+2 C) a+4 b^3 B\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2+3 \left (8 C a^4+32 b B a^3+24 b^2 (2 A+C) a^2+16 b^3 B a+b^4 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {b^2 \sin (c+d x) \cos (c+d x) \left (-\left (a^2 (24 A-26 C)\right )+32 a b B+3 b^2 (4 A+3 C)\right )}{2 d}\right )-\frac {b \sin (c+d x) (12 a A-7 a C-4 b B) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (\int 3 \left (8 (4 A b+a B) a^3+\left (8 C a^4+32 b B a^3+24 b^2 (2 A+C) a^2+16 b^3 B a+b^4 (4 A+3 C)\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {4 b \sin (c+d x) \left (-\left (a^3 (12 A-19 C)\right )+34 a^2 b B+8 a b^2 (3 A+2 C)+4 b^3 B\right )}{d}\right )+\frac {b^2 \sin (c+d x) \cos (c+d x) \left (-\left (a^2 (24 A-26 C)\right )+32 a b B+3 b^2 (4 A+3 C)\right )}{2 d}\right )-\frac {b \sin (c+d x) (12 a A-7 a C-4 b B) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \int \left (8 (4 A b+a B) a^3+\left (8 C a^4+32 b B a^3+24 b^2 (2 A+C) a^2+16 b^3 B a+b^4 (4 A+3 C)\right ) \cos (c+d x)\right ) \sec (c+d x)dx+\frac {4 b \sin (c+d x) \left (-\left (a^3 (12 A-19 C)\right )+34 a^2 b B+8 a b^2 (3 A+2 C)+4 b^3 B\right )}{d}\right )+\frac {b^2 \sin (c+d x) \cos (c+d x) \left (-\left (a^2 (24 A-26 C)\right )+32 a b B+3 b^2 (4 A+3 C)\right )}{2 d}\right )-\frac {b \sin (c+d x) (12 a A-7 a C-4 b B) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \int \frac {8 (4 A b+a B) a^3+\left (8 C a^4+32 b B a^3+24 b^2 (2 A+C) a^2+16 b^3 B a+b^4 (4 A+3 C)\right ) \sin \left (c+d x+\frac {\pi }{2}\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx+\frac {4 b \sin (c+d x) \left (-\left (a^3 (12 A-19 C)\right )+34 a^2 b B+8 a b^2 (3 A+2 C)+4 b^3 B\right )}{d}\right )+\frac {b^2 \sin (c+d x) \cos (c+d x) \left (-\left (a^2 (24 A-26 C)\right )+32 a b B+3 b^2 (4 A+3 C)\right )}{2 d}\right )-\frac {b \sin (c+d x) (12 a A-7 a C-4 b B) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\)

\(\Big \downarrow \) 3214

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \left (8 a^3 (a B+4 A b) \int \sec (c+d x)dx+x \left (8 a^4 C+32 a^3 b B+24 a^2 b^2 (2 A+C)+16 a b^3 B+b^4 (4 A+3 C)\right )\right )+\frac {4 b \sin (c+d x) \left (-\left (a^3 (12 A-19 C)\right )+34 a^2 b B+8 a b^2 (3 A+2 C)+4 b^3 B\right )}{d}\right )+\frac {b^2 \sin (c+d x) \cos (c+d x) \left (-\left (a^2 (24 A-26 C)\right )+32 a b B+3 b^2 (4 A+3 C)\right )}{2 d}\right )-\frac {b \sin (c+d x) (12 a A-7 a C-4 b B) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (3 \left (8 a^3 (a B+4 A b) \int \csc \left (c+d x+\frac {\pi }{2}\right )dx+x \left (8 a^4 C+32 a^3 b B+24 a^2 b^2 (2 A+C)+16 a b^3 B+b^4 (4 A+3 C)\right )\right )+\frac {4 b \sin (c+d x) \left (-\left (a^3 (12 A-19 C)\right )+34 a^2 b B+8 a b^2 (3 A+2 C)+4 b^3 B\right )}{d}\right )+\frac {b^2 \sin (c+d x) \cos (c+d x) \left (-\left (a^2 (24 A-26 C)\right )+32 a b B+3 b^2 (4 A+3 C)\right )}{2 d}\right )-\frac {b \sin (c+d x) (12 a A-7 a C-4 b B) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\)

\(\Big \downarrow \) 4257

\(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {b^2 \sin (c+d x) \cos (c+d x) \left (-\left (a^2 (24 A-26 C)\right )+32 a b B+3 b^2 (4 A+3 C)\right )}{2 d}+\frac {1}{2} \left (\frac {4 b \sin (c+d x) \left (-\left (a^3 (12 A-19 C)\right )+34 a^2 b B+8 a b^2 (3 A+2 C)+4 b^3 B\right )}{d}+3 \left (\frac {8 a^3 (a B+4 A b) \text {arctanh}(\sin (c+d x))}{d}+x \left (8 a^4 C+32 a^3 b B+24 a^2 b^2 (2 A+C)+16 a b^3 B+b^4 (4 A+3 C)\right )\right )\right )\right )-\frac {b \sin (c+d x) (12 a A-7 a C-4 b B) (a+b \cos (c+d x))^2}{3 d}\right )-\frac {b (4 A-C) \sin (c+d x) (a+b \cos (c+d x))^3}{4 d}+\frac {A \tan (c+d x) (a+b \cos (c+d x))^4}{d}\)

Input:

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

Output:

-1/4*(b*(4*A - C)*(a + b*Cos[c + d*x])^3*Sin[c + d*x])/d + (-1/3*(b*(12*a* 
A - 4*b*B - 7*a*C)*(a + b*Cos[c + d*x])^2*Sin[c + d*x])/d + ((b^2*(32*a*b* 
B - a^2*(24*A - 26*C) + 3*b^2*(4*A + 3*C))*Cos[c + d*x]*Sin[c + d*x])/(2*d 
) + (3*((32*a^3*b*B + 16*a*b^3*B + 8*a^4*C + 24*a^2*b^2*(2*A + C) + b^4*(4 
*A + 3*C))*x + (8*a^3*(4*A*b + a*B)*ArcTanh[Sin[c + d*x]])/d) + (4*b*(34*a 
^2*b*B + 4*b^3*B - a^3*(12*A - 19*C) + 8*a*b^2*(3*A + 2*C))*Sin[c + d*x])/ 
d)/2)/3)/4 + (A*(a + b*Cos[c + d*x])^4*Tan[c + d*x])/d
 

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 3526

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/(d*(n + 1)*(c^2 - d^2))   Int[(a + b*Sin[e + f*x])^(m - 
 1)*(c + d*Sin[e + f*x])^(n + 1)*Simp[A*d*(b*d*m + a*c*(n + 1)) + (c*C - B* 
d)*(b*c*m + a*d*(n + 1)) - (d*(A*(a*d*(n + 2) - b*c*(n + 1)) + B*(b*d*(n + 
1) - a*c*(n + 2))) - C*(b*c*d*(n + 1) - a*(c^2 + d^2*(n + 1))))*Sin[e + f*x 
] + b*(d*(B*c - A*d)*(m + n + 2) - C*(c^2*(m + 1) + d^2*(n + 1)))*Sin[e + f 
*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C}, x] && NeQ[b*c - a*d 
, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 0] && LtQ[n, -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 = 13.00 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.86

method	result	size
parts	\(\frac {A \,a^{4} \tan \left (d x +c \right )}{d}+\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (B \,b^{4}+4 C a \,b^{3}\right ) \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3 d}+\frac {\left (A \,b^{4}+4 B a \,b^{3}+6 C \,a^{2} b^{2}\right ) \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {\left (4 a A \,b^{3}+6 B \,a^{2} b^{2}+4 a^{3} b C \right ) \sin \left (d x +c \right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b +a^{4} C \right ) \left (d x +c \right )}{d}+\frac {C \,b^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(235\)
parallelrisch	\(\frac {-768 \cos \left (d x +c \right ) \left (A b +\frac {B a}{4}\right ) a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+768 \cos \left (d x +c \right ) \left (A b +\frac {B a}{4}\right ) a^{3} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+384 b \left (\frac {5 B \,b^{3}}{24}+a \left (A +\frac {5 C}{6}\right ) b^{2}+\frac {3 B \,a^{2} b}{2}+a^{3} C \right ) \sin \left (2 d x +2 c \right )+24 b^{2} \left (\left (A +\frac {9 C}{8}\right ) b^{2}+4 B a b +6 a^{2} C \right ) \sin \left (3 d x +3 c \right )+8 \left (B \,b^{4}+4 C a \,b^{3}\right ) \sin \left (4 d x +4 c \right )+3 C \sin \left (5 d x +5 c \right ) b^{4}+1152 \left (\frac {\left (\frac {A}{3}+\frac {C}{4}\right ) b^{4}}{4}+\frac {B a \,b^{3}}{3}+a^{2} \left (A +\frac {C}{2}\right ) b^{2}+\frac {2 B \,a^{3} b}{3}+\frac {a^{4} C}{6}\right ) x d \cos \left (d x +c \right )+192 \sin \left (d x +c \right ) \left (\frac {\left (A +C \right ) b^{4}}{8}+\frac {B a \,b^{3}}{2}+\frac {3 C \,a^{2} b^{2}}{4}+A \,a^{4}\right )}{192 d \cos \left (d x +c \right )}\)	\(285\)
derivativedivides	\(\frac {A \,a^{4} \tan \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \left (d x +c \right )+4 A \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \,a^{3} b \left (d x +c \right )+4 C \sin \left (d x +c \right ) a^{3} b +6 A \,a^{2} b^{2} \left (d x +c \right )+6 B \sin \left (d x +c \right ) a^{2} b^{2}+6 C \,a^{2} b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 A \sin \left (d x +c \right ) a \,b^{3}+4 B a \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 C a \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+A \,b^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {B \,b^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+C \,b^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(304\)
default	\(\frac {A \,a^{4} \tan \left (d x +c \right )+B \,a^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+a^{4} C \left (d x +c \right )+4 A \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 B \,a^{3} b \left (d x +c \right )+4 C \sin \left (d x +c \right ) a^{3} b +6 A \,a^{2} b^{2} \left (d x +c \right )+6 B \sin \left (d x +c \right ) a^{2} b^{2}+6 C \,a^{2} b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 A \sin \left (d x +c \right ) a \,b^{3}+4 B a \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 C a \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+A \,b^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {B \,b^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+C \,b^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\)	\(304\)
risch	\(\frac {\sin \left (3 d x +3 c \right ) B \,b^{4}}{12 d}+4 x B \,a^{3} b +2 x B a \,b^{3}+6 x A \,a^{2} b^{2}+3 x C \,a^{2} b^{2}+\frac {C \,b^{4} \sin \left (4 d x +4 c \right )}{32 d}+\frac {3 i {\mathrm e}^{-2 i \left (d x +c \right )} C \,a^{2} b^{2}}{4 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} B a \,b^{3}}{2 d}-\frac {3 i {\mathrm e}^{2 i \left (d x +c \right )} C \,a^{2} b^{2}}{4 d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} a A \,b^{3}}{d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{2} b^{2}}{d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} a^{3} b C}{d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} C a \,b^{3}}{2 d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} a A \,b^{3}}{d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{2} b^{2}}{d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} a^{3} b C}{d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} C a \,b^{3}}{2 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} B a \,b^{3}}{2 d}+\frac {3 b^{4} C x}{8}+a^{4} C x +\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B}{d}+\frac {x A \,b^{4}}{2}+\frac {\sin \left (3 d x +3 c \right ) C a \,b^{3}}{3 d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} A \,b^{4}}{8 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} A \,b^{4}}{8 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} C \,b^{4}}{8 d}+\frac {2 i A \,a^{4}}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}-\frac {4 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A b}{d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} C \,b^{4}}{8 d}+\frac {4 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A b}{d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} B \,b^{4}}{8 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B \,b^{4}}{8 d}\)	\(587\)
norman	\(\text {Expression too large to display}\)	\(1160\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.96 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {3 \, {\left (8 \, C a^{4} + 32 \, B a^{3} b + 24 \, {\left (2 \, A + C\right )} a^{2} b^{2} + 16 \, B a b^{3} + {\left (4 \, A + 3 \, C\right )} b^{4}\right )} d x \cos \left (d x + c\right ) + 12 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right ) \log \left (\sin \left (d x + c\right ) + 1\right ) - 12 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right ) \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (6 \, C b^{4} \cos \left (d x + c\right )^{4} + 24 \, A a^{4} + 8 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (24 \, C a^{2} b^{2} + 16 \, B a b^{3} + {\left (4 \, A + 3 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} + 16 \, {\left (6 \, C a^{3} b + 9 \, B a^{2} b^{2} + 2 \, {\left (3 \, A + 2 \, C\right )} a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d \cos \left (d x + c\right )} \] Input:

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

Output:

1/24*(3*(8*C*a^4 + 32*B*a^3*b + 24*(2*A + C)*a^2*b^2 + 16*B*a*b^3 + (4*A + 
 3*C)*b^4)*d*x*cos(d*x + c) + 12*(B*a^4 + 4*A*a^3*b)*cos(d*x + c)*log(sin( 
d*x + c) + 1) - 12*(B*a^4 + 4*A*a^3*b)*cos(d*x + c)*log(-sin(d*x + c) + 1) 
 + (6*C*b^4*cos(d*x + c)^4 + 24*A*a^4 + 8*(4*C*a*b^3 + B*b^4)*cos(d*x + c) 
^3 + 3*(24*C*a^2*b^2 + 16*B*a*b^3 + (4*A + 3*C)*b^4)*cos(d*x + c)^2 + 16*( 
6*C*a^3*b + 9*B*a^2*b^2 + 2*(3*A + 2*C)*a*b^3 + B*b^4)*cos(d*x + c))*sin(d 
*x + c))/(d*cos(d*x + c))
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.12 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {96 \, {\left (d x + c\right )} C a^{4} + 384 \, {\left (d x + c\right )} B a^{3} b + 576 \, {\left (d x + c\right )} A a^{2} b^{2} + 144 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} b^{2} + 96 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b^{3} - 128 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a b^{3} + 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{4} - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B b^{4} + 3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{4} + 48 \, B a^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 192 \, A a^{3} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 384 \, C a^{3} b \sin \left (d x + c\right ) + 576 \, B a^{2} b^{2} \sin \left (d x + c\right ) + 384 \, A a b^{3} \sin \left (d x + c\right ) + 96 \, A a^{4} \tan \left (d x + c\right )}{96 \, d} \] Input:

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

Output:

1/96*(96*(d*x + c)*C*a^4 + 384*(d*x + c)*B*a^3*b + 576*(d*x + c)*A*a^2*b^2 
 + 144*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^2*b^2 + 96*(2*d*x + 2*c + sin( 
2*d*x + 2*c))*B*a*b^3 - 128*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a*b^3 + 24 
*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*b^4 - 32*(sin(d*x + c)^3 - 3*sin(d*x + 
 c))*B*b^4 + 3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*b 
^4 + 48*B*a^4*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 192*A*a^3* 
b*(log(sin(d*x + c) + 1) - log(sin(d*x + c) - 1)) + 384*C*a^3*b*sin(d*x + 
c) + 576*B*a^2*b^2*sin(d*x + c) + 384*A*a*b^3*sin(d*x + c) + 96*A*a^4*tan( 
d*x + c))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 802 vs. \(2 (262) = 524\).

Time = 0.19 (sec) , antiderivative size = 802, normalized size of antiderivative = 2.94 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\text {Too large to display} \] Input:

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

Output:

-1/24*(48*A*a^4*tan(1/2*d*x + 1/2*c)/(tan(1/2*d*x + 1/2*c)^2 - 1) - 3*(8*C 
*a^4 + 32*B*a^3*b + 48*A*a^2*b^2 + 24*C*a^2*b^2 + 16*B*a*b^3 + 4*A*b^4 + 3 
*C*b^4)*(d*x + c) - 24*(B*a^4 + 4*A*a^3*b)*log(abs(tan(1/2*d*x + 1/2*c) + 
1)) + 24*(B*a^4 + 4*A*a^3*b)*log(abs(tan(1/2*d*x + 1/2*c) - 1)) - 2*(96*C* 
a^3*b*tan(1/2*d*x + 1/2*c)^7 + 144*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 72*C 
*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 + 96*A*a*b^3*tan(1/2*d*x + 1/2*c)^7 - 48*B 
*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 96*C*a*b^3*tan(1/2*d*x + 1/2*c)^7 - 12*A*b 
^4*tan(1/2*d*x + 1/2*c)^7 + 24*B*b^4*tan(1/2*d*x + 1/2*c)^7 - 15*C*b^4*tan 
(1/2*d*x + 1/2*c)^7 + 288*C*a^3*b*tan(1/2*d*x + 1/2*c)^5 + 432*B*a^2*b^2*t 
an(1/2*d*x + 1/2*c)^5 - 72*C*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 + 288*A*a*b^3* 
tan(1/2*d*x + 1/2*c)^5 - 48*B*a*b^3*tan(1/2*d*x + 1/2*c)^5 + 160*C*a*b^3*t 
an(1/2*d*x + 1/2*c)^5 - 12*A*b^4*tan(1/2*d*x + 1/2*c)^5 + 40*B*b^4*tan(1/2 
*d*x + 1/2*c)^5 + 9*C*b^4*tan(1/2*d*x + 1/2*c)^5 + 288*C*a^3*b*tan(1/2*d*x 
 + 1/2*c)^3 + 432*B*a^2*b^2*tan(1/2*d*x + 1/2*c)^3 + 72*C*a^2*b^2*tan(1/2* 
d*x + 1/2*c)^3 + 288*A*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 48*B*a*b^3*tan(1/2*d 
*x + 1/2*c)^3 + 160*C*a*b^3*tan(1/2*d*x + 1/2*c)^3 + 12*A*b^4*tan(1/2*d*x 
+ 1/2*c)^3 + 40*B*b^4*tan(1/2*d*x + 1/2*c)^3 - 9*C*b^4*tan(1/2*d*x + 1/2*c 
)^3 + 96*C*a^3*b*tan(1/2*d*x + 1/2*c) + 144*B*a^2*b^2*tan(1/2*d*x + 1/2*c) 
 + 72*C*a^2*b^2*tan(1/2*d*x + 1/2*c) + 96*A*a*b^3*tan(1/2*d*x + 1/2*c) + 4 
8*B*a*b^3*tan(1/2*d*x + 1/2*c) + 96*C*a*b^3*tan(1/2*d*x + 1/2*c) + 12*A...
 

Mupad [B] (verification not implemented)

Time = 3.05 (sec) , antiderivative size = 4781, normalized size of antiderivative = 17.51 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\text {Too large to display} \] Input:

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

Output:

(atan(((((A*b^4*1i)/2 + C*a^4*1i + (C*b^4*3i)/8 + A*a^2*b^2*6i + C*a^2*b^2 
*3i + B*a*b^3*2i + B*a^3*b*4i)*(16*A*b^4 + 32*B*a^4 + 32*C*a^4 + 12*C*b^4 
+ 192*A*a^2*b^2 + 96*C*a^2*b^2 + 128*A*a^3*b + 64*B*a*b^3 + 128*B*a^3*b) + 
 tan(c/2 + (d*x)/2)*(8*A^2*b^8 + 32*B^2*a^8 + 32*C^2*a^8 + (9*C^2*b^8)/2 + 
 192*A^2*a^2*b^6 + 1152*A^2*a^4*b^4 + 512*A^2*a^6*b^2 + 128*B^2*a^2*b^6 + 
512*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 312*C^2*a^4*b^4 + 192 
*C^2*a^6*b^2 + 12*A*C*b^8 + 64*A*B*a*b^7 + 256*A*B*a^7*b + 48*B*C*a*b^7 + 
256*B*C*a^7*b + 896*A*B*a^3*b^5 + 1536*A*B*a^5*b^3 + 240*A*C*a^2*b^6 + 118 
4*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 480*B*C*a^3*b^5 + 896*B*C*a^5*b^3))*((A* 
b^4*1i)/2 + C*a^4*1i + (C*b^4*3i)/8 + A*a^2*b^2*6i + C*a^2*b^2*3i + B*a*b^ 
3*2i + B*a^3*b*4i)*1i - (((A*b^4*1i)/2 + C*a^4*1i + (C*b^4*3i)/8 + A*a^2*b 
^2*6i + C*a^2*b^2*3i + B*a*b^3*2i + B*a^3*b*4i)*(16*A*b^4 + 32*B*a^4 + 32* 
C*a^4 + 12*C*b^4 + 192*A*a^2*b^2 + 96*C*a^2*b^2 + 128*A*a^3*b + 64*B*a*b^3 
 + 128*B*a^3*b) - tan(c/2 + (d*x)/2)*(8*A^2*b^8 + 32*B^2*a^8 + 32*C^2*a^8 
+ (9*C^2*b^8)/2 + 192*A^2*a^2*b^6 + 1152*A^2*a^4*b^4 + 512*A^2*a^6*b^2 + 1 
28*B^2*a^2*b^6 + 512*B^2*a^4*b^4 + 512*B^2*a^6*b^2 + 72*C^2*a^2*b^6 + 312* 
C^2*a^4*b^4 + 192*C^2*a^6*b^2 + 12*A*C*b^8 + 64*A*B*a*b^7 + 256*A*B*a^7*b 
+ 48*B*C*a*b^7 + 256*B*C*a^7*b + 896*A*B*a^3*b^5 + 1536*A*B*a^5*b^3 + 240* 
A*C*a^2*b^6 + 1184*A*C*a^4*b^4 + 384*A*C*a^6*b^2 + 480*B*C*a^3*b^5 + 896*B 
*C*a^5*b^3))*((A*b^4*1i)/2 + C*a^4*1i + (C*b^4*3i)/8 + A*a^2*b^2*6i + C...
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.27 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx=\frac {-6 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{3} b^{4} c +72 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right ) a^{2} b^{2} c +60 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right ) a \,b^{4}+15 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right ) b^{4} c -120 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a^{4} b +120 \cos \left (d x +c \right ) \mathrm {log}\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a^{4} b -32 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a \,b^{3} c -8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{5}+96 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3} b c +240 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2} b^{3}+96 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a \,b^{3} c +24 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{5}+24 \cos \left (d x +c \right ) a^{4} c d x +240 \cos \left (d x +c \right ) a^{3} b^{2} d x +72 \cos \left (d x +c \right ) a^{2} b^{2} c d x +60 \cos \left (d x +c \right ) a \,b^{4} d x +9 \cos \left (d x +c \right ) b^{4} c d x +24 \sin \left (d x +c \right ) a^{5}}{24 \cos \left (d x +c \right ) d} \] Input:

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

Output:

( - 6*cos(c + d*x)**2*sin(c + d*x)**3*b**4*c + 72*cos(c + d*x)**2*sin(c + 
d*x)*a**2*b**2*c + 60*cos(c + d*x)**2*sin(c + d*x)*a*b**4 + 15*cos(c + d*x 
)**2*sin(c + d*x)*b**4*c - 120*cos(c + d*x)*log(tan((c + d*x)/2) - 1)*a**4 
*b + 120*cos(c + d*x)*log(tan((c + d*x)/2) + 1)*a**4*b - 32*cos(c + d*x)*s 
in(c + d*x)**3*a*b**3*c - 8*cos(c + d*x)*sin(c + d*x)**3*b**5 + 96*cos(c + 
 d*x)*sin(c + d*x)*a**3*b*c + 240*cos(c + d*x)*sin(c + d*x)*a**2*b**3 + 96 
*cos(c + d*x)*sin(c + d*x)*a*b**3*c + 24*cos(c + d*x)*sin(c + d*x)*b**5 + 
24*cos(c + d*x)*a**4*c*d*x + 240*cos(c + d*x)*a**3*b**2*d*x + 72*cos(c + d 
*x)*a**2*b**2*c*d*x + 60*cos(c + d*x)*a*b**4*d*x + 9*cos(c + d*x)*b**4*c*d 
*x + 24*sin(c + d*x)*a**5)/(24*cos(c + d*x)*d)