\(\int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx\) [310]

Optimal result

Integrand size = 31, antiderivative size = 309 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {1}{16} \left (5 a^4 A+36 a^2 A b^2+8 A b^4+24 a^3 b B+32 a b^3 B\right ) x+\frac {\left (48 a^3 A b+53 a A b^3+12 a^4 B+87 a^2 b^2 B+15 b^4 B\right ) \sin (c+d x)}{15 d}+\frac {\left (5 a^4 A+36 a^2 A b^2+8 A b^4+24 a^3 b B+32 a b^3 B\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 \left (25 a^2 A+48 A b^2+72 a b B\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {a (3 A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {a A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {a \left (16 a^2 A b+13 A b^3+4 a^3 B+27 a b^2 B\right ) \sin ^3(c+d x)}{15 d} \] Output:

1/16*(5*A*a^4+36*A*a^2*b^2+8*A*b^4+24*B*a^3*b+32*B*a*b^3)*x+1/15*(48*A*a^3 
*b+53*A*a*b^3+12*B*a^4+87*B*a^2*b^2+15*B*b^4)*sin(d*x+c)/d+1/16*(5*A*a^4+3 
6*A*a^2*b^2+8*A*b^4+24*B*a^3*b+32*B*a*b^3)*cos(d*x+c)*sin(d*x+c)/d+1/120*a 
^2*(25*A*a^2+48*A*b^2+72*B*a*b)*cos(d*x+c)^3*sin(d*x+c)/d+1/10*a*(3*A*b+2* 
B*a)*cos(d*x+c)^4*(a+b*sec(d*x+c))^2*sin(d*x+c)/d+1/6*a*A*cos(d*x+c)^5*(a+ 
b*sec(d*x+c))^3*sin(d*x+c)/d-1/15*a*(16*A*a^2*b+13*A*b^3+4*B*a^3+27*B*a*b^ 
2)*sin(d*x+c)^3/d
 

Mathematica [A] (verified)

Time = 2.85 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.08 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {300 a^4 A c+2160 a^2 A b^2 c+480 A b^4 c+1440 a^3 b B c+1920 a b^3 B c+300 a^4 A d x+2160 a^2 A b^2 d x+480 A b^4 d x+1440 a^3 b B d x+1920 a b^3 B d x+120 \left (20 a^3 A b+24 a A b^3+5 a^4 B+36 a^2 b^2 B+8 b^4 B\right ) \sin (c+d x)+15 \left (15 a^4 A+96 a^2 A b^2+16 A b^4+64 a^3 b B+64 a b^3 B\right ) \sin (2 (c+d x))+400 a^3 A b \sin (3 (c+d x))+320 a A b^3 \sin (3 (c+d x))+100 a^4 B \sin (3 (c+d x))+480 a^2 b^2 B \sin (3 (c+d x))+45 a^4 A \sin (4 (c+d x))+180 a^2 A b^2 \sin (4 (c+d x))+120 a^3 b B \sin (4 (c+d x))+48 a^3 A b \sin (5 (c+d x))+12 a^4 B \sin (5 (c+d x))+5 a^4 A \sin (6 (c+d x))}{960 d} \] Input:

Integrate[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]
 

Output:

(300*a^4*A*c + 2160*a^2*A*b^2*c + 480*A*b^4*c + 1440*a^3*b*B*c + 1920*a*b^ 
3*B*c + 300*a^4*A*d*x + 2160*a^2*A*b^2*d*x + 480*A*b^4*d*x + 1440*a^3*b*B* 
d*x + 1920*a*b^3*B*d*x + 120*(20*a^3*A*b + 24*a*A*b^3 + 5*a^4*B + 36*a^2*b 
^2*B + 8*b^4*B)*Sin[c + d*x] + 15*(15*a^4*A + 96*a^2*A*b^2 + 16*A*b^4 + 64 
*a^3*b*B + 64*a*b^3*B)*Sin[2*(c + d*x)] + 400*a^3*A*b*Sin[3*(c + d*x)] + 3 
20*a*A*b^3*Sin[3*(c + d*x)] + 100*a^4*B*Sin[3*(c + d*x)] + 480*a^2*b^2*B*S 
in[3*(c + d*x)] + 45*a^4*A*Sin[4*(c + d*x)] + 180*a^2*A*b^2*Sin[4*(c + d*x 
)] + 120*a^3*b*B*Sin[4*(c + d*x)] + 48*a^3*A*b*Sin[5*(c + d*x)] + 12*a^4*B 
*Sin[5*(c + d*x)] + 5*a^4*A*Sin[6*(c + d*x)])/(960*d)
 

Rubi [A] (verified)

Time = 1.88 (sec) , antiderivative size = 287, normalized size of antiderivative = 0.93, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.548, Rules used = {3042, 4513, 25, 3042, 4582, 3042, 4562, 25, 3042, 4535, 3042, 3115, 24, 4532, 3042, 3492, 2009}

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.

\(\displaystyle \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^4 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx\)

\(\Big \downarrow \) 4513

\(\displaystyle \frac {a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}-\frac {1}{6} \int -\cos ^5(c+d x) (a+b \sec (c+d x))^2 \left (2 b (a A+3 b B) \sec ^2(c+d x)+\left (5 A a^2+12 b B a+6 A b^2\right ) \sec (c+d x)+3 a (3 A b+2 a B)\right )dx\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{6} \int \cos ^5(c+d x) (a+b \sec (c+d x))^2 \left (2 b (a A+3 b B) \sec ^2(c+d x)+\left (5 A a^2+12 b B a+6 A b^2\right ) \sec (c+d x)+3 a (3 A b+2 a B)\right )dx+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^2 \left (2 b (a A+3 b B) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (5 A a^2+12 b B a+6 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a (3 A b+2 a B)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 4582

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (2 b \left (6 B a^2+14 A b a+15 b^2 B\right ) \sec ^2(c+d x)+\left (24 B a^3+71 A b a^2+90 b^2 B a+30 A b^3\right ) \sec (c+d x)+a \left (25 A a^2+72 b B a+48 A b^2\right )\right )dx+\frac {3 a (2 a B+3 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (2 b \left (6 B a^2+14 A b a+15 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (24 B a^3+71 A b a^2+90 b^2 B a+30 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+a \left (25 A a^2+72 b B a+48 A b^2\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {3 a (2 a B+3 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 4562

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {a^2 \left (25 a^2 A+72 a b B+48 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}-\frac {1}{4} \int -\cos ^3(c+d x) \left (8 b^2 \left (6 B a^2+14 A b a+15 b^2 B\right ) \sec ^2(c+d x)+15 \left (5 A a^4+24 b B a^3+36 A b^2 a^2+32 b^3 B a+8 A b^4\right ) \sec (c+d x)+24 a \left (4 B a^3+16 A b a^2+27 b^2 B a+13 A b^3\right )\right )dx\right )+\frac {3 a (2 a B+3 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int \cos ^3(c+d x) \left (8 b^2 \left (6 B a^2+14 A b a+15 b^2 B\right ) \sec ^2(c+d x)+15 \left (5 A a^4+24 b B a^3+36 A b^2 a^2+32 b^3 B a+8 A b^4\right ) \sec (c+d x)+24 a \left (4 B a^3+16 A b a^2+27 b^2 B a+13 A b^3\right )\right )dx+\frac {a^2 \left (25 a^2 A+72 a b B+48 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {3 a (2 a B+3 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \int \frac {8 b^2 \left (6 B a^2+14 A b a+15 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+15 \left (5 A a^4+24 b B a^3+36 A b^2 a^2+32 b^3 B a+8 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+24 a \left (4 B a^3+16 A b a^2+27 b^2 B a+13 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a^2 \left (25 a^2 A+72 a b B+48 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {3 a (2 a B+3 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 4535

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \cos ^3(c+d x) \left (8 b^2 \left (6 B a^2+14 A b a+15 b^2 B\right ) \sec ^2(c+d x)+24 a \left (4 B a^3+16 A b a^2+27 b^2 B a+13 A b^3\right )\right )dx+15 \left (5 a^4 A+24 a^3 b B+36 a^2 A b^2+32 a b^3 B+8 A b^4\right ) \int \cos ^2(c+d x)dx\right )+\frac {a^2 \left (25 a^2 A+72 a b B+48 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {3 a (2 a B+3 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \frac {8 b^2 \left (6 B a^2+14 A b a+15 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+24 a \left (4 B a^3+16 A b a^2+27 b^2 B a+13 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+15 \left (5 a^4 A+24 a^3 b B+36 a^2 A b^2+32 a b^3 B+8 A b^4\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\right )+\frac {a^2 \left (25 a^2 A+72 a b B+48 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {3 a (2 a B+3 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 3115

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \frac {8 b^2 \left (6 B a^2+14 A b a+15 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+24 a \left (4 B a^3+16 A b a^2+27 b^2 B a+13 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+15 \left (5 a^4 A+24 a^3 b B+36 a^2 A b^2+32 a b^3 B+8 A b^4\right ) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )\right )+\frac {a^2 \left (25 a^2 A+72 a b B+48 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {3 a (2 a B+3 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 24

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \frac {8 b^2 \left (6 B a^2+14 A b a+15 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+24 a \left (4 B a^3+16 A b a^2+27 b^2 B a+13 A b^3\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+15 \left (5 a^4 A+24 a^3 b B+36 a^2 A b^2+32 a b^3 B+8 A b^4\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a^2 \left (25 a^2 A+72 a b B+48 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {3 a (2 a B+3 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 4532

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \cos (c+d x) \left (8 \left (6 B a^2+14 A b a+15 b^2 B\right ) b^2+24 a \left (4 B a^3+16 A b a^2+27 b^2 B a+13 A b^3\right ) \cos ^2(c+d x)\right )dx+15 \left (5 a^4 A+24 a^3 b B+36 a^2 A b^2+32 a b^3 B+8 A b^4\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a^2 \left (25 a^2 A+72 a b B+48 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {3 a (2 a B+3 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (\int \sin \left (c+d x+\frac {\pi }{2}\right ) \left (8 \left (6 B a^2+14 A b a+15 b^2 B\right ) b^2+24 a \left (4 B a^3+16 A b a^2+27 b^2 B a+13 A b^3\right ) \sin \left (c+d x+\frac {\pi }{2}\right )^2\right )dx+15 \left (5 a^4 A+24 a^3 b B+36 a^2 A b^2+32 a b^3 B+8 A b^4\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a^2 \left (25 a^2 A+72 a b B+48 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {3 a (2 a B+3 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 3492

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {1}{4} \left (15 \left (5 a^4 A+24 a^3 b B+36 a^2 A b^2+32 a b^3 B+8 A b^4\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {\int \left (8 \left (12 B a^4+48 A b a^3+87 b^2 B a^2+53 A b^3 a+15 b^4 B\right )-24 a \left (4 B a^3+16 A b a^2+27 b^2 B a+13 A b^3\right ) \sin ^2(c+d x)\right )d(-\sin (c+d x))}{d}\right )+\frac {a^2 \left (25 a^2 A+72 a b B+48 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {3 a (2 a B+3 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{6} \left (\frac {1}{5} \left (\frac {a^2 \left (25 a^2 A+72 a b B+48 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {1}{4} \left (15 \left (5 a^4 A+24 a^3 b B+36 a^2 A b^2+32 a b^3 B+8 A b^4\right ) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )-\frac {8 a \left (4 a^3 B+16 a^2 A b+27 a b^2 B+13 A b^3\right ) \sin ^3(c+d x)-8 \left (12 a^4 B+48 a^3 A b+87 a^2 b^2 B+53 a A b^3+15 b^4 B\right ) \sin (c+d x)}{d}\right )\right )+\frac {3 a (2 a B+3 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d}\right )+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}\)

Input:

Int[Cos[c + d*x]^6*(a + b*Sec[c + d*x])^4*(A + B*Sec[c + d*x]),x]
 

Output:

(a*A*Cos[c + d*x]^5*(a + b*Sec[c + d*x])^3*Sin[c + d*x])/(6*d) + ((3*a*(3* 
A*b + 2*a*B)*Cos[c + d*x]^4*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + ( 
(a^2*(25*a^2*A + 48*A*b^2 + 72*a*b*B)*Cos[c + d*x]^3*Sin[c + d*x])/(4*d) + 
 (15*(5*a^4*A + 36*a^2*A*b^2 + 8*A*b^4 + 24*a^3*b*B + 32*a*b^3*B)*(x/2 + ( 
Cos[c + d*x]*Sin[c + d*x])/(2*d)) - (-8*(48*a^3*A*b + 53*a*A*b^3 + 12*a^4* 
B + 87*a^2*b^2*B + 15*b^4*B)*Sin[c + d*x] + 8*a*(16*a^2*A*b + 13*A*b^3 + 4 
*a^3*B + 27*a*b^2*B)*Sin[c + d*x]^3)/d)/4)/5)/6
 

Defintions of rubi rules used

rule 24

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 3115

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

rule 3492

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), 
 x_Symbol] :> Simp[-f^(-1)   Subst[Int[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2 
), x], x, Cos[e + f*x]], x] /; FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2, 0]
 

rule 4513

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( 
a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[a*A*Cot 
[e + f*x]*(a + b*Csc[e + f*x])^(m - 1)*((d*Csc[e + f*x])^n/(f*n)), x] + Sim 
p[1/(d*n)   Int[(a + b*Csc[e + f*x])^(m - 2)*(d*Csc[e + f*x])^(n + 1)*Simp[ 
a*(a*B*n - A*b*(m - n - 1)) + (2*a*b*B*n + A*(b^2*n + a^2*(1 + n)))*Csc[e + 
 f*x] + b*(b*B*n + a*A*(m + n))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, 
d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] & 
& LeQ[n, -1]
 

rule 4532

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), 
 x_Symbol] :> Int[(C + A*Sin[e + f*x]^2)/Sin[e + f*x]^(m + 2), x] /; FreeQ[ 
{e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && ILtQ[(m + 1)/2, 0]
 

rule 4535

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_.)*(x_)]* 
(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.)), x_Symbol] :> Simp[B/b   Int[(b*Cs 
c[e + f*x])^(m + 1), x], x] + Int[(b*Csc[e + f*x])^m*(A + C*Csc[e + f*x]^2) 
, x] /; FreeQ[{b, e, f, A, B, C, m}, x]
 

rule 4562

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a 
_)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Si 
mp[1/(d*n)   Int[(d*Csc[e + f*x])^(n + 1)*Simp[n*(B*a + A*b) + (n*(a*C + B* 
b) + A*a*(n + 1))*Csc[e + f*x] + b*C*n*Csc[e + f*x]^2, x], x], x] /; FreeQ[ 
{a, b, d, e, f, A, B, C}, x] && LtQ[n, -1]
 

rule 4582

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a 
_))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^m*((d*Csc[e 
 + f*x])^n/(f*n)), x] - Simp[1/(d*n)   Int[(a + b*Csc[e + f*x])^(m - 1)*(d* 
Csc[e + f*x])^(n + 1)*Simp[A*b*m - a*B*n - (b*B*n + a*(C*n + A*(n + 1)))*Cs 
c[e + f*x] - b*(C*n + A*(m + n + 1))*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a 
, b, d, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && LeQ[n, -1]
 

Maple [A] (verified)

Time = 1.72 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.80

method	result	size
parallelrisch	\(\frac {\left (225 a^{4} A +1440 A \,a^{2} b^{2}+240 A \,b^{4}+960 a^{3} b B +960 a \,b^{3} B \right ) \sin \left (2 d x +2 c \right )+\left (400 A \,a^{3} b +320 A a \,b^{3}+100 B \,a^{4}+480 B \,a^{2} b^{2}\right ) \sin \left (3 d x +3 c \right )+\left (45 a^{4} A +180 A \,a^{2} b^{2}+120 a^{3} b B \right ) \sin \left (4 d x +4 c \right )+\left (48 A \,a^{3} b +12 B \,a^{4}\right ) \sin \left (5 d x +5 c \right )+5 a^{4} A \sin \left (6 d x +6 c \right )+\left (2400 A \,a^{3} b +2880 A a \,b^{3}+600 B \,a^{4}+4320 B \,a^{2} b^{2}+960 B \,b^{4}\right ) \sin \left (d x +c \right )+300 x \left (a^{4} A +\frac {36}{5} A \,a^{2} b^{2}+\frac {8}{5} A \,b^{4}+\frac {24}{5} a^{3} b B +\frac {32}{5} a \,b^{3} B \right ) d}{960 d}\)	\(247\)
derivativedivides	\(\frac {a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \,a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {4 A \,a^{3} b \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{3} b B \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+6 A \,a^{2} b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 B \,a^{2} b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {4 A a \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 a \,b^{3} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \,b^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,b^{4} \sin \left (d x +c \right )}{d}\)	\(316\)
default	\(\frac {a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \,a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {4 A \,a^{3} b \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 a^{3} b B \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+6 A \,a^{2} b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 B \,a^{2} b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {4 A a \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 a \,b^{3} B \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+A \,b^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,b^{4} \sin \left (d x +c \right )}{d}\)	\(316\)
risch	\(\frac {3 a^{4} A \sin \left (4 d x +4 c \right )}{64 d}+\frac {5 \sin \left (3 d x +3 c \right ) B \,a^{4}}{48 d}+\frac {15 \sin \left (2 d x +2 c \right ) a^{4} A}{64 d}+2 x a \,b^{3} B +\frac {9 A \,a^{2} b^{2} x}{4}+\frac {3 B \,a^{3} b x}{2}+\frac {5 \sin \left (d x +c \right ) A \,a^{3} b}{2 d}+\frac {3 \sin \left (d x +c \right ) A a \,b^{3}}{d}+\frac {9 \sin \left (d x +c \right ) B \,a^{2} b^{2}}{2 d}+\frac {\sin \left (5 d x +5 c \right ) A \,a^{3} b}{20 d}+\frac {3 \sin \left (4 d x +4 c \right ) A \,a^{2} b^{2}}{16 d}+\frac {\sin \left (4 d x +4 c \right ) a^{3} b B}{8 d}+\frac {\sin \left (3 d x +3 c \right ) A a \,b^{3}}{3 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{2} b^{2}}{2 d}+\frac {\sin \left (2 d x +2 c \right ) a \,b^{3} B}{d}+\frac {\sin \left (2 d x +2 c \right ) A \,b^{4}}{4 d}+\frac {\sin \left (5 d x +5 c \right ) B \,a^{4}}{80 d}+\frac {5 \sin \left (d x +c \right ) B \,a^{4}}{8 d}+\frac {\sin \left (d x +c \right ) B \,b^{4}}{d}+\frac {a^{4} A \sin \left (6 d x +6 c \right )}{192 d}+\frac {5 \sin \left (3 d x +3 c \right ) A \,a^{3} b}{12 d}+\frac {3 \sin \left (2 d x +2 c \right ) A \,a^{2} b^{2}}{2 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} b B}{d}+\frac {x A \,b^{4}}{2}+\frac {5 a^{4} A x}{16}\)	\(404\)

Input:

int(cos(d*x+c)^6*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)),x,method=_RETURNVERBO 
SE)
 

Output:

1/960*((225*A*a^4+1440*A*a^2*b^2+240*A*b^4+960*B*a^3*b+960*B*a*b^3)*sin(2* 
d*x+2*c)+(400*A*a^3*b+320*A*a*b^3+100*B*a^4+480*B*a^2*b^2)*sin(3*d*x+3*c)+ 
(45*A*a^4+180*A*a^2*b^2+120*B*a^3*b)*sin(4*d*x+4*c)+(48*A*a^3*b+12*B*a^4)* 
sin(5*d*x+5*c)+5*a^4*A*sin(6*d*x+6*c)+(2400*A*a^3*b+2880*A*a*b^3+600*B*a^4 
+4320*B*a^2*b^2+960*B*b^4)*sin(d*x+c)+300*x*(a^4*A+36/5*A*a^2*b^2+8/5*A*b^ 
4+24/5*a^3*b*B+32/5*a*b^3*B)*d)/d
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.79 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (5 \, A a^{4} + 24 \, B a^{3} b + 36 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} d x + {\left (40 \, A a^{4} \cos \left (d x + c\right )^{5} + 128 \, B a^{4} + 512 \, A a^{3} b + 960 \, B a^{2} b^{2} + 640 \, A a b^{3} + 240 \, B b^{4} + 48 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{4} + 10 \, {\left (5 \, A a^{4} + 24 \, B a^{3} b + 36 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} + 32 \, {\left (2 \, B a^{4} + 8 \, A a^{3} b + 15 \, B a^{2} b^{2} + 10 \, A a b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (5 \, A a^{4} + 24 \, B a^{3} b + 36 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \] Input:

integrate(cos(d*x+c)^6*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)),x, algorithm="f 
ricas")
 

Output:

1/240*(15*(5*A*a^4 + 24*B*a^3*b + 36*A*a^2*b^2 + 32*B*a*b^3 + 8*A*b^4)*d*x 
 + (40*A*a^4*cos(d*x + c)^5 + 128*B*a^4 + 512*A*a^3*b + 960*B*a^2*b^2 + 64 
0*A*a*b^3 + 240*B*b^4 + 48*(B*a^4 + 4*A*a^3*b)*cos(d*x + c)^4 + 10*(5*A*a^ 
4 + 24*B*a^3*b + 36*A*a^2*b^2)*cos(d*x + c)^3 + 32*(2*B*a^4 + 8*A*a^3*b + 
15*B*a^2*b^2 + 10*A*a*b^3)*cos(d*x + c)^2 + 15*(5*A*a^4 + 24*B*a^3*b + 36* 
A*a^2*b^2 + 32*B*a*b^3 + 8*A*b^4)*cos(d*x + c))*sin(d*x + c))/d
 

Sympy [F(-1)]

Timed out. \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\text {Timed out} \] Input:

integrate(cos(d*x+c)**6*(a+b*sec(d*x+c))**4*(A+B*sec(d*x+c)),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.99 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=-\frac {5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} - 256 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{3} b - 120 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} b - 180 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b^{2} + 1920 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} b^{2} + 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a b^{3} - 960 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b^{3} - 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{4} - 960 \, B b^{4} \sin \left (d x + c\right )}{960 \, d} \] Input:

integrate(cos(d*x+c)^6*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)),x, algorithm="m 
axima")
 

Output:

-1/960*(5*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48* 
sin(2*d*x + 2*c))*A*a^4 - 64*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*si 
n(d*x + c))*B*a^4 - 256*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x 
 + c))*A*a^3*b - 120*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c 
))*B*a^3*b - 180*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A 
*a^2*b^2 + 1920*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^2*b^2 + 1280*(sin(d* 
x + c)^3 - 3*sin(d*x + c))*A*a*b^3 - 960*(2*d*x + 2*c + sin(2*d*x + 2*c))* 
B*a*b^3 - 240*(2*d*x + 2*c + sin(2*d*x + 2*c))*A*b^4 - 960*B*b^4*sin(d*x + 
 c))/d
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1127 vs. \(2 (295) = 590\).

Time = 0.20 (sec) , antiderivative size = 1127, normalized size of antiderivative = 3.65 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\text {Too large to display} \] Input:

integrate(cos(d*x+c)^6*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)),x, algorithm="g 
iac")
 

Output:

1/240*(15*(5*A*a^4 + 24*B*a^3*b + 36*A*a^2*b^2 + 32*B*a*b^3 + 8*A*b^4)*(d* 
x + c) - 2*(165*A*a^4*tan(1/2*d*x + 1/2*c)^11 - 240*B*a^4*tan(1/2*d*x + 1/ 
2*c)^11 - 960*A*a^3*b*tan(1/2*d*x + 1/2*c)^11 + 600*B*a^3*b*tan(1/2*d*x + 
1/2*c)^11 + 900*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^11 - 1440*B*a^2*b^2*tan(1/2 
*d*x + 1/2*c)^11 - 960*A*a*b^3*tan(1/2*d*x + 1/2*c)^11 + 480*B*a*b^3*tan(1 
/2*d*x + 1/2*c)^11 + 120*A*b^4*tan(1/2*d*x + 1/2*c)^11 - 240*B*b^4*tan(1/2 
*d*x + 1/2*c)^11 - 25*A*a^4*tan(1/2*d*x + 1/2*c)^9 - 560*B*a^4*tan(1/2*d*x 
 + 1/2*c)^9 - 2240*A*a^3*b*tan(1/2*d*x + 1/2*c)^9 + 840*B*a^3*b*tan(1/2*d* 
x + 1/2*c)^9 + 1260*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^9 - 5280*B*a^2*b^2*tan( 
1/2*d*x + 1/2*c)^9 - 3520*A*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 1440*B*a*b^3*ta 
n(1/2*d*x + 1/2*c)^9 + 360*A*b^4*tan(1/2*d*x + 1/2*c)^9 - 1200*B*b^4*tan(1 
/2*d*x + 1/2*c)^9 + 450*A*a^4*tan(1/2*d*x + 1/2*c)^7 - 1248*B*a^4*tan(1/2* 
d*x + 1/2*c)^7 - 4992*A*a^3*b*tan(1/2*d*x + 1/2*c)^7 + 240*B*a^3*b*tan(1/2 
*d*x + 1/2*c)^7 + 360*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^7 - 8640*B*a^2*b^2*ta 
n(1/2*d*x + 1/2*c)^7 - 5760*A*a*b^3*tan(1/2*d*x + 1/2*c)^7 + 960*B*a*b^3*t 
an(1/2*d*x + 1/2*c)^7 + 240*A*b^4*tan(1/2*d*x + 1/2*c)^7 - 2400*B*b^4*tan( 
1/2*d*x + 1/2*c)^7 - 450*A*a^4*tan(1/2*d*x + 1/2*c)^5 - 1248*B*a^4*tan(1/2 
*d*x + 1/2*c)^5 - 4992*A*a^3*b*tan(1/2*d*x + 1/2*c)^5 - 240*B*a^3*b*tan(1/ 
2*d*x + 1/2*c)^5 - 360*A*a^2*b^2*tan(1/2*d*x + 1/2*c)^5 - 8640*B*a^2*b^2*t 
an(1/2*d*x + 1/2*c)^5 - 5760*A*a*b^3*tan(1/2*d*x + 1/2*c)^5 - 960*B*a*b...
 

Mupad [B] (verification not implemented)

Time = 13.20 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.30 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {5\,A\,a^4\,x}{16}+\frac {A\,b^4\,x}{2}+2\,B\,a\,b^3\,x+\frac {3\,B\,a^3\,b\,x}{2}+\frac {5\,B\,a^4\,\sin \left (c+d\,x\right )}{8\,d}+\frac {B\,b^4\,\sin \left (c+d\,x\right )}{d}+\frac {9\,A\,a^2\,b^2\,x}{4}+\frac {15\,A\,a^4\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {3\,A\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {A\,a^4\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {A\,b^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {5\,B\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {B\,a^4\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {A\,a\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{3\,d}+\frac {5\,A\,a^3\,b\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {A\,a^3\,b\,\sin \left (5\,c+5\,d\,x\right )}{20\,d}+\frac {B\,a\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {B\,a^3\,b\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {B\,a^3\,b\,\sin \left (4\,c+4\,d\,x\right )}{8\,d}+\frac {9\,B\,a^2\,b^2\,\sin \left (c+d\,x\right )}{2\,d}+\frac {3\,A\,a^2\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {3\,A\,a^2\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{16\,d}+\frac {B\,a^2\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{2\,d}+\frac {3\,A\,a\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {5\,A\,a^3\,b\,\sin \left (c+d\,x\right )}{2\,d} \] Input:

int(cos(c + d*x)^6*(A + B/cos(c + d*x))*(a + b/cos(c + d*x))^4,x)
 

Output:

(5*A*a^4*x)/16 + (A*b^4*x)/2 + 2*B*a*b^3*x + (3*B*a^3*b*x)/2 + (5*B*a^4*si 
n(c + d*x))/(8*d) + (B*b^4*sin(c + d*x))/d + (9*A*a^2*b^2*x)/4 + (15*A*a^4 
*sin(2*c + 2*d*x))/(64*d) + (3*A*a^4*sin(4*c + 4*d*x))/(64*d) + (A*a^4*sin 
(6*c + 6*d*x))/(192*d) + (A*b^4*sin(2*c + 2*d*x))/(4*d) + (5*B*a^4*sin(3*c 
 + 3*d*x))/(48*d) + (B*a^4*sin(5*c + 5*d*x))/(80*d) + (A*a*b^3*sin(3*c + 3 
*d*x))/(3*d) + (5*A*a^3*b*sin(3*c + 3*d*x))/(12*d) + (A*a^3*b*sin(5*c + 5* 
d*x))/(20*d) + (B*a*b^3*sin(2*c + 2*d*x))/d + (B*a^3*b*sin(2*c + 2*d*x))/d 
 + (B*a^3*b*sin(4*c + 4*d*x))/(8*d) + (9*B*a^2*b^2*sin(c + d*x))/(2*d) + ( 
3*A*a^2*b^2*sin(2*c + 2*d*x))/(2*d) + (3*A*a^2*b^2*sin(4*c + 4*d*x))/(16*d 
) + (B*a^2*b^2*sin(3*c + 3*d*x))/(2*d) + (3*A*a*b^3*sin(c + d*x))/d + (5*A 
*a^3*b*sin(c + d*x))/(2*d)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.73 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a^{5}-26 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{5}-120 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{3} b^{2}+33 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{5}+300 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{3} b^{2}+120 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a \,b^{4}+48 \sin \left (d x +c \right )^{5} a^{4} b -160 \sin \left (d x +c \right )^{3} a^{4} b -160 \sin \left (d x +c \right )^{3} a^{2} b^{3}+240 \sin \left (d x +c \right ) a^{4} b +480 \sin \left (d x +c \right ) a^{2} b^{3}+48 \sin \left (d x +c \right ) b^{5}+15 a^{5} d x +180 a^{3} b^{2} d x +120 a \,b^{4} d x}{48 d} \] Input:

int(cos(d*x+c)^6*(a+b*sec(d*x+c))^4*(A+B*sec(d*x+c)),x)
 

Output:

(8*cos(c + d*x)*sin(c + d*x)**5*a**5 - 26*cos(c + d*x)*sin(c + d*x)**3*a** 
5 - 120*cos(c + d*x)*sin(c + d*x)**3*a**3*b**2 + 33*cos(c + d*x)*sin(c + d 
*x)*a**5 + 300*cos(c + d*x)*sin(c + d*x)*a**3*b**2 + 120*cos(c + d*x)*sin( 
c + d*x)*a*b**4 + 48*sin(c + d*x)**5*a**4*b - 160*sin(c + d*x)**3*a**4*b - 
 160*sin(c + d*x)**3*a**2*b**3 + 240*sin(c + d*x)*a**4*b + 480*sin(c + d*x 
)*a**2*b**3 + 48*sin(c + d*x)*b**5 + 15*a**5*d*x + 180*a**3*b**2*d*x + 120 
*a*b**4*d*x)/(48*d)