Integrand size = 29, antiderivative size = 132 \[ \int \cos ^6(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{16} (5 A+6 C) x+\frac {B \sin (c+d x)}{d}+\frac {(5 A+6 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {(5 A+6 C) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {A \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 B \sin ^3(c+d x)}{3 d}+\frac {B \sin ^5(c+d x)}{5 d} \] Output:
1/16*(5*A+6*C)*x+B*sin(d*x+c)/d+1/16*(5*A+6*C)*cos(d*x+c)*sin(d*x+c)/d+1/2 4*(5*A+6*C)*cos(d*x+c)^3*sin(d*x+c)/d+1/6*A*cos(d*x+c)^5*sin(d*x+c)/d-2/3* B*sin(d*x+c)^3/d+1/5*B*sin(d*x+c)^5/d
Time = 0.17 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.77 \[ \int \cos ^6(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {960 B \sin (c+d x)-640 B \sin ^3(c+d x)+192 B \sin ^5(c+d x)+5 (60 A c+72 c C+60 A d x+72 C d x+(45 A+48 C) \sin (2 (c+d x))+(9 A+6 C) \sin (4 (c+d x))+A \sin (6 (c+d x)))}{960 d} \] Input:
Integrate[Cos[c + d*x]^6*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Output:
(960*B*Sin[c + d*x] - 640*B*Sin[c + d*x]^3 + 192*B*Sin[c + d*x]^5 + 5*(60* A*c + 72*c*C + 60*A*d*x + 72*C*d*x + (45*A + 48*C)*Sin[2*(c + d*x)] + (9*A + 6*C)*Sin[4*(c + d*x)] + A*Sin[6*(c + d*x)]))/(960*d)
Time = 0.54 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.94, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.379, Rules used = {3042, 4535, 3042, 3113, 2009, 4533, 3042, 3115, 3042, 3115, 24}
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.
| \(\displaystyle \int \cos ^6(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \int \cos ^6(c+d x) \left (C \sec ^2(c+d x)+A\right )dx+B \int \cos ^5(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {C \csc \left (c+d x+\frac {\pi }{2}\right )^2+A}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx+B \int \sin \left (c+d x+\frac {\pi }{2}\right )^5dx\) |
| \(\Big \downarrow \) 3113 |
| \(\displaystyle \int \frac {C \csc \left (c+d x+\frac {\pi }{2}\right )^2+A}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx-\frac {B \int \left (\sin ^4(c+d x)-2 \sin ^2(c+d x)+1\right )d(-\sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {C \csc \left (c+d x+\frac {\pi }{2}\right )^2+A}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx-\frac {B \left (-\frac {1}{5} \sin ^5(c+d x)+\frac {2}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 4533 |
| \(\displaystyle \frac {1}{6} (5 A+6 C) \int \cos ^4(c+d x)dx+\frac {A \sin (c+d x) \cos ^5(c+d x)}{6 d}-\frac {B \left (-\frac {1}{5} \sin ^5(c+d x)+\frac {2}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} (5 A+6 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {A \sin (c+d x) \cos ^5(c+d x)}{6 d}-\frac {B \left (-\frac {1}{5} \sin ^5(c+d x)+\frac {2}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{6} (5 A+6 C) \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x)}{6 d}-\frac {B \left (-\frac {1}{5} \sin ^5(c+d x)+\frac {2}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{6} (5 A+6 C) \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x)}{6 d}-\frac {B \left (-\frac {1}{5} \sin ^5(c+d x)+\frac {2}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{6} (5 A+6 C) \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {A \sin (c+d x) \cos ^5(c+d x)}{6 d}-\frac {B \left (-\frac {1}{5} \sin ^5(c+d x)+\frac {2}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{6} (5 A+6 C) \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {A \sin (c+d x) \cos ^5(c+d x)}{6 d}-\frac {B \left (-\frac {1}{5} \sin ^5(c+d x)+\frac {2}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{d}\) |
Input:
Int[Cos[c + d*x]^6*(A + B*Sec[c + d*x] + C*Sec[c + d*x]^2),x]
Output:
(A*Cos[c + d*x]^5*Sin[c + d*x])/(6*d) - (B*(-Sin[c + d*x] + (2*Sin[c + d*x ]^3)/3 - Sin[c + d*x]^5/5))/d + ((5*A + 6*C)*((Cos[c + d*x]^3*Sin[c + d*x] )/(4*d) + (3*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4))/6
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
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]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*(csc[(e_.) + (f_.)*(x_)]^2*(C_.) + (A_)), x_Symbol] :> Simp[A*Cot[e + f*x]*((b*Csc[e + f*x])^m/(f*m)), x] + Simp[(C*m + A*(m + 1))/(b^2*m) Int[(b*Csc[e + f*x])^(m + 2), x], x] /; Fr eeQ[{b, e, f, A, C}, x] && NeQ[C*m + A*(m + 1), 0] && LeQ[m, -1]
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]
Time = 0.78 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.72
| method | result | size |
| parallelrisch | \(\frac {\left (225 A +240 C \right ) \sin \left (2 d x +2 c \right )+\left (45 A +30 C \right ) \sin \left (4 d x +4 c \right )+100 B \sin \left (3 d x +3 c \right )+12 B \sin \left (5 d x +5 c \right )+5 A \sin \left (6 d x +6 c \right )+600 B \sin \left (d x +c \right )+300 x d \left (A +\frac {6 C}{5}\right )}{960 d}\) | \(95\) |
| derivativedivides | \(\frac {A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(115\) |
| default | \(\frac {A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(115\) |
| risch | \(\frac {5 A x}{16}+\frac {3 C x}{8}+\frac {5 B \sin \left (d x +c \right )}{8 d}+\frac {A \sin \left (6 d x +6 c \right )}{192 d}+\frac {B \sin \left (5 d x +5 c \right )}{80 d}+\frac {3 A \sin \left (4 d x +4 c \right )}{64 d}+\frac {\sin \left (4 d x +4 c \right ) C}{32 d}+\frac {5 B \sin \left (3 d x +3 c \right )}{48 d}+\frac {15 A \sin \left (2 d x +2 c \right )}{64 d}+\frac {\sin \left (2 d x +2 c \right ) C}{4 d}\) | \(127\) |
| norman | \(\frac {\left (-\frac {5 A}{16}-\frac {3 C}{8}\right ) x +\left (-\frac {45 A}{16}-\frac {27 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {25 A}{16}-\frac {15 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {25 A}{16}-\frac {15 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {5 A}{16}+\frac {3 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {25 A}{16}+\frac {15 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (\frac {25 A}{16}+\frac {15 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {45 A}{16}+\frac {27 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\frac {\left (15 A +2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{2 d}-\frac {\left (11 A -16 B +10 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{8 d}-\frac {\left (11 A +16 B +10 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {\left (19 A -32 B -6 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}+\frac {\left (19 A +32 B -6 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{12 d}-\frac {\left (475 A -688 B -150 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{120 d}-\frac {\left (475 A +688 B -150 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{120 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\) | \(359\) |
Input:
int(cos(d*x+c)^6*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x,method=_RETURNVERBOSE)
Output:
1/960*((225*A+240*C)*sin(2*d*x+2*c)+(45*A+30*C)*sin(4*d*x+4*c)+100*B*sin(3 *d*x+3*c)+12*B*sin(5*d*x+5*c)+5*A*sin(6*d*x+6*c)+600*B*sin(d*x+c)+300*x*d* (A+6/5*C))/d
Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.70 \[ \int \cos ^6(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (5 \, A + 6 \, C\right )} d x + {\left (40 \, A \cos \left (d x + c\right )^{5} + 48 \, B \cos \left (d x + c\right )^{4} + 10 \, {\left (5 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{3} + 64 \, B \cos \left (d x + c\right )^{2} + 15 \, {\left (5 \, A + 6 \, C\right )} \cos \left (d x + c\right ) + 128 \, B\right )} \sin \left (d x + c\right )}{240 \, d} \] Input:
integrate(cos(d*x+c)^6*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="frica s")
Output:
1/240*(15*(5*A + 6*C)*d*x + (40*A*cos(d*x + c)^5 + 48*B*cos(d*x + c)^4 + 1 0*(5*A + 6*C)*cos(d*x + c)^3 + 64*B*cos(d*x + c)^2 + 15*(5*A + 6*C)*cos(d* x + c) + 128*B)*sin(d*x + c))/d
Timed out. \[ \int \cos ^6(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \] Input:
integrate(cos(d*x+c)**6*(A+B*sec(d*x+c)+C*sec(d*x+c)**2),x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.87 \[ \int \cos ^6(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A - 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C}{960 \, d} \] Input:
integrate(cos(d*x+c)^6*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxim a")
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 - 64*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d* x + c))*B - 30*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C)/ d
Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (120) = 240\).
Time = 0.33 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.15 \[ \int \cos ^6(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (d x + c\right )} {\left (5 \, A + 6 \, C\right )} - \frac {2 \, {\left (165 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 240 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 150 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 25 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 560 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 210 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 450 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1248 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 60 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 450 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1248 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 25 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 560 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 210 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 165 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 240 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 150 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \] Input:
integrate(cos(d*x+c)^6*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="giac" )
Output:
1/240*(15*(d*x + c)*(5*A + 6*C) - 2*(165*A*tan(1/2*d*x + 1/2*c)^11 - 240*B *tan(1/2*d*x + 1/2*c)^11 + 150*C*tan(1/2*d*x + 1/2*c)^11 - 25*A*tan(1/2*d* x + 1/2*c)^9 - 560*B*tan(1/2*d*x + 1/2*c)^9 + 210*C*tan(1/2*d*x + 1/2*c)^9 + 450*A*tan(1/2*d*x + 1/2*c)^7 - 1248*B*tan(1/2*d*x + 1/2*c)^7 + 60*C*tan (1/2*d*x + 1/2*c)^7 - 450*A*tan(1/2*d*x + 1/2*c)^5 - 1248*B*tan(1/2*d*x + 1/2*c)^5 - 60*C*tan(1/2*d*x + 1/2*c)^5 + 25*A*tan(1/2*d*x + 1/2*c)^3 - 560 *B*tan(1/2*d*x + 1/2*c)^3 - 210*C*tan(1/2*d*x + 1/2*c)^3 - 165*A*tan(1/2*d *x + 1/2*c) - 240*B*tan(1/2*d*x + 1/2*c) - 150*C*tan(1/2*d*x + 1/2*c))/(ta n(1/2*d*x + 1/2*c)^2 + 1)^6)/d
Time = 11.45 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.95 \[ \int \cos ^6(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {5\,A\,x}{16}+\frac {3\,C\,x}{8}+\frac {15\,A\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {3\,A\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {A\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {5\,B\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {B\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {C\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {5\,B\,\sin \left (c+d\,x\right )}{8\,d} \] Input:
int(cos(c + d*x)^6*(A + B/cos(c + d*x) + C/cos(c + d*x)^2),x)
Output:
(5*A*x)/16 + (3*C*x)/8 + (15*A*sin(2*c + 2*d*x))/(64*d) + (3*A*sin(4*c + 4 *d*x))/(64*d) + (A*sin(6*c + 6*d*x))/(192*d) + (5*B*sin(3*c + 3*d*x))/(48* d) + (B*sin(5*c + 5*d*x))/(80*d) + (C*sin(2*c + 2*d*x))/(4*d) + (C*sin(4*c + 4*d*x))/(32*d) + (5*B*sin(c + d*x))/(8*d)
Time = 0.19 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.97 \[ \int \cos ^6(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {40 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a -130 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a -60 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} c +165 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +150 \cos \left (d x +c \right ) \sin \left (d x +c \right ) c +48 \sin \left (d x +c \right )^{5} b -160 \sin \left (d x +c \right )^{3} b +240 \sin \left (d x +c \right ) b +75 a d x +90 c d x}{240 d} \] Input:
int(cos(d*x+c)^6*(A+B*sec(d*x+c)+C*sec(d*x+c)^2),x)
Output:
(40*cos(c + d*x)*sin(c + d*x)**5*a - 130*cos(c + d*x)*sin(c + d*x)**3*a - 60*cos(c + d*x)*sin(c + d*x)**3*c + 165*cos(c + d*x)*sin(c + d*x)*a + 150* cos(c + d*x)*sin(c + d*x)*c + 48*sin(c + d*x)**5*b - 160*sin(c + d*x)**3*b + 240*sin(c + d*x)*b + 75*a*d*x + 90*c*d*x)/(240*d)