Integrand size = 40, antiderivative size = 176 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{8} a^3 (13 B+15 C) x+\frac {a^3 (38 B+45 C) \sin (c+d x)}{15 d}+\frac {a^3 (13 B+15 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^3 (43 B+45 C) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac {a B \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {(7 B+5 C) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{20 d} \]
1/8*a^3*(13*B+15*C)*x+1/15*a^3*(38*B+45*C)*sin(d*x+c)/d+1/8*a^3*(13*B+15*C )*cos(d*x+c)*sin(d*x+c)/d+1/60*a^3*(43*B+45*C)*cos(d*x+c)^2*sin(d*x+c)/d+1 /5*a*B*cos(d*x+c)^4*(a+a*sec(d*x+c))^2*sin(d*x+c)/d+1/20*(7*B+5*C)*cos(d*x +c)^3*(a^3+a^3*sec(d*x+c))*sin(d*x+c)/d
Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.61 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 (780 B c+780 B d x+900 C d x+60 (23 B+26 C) \sin (c+d x)+480 (B+C) \sin (2 (c+d x))+170 B \sin (3 (c+d x))+120 C \sin (3 (c+d x))+45 B \sin (4 (c+d x))+15 C \sin (4 (c+d x))+6 B \sin (5 (c+d x)))}{480 d} \]
(a^3*(780*B*c + 780*B*d*x + 900*C*d*x + 60*(23*B + 26*C)*Sin[c + d*x] + 48 0*(B + C)*Sin[2*(c + d*x)] + 170*B*Sin[3*(c + d*x)] + 120*C*Sin[3*(c + d*x )] + 45*B*Sin[4*(c + d*x)] + 15*C*Sin[4*(c + d*x)] + 6*B*Sin[5*(c + d*x)]) )/(480*d)
Time = 1.13 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3042, 4560, 3042, 4505, 3042, 4505, 3042, 4484, 25, 3042, 4274, 3042, 3115, 24, 3117}
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.
| \(\displaystyle \int \cos ^6(c+d x) (a \sec (c+d x)+a)^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^6}dx\) |
| \(\Big \downarrow \) 4560 |
| \(\displaystyle \int \cos ^5(c+d x) (a \sec (c+d x)+a)^3 (B+C \sec (c+d x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (B+C \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {1}{5} \int \cos ^4(c+d x) (\sec (c+d x) a+a)^2 (a (7 B+5 C)+a (2 B+5 C) \sec (c+d x))dx+\frac {a B \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (a (7 B+5 C)+a (2 B+5 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {a B \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int \cos ^3(c+d x) (\sec (c+d x) a+a) \left ((43 B+45 C) a^2+2 (11 B+15 C) \sec (c+d x) a^2\right )dx+\frac {(7 B+5 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((43 B+45 C) a^2+2 (11 B+15 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {(7 B+5 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 4484 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {a^3 (43 B+45 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}-\frac {1}{3} \int -\cos ^2(c+d x) \left (15 (13 B+15 C) a^3+4 (38 B+45 C) \sec (c+d x) a^3\right )dx\right )+\frac {(7 B+5 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int \cos ^2(c+d x) \left (15 (13 B+15 C) a^3+4 (38 B+45 C) \sec (c+d x) a^3\right )dx+\frac {a^3 (43 B+45 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(7 B+5 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int \frac {15 (13 B+15 C) a^3+4 (38 B+45 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a^3 (43 B+45 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(7 B+5 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (15 a^3 (13 B+15 C) \int \cos ^2(c+d x)dx+4 a^3 (38 B+45 C) \int \cos (c+d x)dx\right )+\frac {a^3 (43 B+45 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(7 B+5 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (4 a^3 (38 B+45 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+15 a^3 (13 B+15 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\right )+\frac {a^3 (43 B+45 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(7 B+5 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (4 a^3 (38 B+45 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+15 a^3 (13 B+15 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )\right )+\frac {a^3 (43 B+45 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(7 B+5 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (4 a^3 (38 B+45 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+15 a^3 (13 B+15 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a^3 (43 B+45 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(7 B+5 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {a^3 (43 B+45 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac {1}{3} \left (\frac {4 a^3 (38 B+45 C) \sin (c+d x)}{d}+15 a^3 (13 B+15 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )+\frac {(7 B+5 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a B \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
(a*B*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + (((7*B + 5*C)*Cos[c + d*x]^3*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(4*d) + ((a^3*( 43*B + 45*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + ((4*a^3*(38*B + 45*C)*Si n[c + d*x])/d + 15*a^3*(13*B + 15*C)*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2 *d)))/3)/4)/5
3.4.30.3.1 Defintions of rubi rules used
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[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[a Int[(d*Csc[e + f*x])^n, x], x] + Simp[b/d In t[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( a_))*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[A*a*Cot[e + f*x]*((d*Csc[e + f*x])^n/(f*n)), x] + Simp[1/(d*n) Int[(d*Csc[e + f*x])^( n + 1)*Simp[n*(B*a + A*b) + (B*b*n + A*a*(n + 1))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && LeQ[n, -1]
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[b/(a*d*n) Int[(a + b*Csc[e + f*x])^(m - 1)*(d*Csc[e + f*x])^(n + 1)*Sim p[a*A*(m - n - 1) - b*B*n - (a*B*n + A*b*(m + n))*Csc[e + f*x], x], x], x] /; FreeQ[{a, b, d, e, f, A, B}, x] && NeQ[A*b - a*B, 0] && EqQ[a^2 - b^2, 0 ] && GtQ[m, 1/2] && LtQ[n, -1]
Int[((a_.) + csc[(e_.) + (f_.)*(x_)]*(b_.))^(m_.)*((A_.) + csc[(e_.) + (f_. )*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*((c_.) + csc[(e_.) + (f_.) *(x_)]*(d_.))^(n_.), x_Symbol] :> Simp[1/b^2 Int[(a + b*Csc[e + f*x])^(m + 1)*(c + d*Csc[e + f*x])^n*(b*B - a*C + b*C*Csc[e + f*x]), x], x] /; FreeQ [{a, b, c, d, e, f, A, B, C, m, n}, x] && EqQ[A*b^2 - a*b*B + a^2*C, 0]
Time = 0.33 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.53
| method | result | size |
| parallelrisch | \(\frac {3 \left (\frac {32 \left (B +C \right ) \sin \left (2 d x +2 c \right )}{3}+\frac {2 \left (\frac {17 B}{3}+4 C \right ) \sin \left (3 d x +3 c \right )}{3}+\left (B +\frac {C}{3}\right ) \sin \left (4 d x +4 c \right )+\frac {2 B \sin \left (5 d x +5 c \right )}{15}+\frac {4 \left (23 B +26 C \right ) \sin \left (d x +c \right )}{3}+\frac {52 \left (B +\frac {15 C}{13}\right ) x d}{3}\right ) a^{3}}{32 d}\) | \(93\) |
| risch | \(\frac {13 a^{3} B x}{8}+\frac {15 a^{3} x C}{8}+\frac {23 a^{3} B \sin \left (d x +c \right )}{8 d}+\frac {13 a^{3} C \sin \left (d x +c \right )}{4 d}+\frac {B \,a^{3} \sin \left (5 d x +5 c \right )}{80 d}+\frac {3 B \,a^{3} \sin \left (4 d x +4 c \right )}{32 d}+\frac {\sin \left (4 d x +4 c \right ) a^{3} C}{32 d}+\frac {17 B \,a^{3} \sin \left (3 d x +3 c \right )}{48 d}+\frac {\sin \left (3 d x +3 c \right ) a^{3} C}{4 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{3}}{d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} C}{d}\) | \(170\) |
| derivativedivides | \(\frac {B \,a^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{3} C \sin \left (d x +c \right )+B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 a^{3} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {B \,a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{3} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(223\) |
| default | \(\frac {B \,a^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{3} C \sin \left (d x +c \right )+B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 a^{3} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {B \,a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{3} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(223\) |
3/32*(32/3*(B+C)*sin(2*d*x+2*c)+2/3*(17/3*B+4*C)*sin(3*d*x+3*c)+(B+1/3*C)* sin(4*d*x+4*c)+2/15*B*sin(5*d*x+5*c)+4/3*(23*B+26*C)*sin(d*x+c)+52/3*(B+15 /13*C)*x*d)*a^3/d
Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.62 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (13 \, B + 15 \, C\right )} a^{3} d x + {\left (24 \, B a^{3} \cos \left (d x + c\right )^{4} + 30 \, {\left (3 \, B + C\right )} a^{3} \cos \left (d x + c\right )^{3} + 8 \, {\left (19 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \, {\left (13 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right ) + 8 \, {\left (38 \, B + 45 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{120 \, d} \]
integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^3*(B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")
1/120*(15*(13*B + 15*C)*a^3*d*x + (24*B*a^3*cos(d*x + c)^4 + 30*(3*B + C)* a^3*cos(d*x + c)^3 + 8*(19*B + 15*C)*a^3*cos(d*x + c)^2 + 15*(13*B + 15*C) *a^3*cos(d*x + c) + 8*(38*B + 45*C)*a^3)*sin(d*x + c))/d
Timed out. \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Time = 0.25 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.21 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{3} - 480 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} + 45 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 480 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 360 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 480 \, C a^{3} \sin \left (d x + c\right )}{480 \, d} \]
integrate(cos(d*x+c)^6*(a+a*sec(d*x+c))^3*(B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")
1/480*(32*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*B*a^3 - 480*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^3 + 45*(12*d*x + 12*c + sin(4*d *x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^3 + 120*(2*d*x + 2*c + sin(2*d*x + 2*c ))*B*a^3 - 480*(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^3 + 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*C*a^3 + 360*(2*d*x + 2*c + sin(2 *d*x + 2*c))*C*a^3 + 480*C*a^3*sin(d*x + c))/d
Time = 0.33 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.19 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (13 \, B a^{3} + 15 \, C a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (195 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 225 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 910 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1050 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1664 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1920 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1330 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1830 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 765 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 735 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \]
1/120*(15*(13*B*a^3 + 15*C*a^3)*(d*x + c) + 2*(195*B*a^3*tan(1/2*d*x + 1/2 *c)^9 + 225*C*a^3*tan(1/2*d*x + 1/2*c)^9 + 910*B*a^3*tan(1/2*d*x + 1/2*c)^ 7 + 1050*C*a^3*tan(1/2*d*x + 1/2*c)^7 + 1664*B*a^3*tan(1/2*d*x + 1/2*c)^5 + 1920*C*a^3*tan(1/2*d*x + 1/2*c)^5 + 1330*B*a^3*tan(1/2*d*x + 1/2*c)^3 + 1830*C*a^3*tan(1/2*d*x + 1/2*c)^3 + 765*B*a^3*tan(1/2*d*x + 1/2*c) + 735*C *a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^5)/d
Time = 19.19 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.40 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (\frac {13\,B\,a^3}{4}+\frac {15\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {91\,B\,a^3}{6}+\frac {35\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {416\,B\,a^3}{15}+32\,C\,a^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {133\,B\,a^3}{6}+\frac {61\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {51\,B\,a^3}{4}+\frac {49\,C\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (13\,B+15\,C\right )}{4\,\left (\frac {13\,B\,a^3}{4}+\frac {15\,C\,a^3}{4}\right )}\right )\,\left (13\,B+15\,C\right )}{4\,d} \]
(tan(c/2 + (d*x)/2)*((51*B*a^3)/4 + (49*C*a^3)/4) + tan(c/2 + (d*x)/2)^9*( (13*B*a^3)/4 + (15*C*a^3)/4) + tan(c/2 + (d*x)/2)^7*((91*B*a^3)/6 + (35*C* a^3)/2) + tan(c/2 + (d*x)/2)^3*((133*B*a^3)/6 + (61*C*a^3)/2) + tan(c/2 + (d*x)/2)^5*((416*B*a^3)/15 + 32*C*a^3))/(d*(5*tan(c/2 + (d*x)/2)^2 + 10*ta n(c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x)/2)^6 + 5*tan(c/2 + (d*x)/2)^8 + ta n(c/2 + (d*x)/2)^10 + 1)) + (a^3*atan((a^3*tan(c/2 + (d*x)/2)*(13*B + 15*C ))/(4*((13*B*a^3)/4 + (15*C*a^3)/4)))*(13*B + 15*C))/(4*d)