Integrand size = 31, antiderivative size = 176 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {1}{8} a^3 (13 A+15 B) x+\frac {a^3 (38 A+45 B) \sin (c+d x)}{15 d}+\frac {a^3 (13 A+15 B) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^3 (43 A+45 B) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac {a A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {(7 A+5 B) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{20 d} \]
1/8*a^3*(13*A+15*B)*x+1/15*a^3*(38*A+45*B)*sin(d*x+c)/d+1/8*a^3*(13*A+15*B )*cos(d*x+c)*sin(d*x+c)/d+1/60*a^3*(43*A+45*B)*cos(d*x+c)^2*sin(d*x+c)/d+1 /5*a*A*cos(d*x+c)^4*(a+a*sec(d*x+c))^2*sin(d*x+c)/d+1/20*(7*A+5*B)*cos(d*x +c)^3*(a^3+a^3*sec(d*x+c))*sin(d*x+c)/d
Time = 0.45 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.61 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {a^3 (780 A c+780 A d x+900 B d x+60 (23 A+26 B) \sin (c+d x)+480 (A+B) \sin (2 (c+d x))+170 A \sin (3 (c+d x))+120 B \sin (3 (c+d x))+45 A \sin (4 (c+d x))+15 B \sin (4 (c+d x))+6 A \sin (5 (c+d x)))}{480 d} \]
(a^3*(780*A*c + 780*A*d*x + 900*B*d*x + 60*(23*A + 26*B)*Sin[c + d*x] + 48 0*(A + B)*Sin[2*(c + d*x)] + 170*A*Sin[3*(c + d*x)] + 120*B*Sin[3*(c + d*x )] + 45*A*Sin[4*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 6*A*Sin[5*(c + d*x)]) )/(480*d)
Time = 1.05 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {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.
| \(\displaystyle \int \cos ^5(c+d x) (a \sec (c+d x)+a)^3 (A+B \sec (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^3 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {1}{5} \int \cos ^4(c+d x) (\sec (c+d x) a+a)^2 (a (7 A+5 B)+a (2 A+5 B) \sec (c+d x))dx+\frac {a A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (a (7 A+5 B)+a (2 A+5 B) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx+\frac {a A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int \cos ^3(c+d x) (\sec (c+d x) a+a) \left ((43 A+45 B) a^2+2 (11 A+15 B) \sec (c+d x) a^2\right )dx+\frac {(7 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((43 A+45 B) a^2+2 (11 A+15 B) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {(7 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 4484 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {a^3 (43 A+45 B) \sin (c+d x) \cos ^2(c+d x)}{3 d}-\frac {1}{3} \int -\cos ^2(c+d x) \left (15 (13 A+15 B) a^3+4 (38 A+45 B) \sec (c+d x) a^3\right )dx\right )+\frac {(7 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int \cos ^2(c+d x) \left (15 (13 A+15 B) a^3+4 (38 A+45 B) \sec (c+d x) a^3\right )dx+\frac {a^3 (43 A+45 B) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(7 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \int \frac {15 (13 A+15 B) a^3+4 (38 A+45 B) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a^3 (43 A+45 B) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(7 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (15 a^3 (13 A+15 B) \int \cos ^2(c+d x)dx+4 a^3 (38 A+45 B) \int \cos (c+d x)dx\right )+\frac {a^3 (43 A+45 B) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(7 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (4 a^3 (38 A+45 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+15 a^3 (13 A+15 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\right )+\frac {a^3 (43 A+45 B) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(7 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (4 a^3 (38 A+45 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+15 a^3 (13 A+15 B) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )\right )+\frac {a^3 (43 A+45 B) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(7 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {1}{3} \left (4 a^3 (38 A+45 B) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+15 a^3 (13 A+15 B) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a^3 (43 A+45 B) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(7 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{4} \left (\frac {a^3 (43 A+45 B) \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac {1}{3} \left (\frac {4 a^3 (38 A+45 B) \sin (c+d x)}{d}+15 a^3 (13 A+15 B) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )+\frac {(7 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}\right )+\frac {a A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
(a*A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + (((7*A + 5*B)*Cos[c + d*x]^3*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(4*d) + ((a^3*( 43*A + 45*B)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + ((4*a^3*(38*A + 45*B)*Si n[c + d*x])/d + 15*a^3*(13*A + 15*B)*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2 *d)))/3)/4)/5
3.1.70.3.1 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]
Time = 3.21 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.53
| method | result | size |
| parallelrisch | \(\frac {3 \left (\frac {32 \left (A +B \right ) \sin \left (2 d x +2 c \right )}{3}+\frac {2 \left (\frac {17 A}{3}+4 B \right ) \sin \left (3 d x +3 c \right )}{3}+\left (A +\frac {B}{3}\right ) \sin \left (4 d x +4 c \right )+\frac {2 A \sin \left (5 d x +5 c \right )}{15}+\frac {4 \left (23 A +26 B \right ) \sin \left (d x +c \right )}{3}+\frac {52 d \left (A +\frac {15 B}{13}\right ) x}{3}\right ) a^{3}}{32 d}\) | \(93\) |
| risch | \(\frac {13 a^{3} A x}{8}+\frac {15 a^{3} x B}{8}+\frac {23 a^{3} A \sin \left (d x +c \right )}{8 d}+\frac {13 a^{3} B \sin \left (d x +c \right )}{4 d}+\frac {a^{3} A \sin \left (5 d x +5 c \right )}{80 d}+\frac {3 a^{3} A \sin \left (4 d x +4 c \right )}{32 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{3}}{32 d}+\frac {17 a^{3} A \sin \left (3 d x +3 c \right )}{48 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{3}}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} A}{d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{3}}{d}\) | \(170\) |
| derivativedivides | \(\frac {a^{3} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{3} \sin \left (d x +c \right )+a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {a^{3} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(223\) |
| default | \(\frac {a^{3} A \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{3} \sin \left (d x +c \right )+a^{3} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 B \,a^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {a^{3} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(223\) |
| norman | \(\frac {-\frac {a^{3} \left (13 A +15 B \right ) x}{8}-\frac {37 a^{3} \left (13 A +15 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{60 d}+\frac {5 a^{3} \left (13 A +15 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{12 d}+\frac {a^{3} \left (13 A +15 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{4 d}-\frac {a^{3} \left (13 A +15 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4}+\frac {a^{3} \left (13 A +15 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}+\frac {3 a^{3} \left (13 A +15 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4}-\frac {3 a^{3} \left (13 A +15 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{4}-\frac {a^{3} \left (13 A +15 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{4}+\frac {a^{3} \left (13 A +15 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{4}+\frac {a^{3} \left (13 A +15 B \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{8}+\frac {a^{3} \left (31 A +1365 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{60 d}-\frac {a^{3} \left (51 A +49 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{3} \left (193 A +75 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}+\frac {a^{3} \left (857 A -45 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{60 d}-\frac {a^{3} \left (1127 A +1005 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{60 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}\) | \(420\) |
3/32*(32/3*(A+B)*sin(2*d*x+2*c)+2/3*(17/3*A+4*B)*sin(3*d*x+3*c)+(A+1/3*B)* sin(4*d*x+4*c)+2/15*A*sin(5*d*x+5*c)+4/3*(23*A+26*B)*sin(d*x+c)+52/3*d*(A+ 15/13*B)*x)*a^3/d
Time = 0.26 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.62 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (13 \, A + 15 \, B\right )} a^{3} d x + {\left (24 \, A a^{3} \cos \left (d x + c\right )^{4} + 30 \, {\left (3 \, A + B\right )} a^{3} \cos \left (d x + c\right )^{3} + 8 \, {\left (19 \, A + 15 \, B\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \, {\left (13 \, A + 15 \, B\right )} a^{3} \cos \left (d x + c\right ) + 8 \, {\left (38 \, A + 45 \, B\right )} a^{3}\right )} \sin \left (d x + c\right )}{120 \, d} \]
1/120*(15*(13*A + 15*B)*a^3*d*x + (24*A*a^3*cos(d*x + c)^4 + 30*(3*A + B)* a^3*cos(d*x + c)^3 + 8*(19*A + 15*B)*a^3*cos(d*x + c)^2 + 15*(13*A + 15*B) *a^3*cos(d*x + c) + 8*(38*A + 45*B)*a^3)*sin(d*x + c))/d
Timed out. \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
Time = 0.21 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.21 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{3} - 480 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{3} + 45 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} + 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 480 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 360 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 480 \, B a^{3} \sin \left (d x + c\right )}{480 \, d} \]
1/480*(32*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*A*a^3 - 480*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^3 + 45*(12*d*x + 12*c + sin(4*d *x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^3 + 120*(2*d*x + 2*c + sin(2*d*x + 2*c ))*A*a^3 - 480*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^3 + 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^3 + 360*(2*d*x + 2*c + sin(2 *d*x + 2*c))*B*a^3 + 480*B*a^3*sin(d*x + c))/d
Time = 0.33 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.19 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (13 \, A a^{3} + 15 \, B a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (195 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 225 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 910 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1050 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1664 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1920 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1330 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1830 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 765 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 735 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \]
1/120*(15*(13*A*a^3 + 15*B*a^3)*(d*x + c) + 2*(195*A*a^3*tan(1/2*d*x + 1/2 *c)^9 + 225*B*a^3*tan(1/2*d*x + 1/2*c)^9 + 910*A*a^3*tan(1/2*d*x + 1/2*c)^ 7 + 1050*B*a^3*tan(1/2*d*x + 1/2*c)^7 + 1664*A*a^3*tan(1/2*d*x + 1/2*c)^5 + 1920*B*a^3*tan(1/2*d*x + 1/2*c)^5 + 1330*A*a^3*tan(1/2*d*x + 1/2*c)^3 + 1830*B*a^3*tan(1/2*d*x + 1/2*c)^3 + 765*A*a^3*tan(1/2*d*x + 1/2*c) + 735*B *a^3*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^5)/d
Time = 16.07 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.40 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {\left (\frac {13\,A\,a^3}{4}+\frac {15\,B\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {91\,A\,a^3}{6}+\frac {35\,B\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {416\,A\,a^3}{15}+32\,B\,a^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {133\,A\,a^3}{6}+\frac {61\,B\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {51\,A\,a^3}{4}+\frac {49\,B\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (13\,A+15\,B\right )}{4\,\left (\frac {13\,A\,a^3}{4}+\frac {15\,B\,a^3}{4}\right )}\right )\,\left (13\,A+15\,B\right )}{4\,d} \]
(tan(c/2 + (d*x)/2)*((51*A*a^3)/4 + (49*B*a^3)/4) + tan(c/2 + (d*x)/2)^9*( (13*A*a^3)/4 + (15*B*a^3)/4) + tan(c/2 + (d*x)/2)^7*((91*A*a^3)/6 + (35*B* a^3)/2) + tan(c/2 + (d*x)/2)^3*((133*A*a^3)/6 + (61*B*a^3)/2) + tan(c/2 + (d*x)/2)^5*((416*A*a^3)/15 + 32*B*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*A + 15*B ))/(4*((13*A*a^3)/4 + (15*B*a^3)/4)))*(13*A + 15*B))/(4*d)