Integrand size = 41, antiderivative size = 187 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{8} a^2 (6 A+7 B+8 C) x+\frac {a^2 (18 A+20 B+25 C) \sin (c+d x)}{15 d}+\frac {a^2 (6 A+7 B+8 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 (18 A+25 B+20 C) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac {A \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {(2 A+5 B) \cos ^3(c+d x) \left (a^2+a^2 \sec (c+d x)\right ) \sin (c+d x)}{20 d} \]
1/8*a^2*(6*A+7*B+8*C)*x+1/15*a^2*(18*A+20*B+25*C)*sin(d*x+c)/d+1/8*a^2*(6* A+7*B+8*C)*cos(d*x+c)*sin(d*x+c)/d+1/60*a^2*(18*A+25*B+20*C)*cos(d*x+c)^2* sin(d*x+c)/d+1/5*A*cos(d*x+c)^4*(a+a*sec(d*x+c))^2*sin(d*x+c)/d+1/20*(2*A+ 5*B)*cos(d*x+c)^3*(a^2+a^2*sec(d*x+c))*sin(d*x+c)/d
Time = 0.39 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.71 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^2 (240 A c+420 B c+360 A d x+420 B d x+480 C d x+60 (11 A+12 B+14 C) \sin (c+d x)+240 (A+B+C) \sin (2 (c+d x))+90 A \sin (3 (c+d x))+80 B \sin (3 (c+d x))+40 C \sin (3 (c+d x))+30 A \sin (4 (c+d x))+15 B \sin (4 (c+d x))+6 A \sin (5 (c+d x)))}{480 d} \]
(a^2*(240*A*c + 420*B*c + 360*A*d*x + 420*B*d*x + 480*C*d*x + 60*(11*A + 1 2*B + 14*C)*Sin[c + d*x] + 240*(A + B + C)*Sin[2*(c + d*x)] + 90*A*Sin[3*( c + d*x)] + 80*B*Sin[3*(c + d*x)] + 40*C*Sin[3*(c + d*x)] + 30*A*Sin[4*(c + d*x)] + 15*B*Sin[4*(c + d*x)] + 6*A*Sin[5*(c + d*x)]))/(480*d)
Time = 1.10 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.317, Rules used = {3042, 4574, 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.
| \(\displaystyle \int \cos ^5(c+d x) (a \sec (c+d x)+a)^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a \csc \left (c+d x+\frac {\pi }{2}\right )+a\right )^2 \left (A+B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 4574 |
| \(\displaystyle \frac {\int \cos ^4(c+d x) (\sec (c+d x) a+a)^2 (a (2 A+5 B)+a (2 A+5 C) \sec (c+d x))dx}{5 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right )^2 \left (a (2 A+5 B)+a (2 A+5 C) \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx}{5 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 4505 |
| \(\displaystyle \frac {\frac {1}{4} \int \cos ^3(c+d x) (\sec (c+d x) a+a) \left ((18 A+25 B+20 C) a^2+2 (6 A+5 B+10 C) \sec (c+d x) a^2\right )dx+\frac {(2 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}}{5 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \int \frac {\left (\csc \left (c+d x+\frac {\pi }{2}\right ) a+a\right ) \left ((18 A+25 B+20 C) a^2+2 (6 A+5 B+10 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {(2 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}}{5 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 4484 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {a^3 (18 A+25 B+20 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}-\frac {1}{3} \int -\cos ^2(c+d x) \left (15 (6 A+7 B+8 C) a^3+4 (18 A+20 B+25 C) \sec (c+d x) a^3\right )dx\right )+\frac {(2 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}}{5 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \int \cos ^2(c+d x) \left (15 (6 A+7 B+8 C) a^3+4 (18 A+20 B+25 C) \sec (c+d x) a^3\right )dx+\frac {a^3 (18 A+25 B+20 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(2 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}}{5 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \int \frac {15 (6 A+7 B+8 C) a^3+4 (18 A+20 B+25 C) \csc \left (c+d x+\frac {\pi }{2}\right ) a^3}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {a^3 (18 A+25 B+20 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(2 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}}{5 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 4274 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (15 a^3 (6 A+7 B+8 C) \int \cos ^2(c+d x)dx+4 a^3 (18 A+20 B+25 C) \int \cos (c+d x)dx\right )+\frac {a^3 (18 A+25 B+20 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(2 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}}{5 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (4 a^3 (18 A+20 B+25 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+15 a^3 (6 A+7 B+8 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx\right )+\frac {a^3 (18 A+25 B+20 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(2 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}}{5 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (4 a^3 (18 A+20 B+25 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+15 a^3 (6 A+7 B+8 C) \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )\right )+\frac {a^3 (18 A+25 B+20 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(2 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}}{5 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{3} \left (4 a^3 (18 A+20 B+25 C) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx+15 a^3 (6 A+7 B+8 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )+\frac {a^3 (18 A+25 B+20 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {(2 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}}{5 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {a^3 (18 A+25 B+20 C) \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac {1}{3} \left (\frac {4 a^3 (18 A+20 B+25 C) \sin (c+d x)}{d}+15 a^3 (6 A+7 B+8 C) \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )+\frac {(2 A+5 B) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{4 d}}{5 a}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d}\) |
(A*Cos[c + d*x]^4*(a + a*Sec[c + d*x])^2*Sin[c + d*x])/(5*d) + (((2*A + 5* B)*Cos[c + d*x]^3*(a^3 + a^3*Sec[c + d*x])*Sin[c + d*x])/(4*d) + ((a^3*(18 *A + 25*B + 20*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + ((4*a^3*(18*A + 20* B + 25*C)*Sin[c + d*x])/d + 15*a^3*(6*A + 7*B + 8*C)*(x/2 + (Cos[c + d*x]* Sin[c + d*x])/(2*d)))/3)/4)/(5*a)
3.5.25.3.1 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_.) + 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/(b*d*n) Int[(a + b*Csc[e + f*x])^m*(d*Csc[ e + f*x])^(n + 1)*Simp[a*A*m - b*B*n - b*(A*(m + n + 1) + C*n)*Csc[e + f*x] , x], x], x] /; FreeQ[{a, b, d, e, f, A, B, C, m}, x] && EqQ[a^2 - b^2, 0] && !LtQ[m, -2^(-1)] && (LtQ[n, -2^(-1)] || EqQ[m + n + 1, 0])
Time = 0.47 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.54
| method | result | size |
| parallelrisch | \(\frac {3 a^{2} \left (\frac {8 \left (A +B +C \right ) \sin \left (2 d x +2 c \right )}{3}+\left (A +\frac {8 B}{9}+\frac {4 C}{9}\right ) \sin \left (3 d x +3 c \right )+\frac {\left (A +\frac {B}{2}\right ) \sin \left (4 d x +4 c \right )}{3}+\frac {A \sin \left (5 d x +5 c \right )}{15}+2 \left (\frac {11 A}{3}+4 B +\frac {14 C}{3}\right ) \sin \left (d x +c \right )+4 x d \left (A +\frac {7 B}{6}+\frac {4 C}{3}\right )\right )}{16 d}\) | \(101\) |
| risch | \(\frac {3 a^{2} A x}{4}+\frac {7 a^{2} B x}{8}+a^{2} x C +\frac {11 \sin \left (d x +c \right ) a^{2} A}{8 d}+\frac {3 a^{2} B \sin \left (d x +c \right )}{2 d}+\frac {7 \sin \left (d x +c \right ) C \,a^{2}}{4 d}+\frac {a^{2} A \sin \left (5 d x +5 c \right )}{80 d}+\frac {a^{2} A \sin \left (4 d x +4 c \right )}{16 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{2}}{32 d}+\frac {3 a^{2} A \sin \left (3 d x +3 c \right )}{16 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{2}}{6 d}+\frac {\sin \left (3 d x +3 c \right ) C \,a^{2}}{12 d}+\frac {a^{2} A \sin \left (2 d x +2 c \right )}{2 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{2}}{2 d}+\frac {\sin \left (2 d x +2 c \right ) C \,a^{2}}{2 d}\) | \(229\) |
| derivativedivides | \(\frac {\frac {a^{2} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,a^{2} \sin \left (d x +c \right )+2 a^{2} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {2 B \,a^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+2 C \,a^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{2} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {C \,a^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(247\) |
| default | \(\frac {\frac {a^{2} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \,a^{2} \sin \left (d x +c \right )+2 a^{2} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {2 B \,a^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+2 C \,a^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {a^{2} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {C \,a^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}}{d}\) | \(247\) |
| norman | \(\frac {-\frac {a^{2} \left (6 A +7 B +8 C \right ) x}{8}+\frac {5 a^{2} \left (6 A +7 B +8 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{12 d}+\frac {a^{2} \left (6 A +7 B +8 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{4 d}-\frac {a^{2} \left (6 A +7 B +8 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{4}+\frac {a^{2} \left (6 A +7 B +8 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}+\frac {3 a^{2} \left (6 A +7 B +8 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4}-\frac {3 a^{2} \left (6 A +7 B +8 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{4}-\frac {a^{2} \left (6 A +7 B +8 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{4}+\frac {a^{2} \left (6 A +7 B +8 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{4}+\frac {a^{2} \left (6 A +7 B +8 C \right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{8}-\frac {a^{2} \left (26 A +25 B +24 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {a^{2} \left (126 A +67 B +8 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}-\frac {a^{2} \left (126 A +355 B +200 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{60 d}-\frac {a^{2} \left (414 A -445 B -920 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{60 d}-\frac {a^{2} \left (882 A +245 B +760 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{60 d}+\frac {a^{2} \left (942 A -85 B +40 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{60 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{3}}\) | \(468\) |
3/16*a^2*(8/3*(A+B+C)*sin(2*d*x+2*c)+(A+8/9*B+4/9*C)*sin(3*d*x+3*c)+1/3*(A +1/2*B)*sin(4*d*x+4*c)+1/15*A*sin(5*d*x+5*c)+2*(11/3*A+4*B+14/3*C)*sin(d*x +c)+4*x*d*(A+7/6*B+4/3*C))/d
Time = 0.26 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.65 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (6 \, A + 7 \, B + 8 \, C\right )} a^{2} d x + {\left (24 \, A a^{2} \cos \left (d x + c\right )^{4} + 30 \, {\left (2 \, A + B\right )} a^{2} \cos \left (d x + c\right )^{3} + 8 \, {\left (9 \, A + 10 \, B + 5 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 15 \, {\left (6 \, A + 7 \, B + 8 \, C\right )} a^{2} \cos \left (d x + c\right ) + 8 \, {\left (18 \, A + 20 \, B + 25 \, C\right )} a^{2}\right )} \sin \left (d x + c\right )}{120 \, d} \]
integrate(cos(d*x+c)^5*(a+a*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="fricas")
1/120*(15*(6*A + 7*B + 8*C)*a^2*d*x + (24*A*a^2*cos(d*x + c)^4 + 30*(2*A + B)*a^2*cos(d*x + c)^3 + 8*(9*A + 10*B + 5*C)*a^2*cos(d*x + c)^2 + 15*(6*A + 7*B + 8*C)*a^2*cos(d*x + c) + 8*(18*A + 20*B + 25*C)*a^2)*sin(d*x + c)) /d
Timed out. \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Time = 0.25 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.26 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{2} - 160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} + 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} + 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} - 160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} + 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} + 480 \, C a^{2} \sin \left (d x + c\right )}{480 \, d} \]
integrate(cos(d*x+c)^5*(a+a*sec(d*x+c))^2*(A+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))*A*a^2 - 160*(sin(d*x + c)^3 - 3*sin(d*x + c))*A*a^2 + 30*(12*d*x + 12*c + sin(4*d *x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^2 - 320*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^2 + 15*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a ^2 + 120*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a^2 - 160*(sin(d*x + c)^3 - 3* sin(d*x + c))*C*a^2 + 240*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^2 + 480*C*a ^2*sin(d*x + c))/d
Time = 0.33 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.60 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (6 \, A a^{2} + 7 \, B a^{2} + 8 \, C a^{2}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (90 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 105 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 120 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 420 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 490 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 560 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 864 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 800 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1120 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 540 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 790 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1040 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 390 \, A a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 375 \, B a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 360 \, C a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \]
integrate(cos(d*x+c)^5*(a+a*sec(d*x+c))^2*(A+B*sec(d*x+c)+C*sec(d*x+c)^2), x, algorithm="giac")
1/120*(15*(6*A*a^2 + 7*B*a^2 + 8*C*a^2)*(d*x + c) + 2*(90*A*a^2*tan(1/2*d* x + 1/2*c)^9 + 105*B*a^2*tan(1/2*d*x + 1/2*c)^9 + 120*C*a^2*tan(1/2*d*x + 1/2*c)^9 + 420*A*a^2*tan(1/2*d*x + 1/2*c)^7 + 490*B*a^2*tan(1/2*d*x + 1/2* c)^7 + 560*C*a^2*tan(1/2*d*x + 1/2*c)^7 + 864*A*a^2*tan(1/2*d*x + 1/2*c)^5 + 800*B*a^2*tan(1/2*d*x + 1/2*c)^5 + 1120*C*a^2*tan(1/2*d*x + 1/2*c)^5 + 540*A*a^2*tan(1/2*d*x + 1/2*c)^3 + 790*B*a^2*tan(1/2*d*x + 1/2*c)^3 + 1040 *C*a^2*tan(1/2*d*x + 1/2*c)^3 + 390*A*a^2*tan(1/2*d*x + 1/2*c) + 375*B*a^2 *tan(1/2*d*x + 1/2*c) + 360*C*a^2*tan(1/2*d*x + 1/2*c))/(tan(1/2*d*x + 1/2 *c)^2 + 1)^5)/d
Time = 18.99 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.55 \[ \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (\frac {3\,A\,a^2}{2}+\frac {7\,B\,a^2}{4}+2\,C\,a^2\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (7\,A\,a^2+\frac {49\,B\,a^2}{6}+\frac {28\,C\,a^2}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {72\,A\,a^2}{5}+\frac {40\,B\,a^2}{3}+\frac {56\,C\,a^2}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (9\,A\,a^2+\frac {79\,B\,a^2}{6}+\frac {52\,C\,a^2}{3}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {13\,A\,a^2}{2}+\frac {25\,B\,a^2}{4}+6\,C\,a^2\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (6\,A+7\,B+8\,C\right )}{4\,\left (\frac {3\,A\,a^2}{2}+\frac {7\,B\,a^2}{4}+2\,C\,a^2\right )}\right )\,\left (6\,A+7\,B+8\,C\right )}{4\,d} \]
(tan(c/2 + (d*x)/2)^9*((3*A*a^2)/2 + (7*B*a^2)/4 + 2*C*a^2) + tan(c/2 + (d *x)/2)^7*(7*A*a^2 + (49*B*a^2)/6 + (28*C*a^2)/3) + tan(c/2 + (d*x)/2)^3*(9 *A*a^2 + (79*B*a^2)/6 + (52*C*a^2)/3) + tan(c/2 + (d*x)/2)^5*((72*A*a^2)/5 + (40*B*a^2)/3 + (56*C*a^2)/3) + tan(c/2 + (d*x)/2)*((13*A*a^2)/2 + (25*B *a^2)/4 + 6*C*a^2))/(d*(5*tan(c/2 + (d*x)/2)^2 + 10*tan(c/2 + (d*x)/2)^4 + 10*tan(c/2 + (d*x)/2)^6 + 5*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 + 1)) + (a^2*atan((a^2*tan(c/2 + (d*x)/2)*(6*A + 7*B + 8*C))/(4*((3*A*a^2) /2 + (7*B*a^2)/4 + 2*C*a^2)))*(6*A + 7*B + 8*C))/(4*d)