Integrand size = 40, antiderivative size = 179 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{8} \left (3 a^3 B+12 a b^2 B+12 a^2 b C+8 b^3 C\right ) x+\frac {\left (6 a^2 b B+3 b^3 B+2 a^3 C+9 a b^2 C\right ) \sin (c+d x)}{3 d}+\frac {a \left (3 a^2 B+10 b^2 B+12 a b C\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 (3 b B+2 a C) \cos ^2(c+d x) \sin (c+d x)}{6 d}+\frac {a B \cos ^3(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{4 d} \]
1/8*(3*B*a^3+12*B*a*b^2+12*C*a^2*b+8*C*b^3)*x+1/3*(6*B*a^2*b+3*B*b^3+2*C*a ^3+9*C*a*b^2)*sin(d*x+c)/d+1/8*a*(3*B*a^2+10*B*b^2+12*C*a*b)*cos(d*x+c)*si n(d*x+c)/d+1/6*a^2*(3*B*b+2*C*a)*cos(d*x+c)^2*sin(d*x+c)/d+1/4*a*B*cos(d*x +c)^3*(a+b*sec(d*x+c))^2*sin(d*x+c)/d
Time = 1.37 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.78 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {12 \left (3 a^3 B+12 a b^2 B+12 a^2 b C+8 b^3 C\right ) (c+d x)+24 \left (9 a^2 b B+4 b^3 B+3 a^3 C+12 a b^2 C\right ) \sin (c+d x)+24 a \left (a^2 B+3 b^2 B+3 a b C\right ) \sin (2 (c+d x))+8 a^2 (3 b B+a C) \sin (3 (c+d x))+3 a^3 B \sin (4 (c+d x))}{96 d} \]
(12*(3*a^3*B + 12*a*b^2*B + 12*a^2*b*C + 8*b^3*C)*(c + d*x) + 24*(9*a^2*b* B + 4*b^3*B + 3*a^3*C + 12*a*b^2*C)*Sin[c + d*x] + 24*a*(a^2*B + 3*b^2*B + 3*a*b*C)*Sin[2*(c + d*x)] + 8*a^2*(3*b*B + a*C)*Sin[3*(c + d*x)] + 3*a^3* B*Sin[4*(c + d*x)])/(96*d)
Time = 1.19 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.04, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {3042, 4560, 3042, 4513, 25, 3042, 4562, 25, 3042, 4535, 3042, 3117, 4533, 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.
| \(\displaystyle \int \cos ^5(c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (B \csc \left (c+d x+\frac {\pi }{2}\right )+C \csc \left (c+d x+\frac {\pi }{2}\right )^2\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 4560 |
| \(\displaystyle \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 (B+C \sec (c+d x))dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (B+C \csc \left (c+d x+\frac {\pi }{2}\right )\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^4}dx\) |
| \(\Big \downarrow \) 4513 |
| \(\displaystyle \frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}-\frac {1}{4} \int -\cos ^3(c+d x) (a+b \sec (c+d x)) \left (b (a B+4 b C) \sec ^2(c+d x)+\left (3 B a^2+8 b C a+4 b^2 B\right ) \sec (c+d x)+2 a (3 b B+2 a C)\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \int \cos ^3(c+d x) (a+b \sec (c+d x)) \left (b (a B+4 b C) \sec ^2(c+d x)+\left (3 B a^2+8 b C a+4 b^2 B\right ) \sec (c+d x)+2 a (3 b B+2 a C)\right )dx+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \int \frac {\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right ) \left (b (a B+4 b C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+\left (3 B a^2+8 b C a+4 b^2 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+2 a (3 b B+2 a C)\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3}dx+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 4562 |
| \(\displaystyle \frac {1}{4} \left (\frac {2 a^2 (2 a C+3 b B) \sin (c+d x) \cos ^2(c+d x)}{3 d}-\frac {1}{3} \int -\cos ^2(c+d x) \left (3 b^2 (a B+4 b C) \sec ^2(c+d x)+4 \left (2 C a^3+6 b B a^2+9 b^2 C a+3 b^3 B\right ) \sec (c+d x)+3 a \left (3 B a^2+12 b C a+10 b^2 B\right )\right )dx\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \cos ^2(c+d x) \left (3 b^2 (a B+4 b C) \sec ^2(c+d x)+4 \left (2 C a^3+6 b B a^2+9 b^2 C a+3 b^3 B\right ) \sec (c+d x)+3 a \left (3 B a^2+12 b C a+10 b^2 B\right )\right )dx+\frac {2 a^2 (2 a C+3 b B) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \int \frac {3 b^2 (a B+4 b C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+4 \left (2 C a^3+6 b B a^2+9 b^2 C a+3 b^3 B\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 a \left (3 B a^2+12 b C a+10 b^2 B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {2 a^2 (2 a C+3 b B) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 4535 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\int \cos ^2(c+d x) \left (3 b^2 (a B+4 b C) \sec ^2(c+d x)+3 a \left (3 B a^2+12 b C a+10 b^2 B\right )\right )dx+4 \left (2 a^3 C+6 a^2 b B+9 a b^2 C+3 b^3 B\right ) \int \cos (c+d x)dx\right )+\frac {2 a^2 (2 a C+3 b B) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\int \frac {3 b^2 (a B+4 b C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+3 a \left (3 B a^2+12 b C a+10 b^2 B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+4 \left (2 a^3 C+6 a^2 b B+9 a b^2 C+3 b^3 B\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )dx\right )+\frac {2 a^2 (2 a C+3 b B) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 3117 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\int \frac {3 b^2 (a B+4 b C) \csc \left (c+d x+\frac {\pi }{2}\right )^2+3 a \left (3 B a^2+12 b C a+10 b^2 B\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^2}dx+\frac {4 \left (2 a^3 C+6 a^2 b B+9 a b^2 C+3 b^3 B\right ) \sin (c+d x)}{d}\right )+\frac {2 a^2 (2 a C+3 b B) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 4533 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{3} \left (\frac {3}{2} \left (3 a^3 B+12 a^2 b C+12 a b^2 B+8 b^3 C\right ) \int 1dx+\frac {3 a \left (3 a^2 B+12 a b C+10 b^2 B\right ) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {4 \left (2 a^3 C+6 a^2 b B+9 a b^2 C+3 b^3 B\right ) \sin (c+d x)}{d}\right )+\frac {2 a^2 (2 a C+3 b B) \sin (c+d x) \cos ^2(c+d x)}{3 d}\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{4} \left (\frac {2 a^2 (2 a C+3 b B) \sin (c+d x) \cos ^2(c+d x)}{3 d}+\frac {1}{3} \left (\frac {3 a \left (3 a^2 B+12 a b C+10 b^2 B\right ) \sin (c+d x) \cos (c+d x)}{2 d}+\frac {4 \left (2 a^3 C+6 a^2 b B+9 a b^2 C+3 b^3 B\right ) \sin (c+d x)}{d}+\frac {3}{2} x \left (3 a^3 B+12 a^2 b C+12 a b^2 B+8 b^3 C\right )\right )\right )+\frac {a B \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^2}{4 d}\) |
(a*B*Cos[c + d*x]^3*(a + b*Sec[c + d*x])^2*Sin[c + d*x])/(4*d) + ((2*a^2*( 3*b*B + 2*a*C)*Cos[c + d*x]^2*Sin[c + d*x])/(3*d) + ((3*(3*a^3*B + 12*a*b^ 2*B + 12*a^2*b*C + 8*b^3*C)*x)/2 + (4*(6*a^2*b*B + 3*b^3*B + 2*a^3*C + 9*a *b^2*C)*Sin[c + d*x])/d + (3*a*(3*a^2*B + 10*b^2*B + 12*a*b*C)*Cos[c + d*x ]*Sin[c + d*x])/(2*d))/3)/4
3.8.91.3.1 Defintions of rubi rules used
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_))^(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]
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]
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]
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]
Time = 0.54 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.78
| method | result | size |
| parallelrisch | \(\frac {24 a \left (B \,a^{2}+3 B \,b^{2}+3 C a b \right ) \sin \left (2 d x +2 c \right )+\left (24 B \,a^{2} b +8 a^{3} C \right ) \sin \left (3 d x +3 c \right )+3 B \,a^{3} \sin \left (4 d x +4 c \right )+\left (216 B \,a^{2} b +96 B \,b^{3}+72 a^{3} C +288 C a \,b^{2}\right ) \sin \left (d x +c \right )+36 x d \left (B \,a^{3}+4 B a \,b^{2}+4 a^{2} b C +\frac {8}{3} C \,b^{3}\right )}{96 d}\) | \(139\) |
| derivativedivides | \(\frac {B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{2} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {a^{3} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+3 B a \,b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{2} b C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,b^{3} \sin \left (d x +c \right )+3 C a \,b^{2} \sin \left (d x +c \right )+C \,b^{3} \left (d x +c \right )}{d}\) | \(180\) |
| default | \(\frac {B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{2} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {a^{3} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+3 B a \,b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 a^{2} b C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,b^{3} \sin \left (d x +c \right )+3 C a \,b^{2} \sin \left (d x +c \right )+C \,b^{3} \left (d x +c \right )}{d}\) | \(180\) |
| risch | \(\frac {3 a^{3} B x}{8}+\frac {3 B a \,b^{2} x}{2}+\frac {3 C \,a^{2} b x}{2}+x C \,b^{3}+\frac {9 \sin \left (d x +c \right ) B \,a^{2} b}{4 d}+\frac {\sin \left (d x +c \right ) B \,b^{3}}{d}+\frac {3 a^{3} C \sin \left (d x +c \right )}{4 d}+\frac {3 \sin \left (d x +c \right ) C a \,b^{2}}{d}+\frac {B \,a^{3} \sin \left (4 d x +4 c \right )}{32 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{2} b}{4 d}+\frac {\sin \left (3 d x +3 c \right ) a^{3} C}{12 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{3}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) B a \,b^{2}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) a^{2} b C}{4 d}\) | \(203\) |
1/96*(24*a*(B*a^2+3*B*b^2+3*C*a*b)*sin(2*d*x+2*c)+(24*B*a^2*b+8*C*a^3)*sin (3*d*x+3*c)+3*B*a^3*sin(4*d*x+4*c)+(216*B*a^2*b+96*B*b^3+72*C*a^3+288*C*a* b^2)*sin(d*x+c)+36*x*d*(B*a^3+4*B*a*b^2+4*a^2*b*C+8/3*C*b^3))/d
Time = 0.27 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.76 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (3 \, B a^{3} + 12 \, C a^{2} b + 12 \, B a b^{2} + 8 \, C b^{3}\right )} d x + {\left (6 \, B a^{3} \cos \left (d x + c\right )^{3} + 16 \, C a^{3} + 48 \, B a^{2} b + 72 \, C a b^{2} + 24 \, B b^{3} + 8 \, {\left (C a^{3} + 3 \, B a^{2} b\right )} \cos \left (d x + c\right )^{2} + 9 \, {\left (B a^{3} + 4 \, C a^{2} b + 4 \, B a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \]
integrate(cos(d*x+c)^5*(a+b*sec(d*x+c))^3*(B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="fricas")
1/24*(3*(3*B*a^3 + 12*C*a^2*b + 12*B*a*b^2 + 8*C*b^3)*d*x + (6*B*a^3*cos(d *x + c)^3 + 16*C*a^3 + 48*B*a^2*b + 72*C*a*b^2 + 24*B*b^3 + 8*(C*a^3 + 3*B *a^2*b)*cos(d*x + c)^2 + 9*(B*a^3 + 4*C*a^2*b + 4*B*a*b^2)*cos(d*x + c))*s in(d*x + c))/d
Timed out. \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
Time = 0.24 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.96 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 32 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} - 96 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} b + 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} b + 72 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b^{2} + 96 \, {\left (d x + c\right )} C b^{3} + 288 \, C a b^{2} \sin \left (d x + c\right ) + 96 \, B b^{3} \sin \left (d x + c\right )}{96 \, d} \]
integrate(cos(d*x+c)^5*(a+b*sec(d*x+c))^3*(B*sec(d*x+c)+C*sec(d*x+c)^2),x, algorithm="maxima")
1/96*(3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*B*a^3 - 32 *(sin(d*x + c)^3 - 3*sin(d*x + c))*C*a^3 - 96*(sin(d*x + c)^3 - 3*sin(d*x + c))*B*a^2*b + 72*(2*d*x + 2*c + sin(2*d*x + 2*c))*C*a^2*b + 72*(2*d*x + 2*c + sin(2*d*x + 2*c))*B*a*b^2 + 96*(d*x + c)*C*b^3 + 288*C*a*b^2*sin(d*x + c) + 96*B*b^3*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (169) = 338\).
Time = 0.33 (sec) , antiderivative size = 536, normalized size of antiderivative = 2.99 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (3 \, B a^{3} + 12 \, C a^{2} b + 12 \, B a b^{2} + 8 \, C b^{3}\right )} {\left (d x + c\right )} - \frac {2 \, {\left (15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 36 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 36 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 72 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 9 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 120 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 216 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 120 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 216 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 72 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 15 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, C a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 36 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 72 \, C a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 24 \, B b^{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 )}^{4}}}{24 \, d} \]
1/24*(3*(3*B*a^3 + 12*C*a^2*b + 12*B*a*b^2 + 8*C*b^3)*(d*x + c) - 2*(15*B* a^3*tan(1/2*d*x + 1/2*c)^7 - 24*C*a^3*tan(1/2*d*x + 1/2*c)^7 - 72*B*a^2*b* tan(1/2*d*x + 1/2*c)^7 + 36*C*a^2*b*tan(1/2*d*x + 1/2*c)^7 + 36*B*a*b^2*ta n(1/2*d*x + 1/2*c)^7 - 72*C*a*b^2*tan(1/2*d*x + 1/2*c)^7 - 24*B*b^3*tan(1/ 2*d*x + 1/2*c)^7 - 9*B*a^3*tan(1/2*d*x + 1/2*c)^5 - 40*C*a^3*tan(1/2*d*x + 1/2*c)^5 - 120*B*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 36*C*a^2*b*tan(1/2*d*x + 1/2*c)^5 + 36*B*a*b^2*tan(1/2*d*x + 1/2*c)^5 - 216*C*a*b^2*tan(1/2*d*x + 1 /2*c)^5 - 72*B*b^3*tan(1/2*d*x + 1/2*c)^5 + 9*B*a^3*tan(1/2*d*x + 1/2*c)^3 - 40*C*a^3*tan(1/2*d*x + 1/2*c)^3 - 120*B*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 36*C*a^2*b*tan(1/2*d*x + 1/2*c)^3 - 36*B*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 21 6*C*a*b^2*tan(1/2*d*x + 1/2*c)^3 - 72*B*b^3*tan(1/2*d*x + 1/2*c)^3 - 15*B* a^3*tan(1/2*d*x + 1/2*c) - 24*C*a^3*tan(1/2*d*x + 1/2*c) - 72*B*a^2*b*tan( 1/2*d*x + 1/2*c) - 36*C*a^2*b*tan(1/2*d*x + 1/2*c) - 36*B*a*b^2*tan(1/2*d* x + 1/2*c) - 72*C*a*b^2*tan(1/2*d*x + 1/2*c) - 24*B*b^3*tan(1/2*d*x + 1/2* c))/(tan(1/2*d*x + 1/2*c)^2 + 1)^4)/d
Time = 17.14 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.13 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {3\,B\,a^3\,x}{8}+C\,b^3\,x+\frac {3\,B\,a\,b^2\,x}{2}+\frac {3\,C\,a^2\,b\,x}{2}+\frac {B\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {3\,C\,a^3\,\sin \left (c+d\,x\right )}{4\,d}+\frac {B\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {C\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {3\,B\,a\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a^2\,b\,\sin \left (3\,c+3\,d\,x\right )}{4\,d}+\frac {3\,C\,a^2\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {9\,B\,a^2\,b\,\sin \left (c+d\,x\right )}{4\,d}+\frac {3\,C\,a\,b^2\,\sin \left (c+d\,x\right )}{d} \]
(3*B*a^3*x)/8 + C*b^3*x + (3*B*a*b^2*x)/2 + (3*C*a^2*b*x)/2 + (B*b^3*sin(c + d*x))/d + (3*C*a^3*sin(c + d*x))/(4*d) + (B*a^3*sin(2*c + 2*d*x))/(4*d) + (B*a^3*sin(4*c + 4*d*x))/(32*d) + (C*a^3*sin(3*c + 3*d*x))/(12*d) + (3* B*a*b^2*sin(2*c + 2*d*x))/(4*d) + (B*a^2*b*sin(3*c + 3*d*x))/(4*d) + (3*C* a^2*b*sin(2*c + 2*d*x))/(4*d) + (9*B*a^2*b*sin(c + d*x))/(4*d) + (3*C*a*b^ 2*sin(c + d*x))/d