Integrand size = 41, antiderivative size = 314 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{8} \left (3 a^4 B+24 a^2 b^2 B+8 b^4 B+16 a b^3 (A+2 C)+4 a^3 b (3 A+4 C)\right ) x+\frac {b^4 C \text {arctanh}(\sin (c+d x))}{d}+\frac {\left (12 A b^4+80 a^3 b B+95 a b^3 B+4 a^4 (4 A+5 C)+2 a^2 b^2 (56 A+85 C)\right ) \sin (c+d x)}{30 d}+\frac {a \left (24 A b^3+45 a^3 B+130 a b^2 B+4 a^2 b (29 A+40 C)\right ) \cos (c+d x) \sin (c+d x)}{120 d}+\frac {\left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{60 d}+\frac {(4 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{5 d} \]
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Time = 1.17 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {4179, 4159, 4132, 8, 4130, 3855} \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\sin (c+d x) \cos ^2(c+d x) \left (4 a^2 (4 A+5 C)+35 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{60 d}+\frac {a \sin (c+d x) \cos (c+d x) \left (45 a^3 B+4 a^2 b (29 A+40 C)+130 a b^2 B+24 A b^3\right )}{120 d}+\frac {\sin (c+d x) \left (4 a^4 (4 A+5 C)+80 a^3 b B+2 a^2 b^2 (56 A+85 C)+95 a b^3 B+12 A b^4\right )}{30 d}+\frac {1}{8} x \left (3 a^4 B+4 a^3 b (3 A+4 C)+24 a^2 b^2 B+16 a b^3 (A+2 C)+8 b^4 B\right )+\frac {(5 a B+4 A b) \sin (c+d x) \cos ^3(c+d x) (a+b \sec (c+d x))^3}{20 d}+\frac {A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^4}{5 d}+\frac {b^4 C \text {arctanh}(\sin (c+d x))}{d} \]
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Rule 8
Rule 3855
Rule 4130
Rule 4132
Rule 4159
Rule 4179
Rubi steps \begin{align*} \text {integral}& = \frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {1}{5} \int \cos ^4(c+d x) (a+b \sec (c+d x))^3 \left (4 A b+5 a B+(4 a A+5 b B+5 a C) \sec (c+d x)+5 b C \sec ^2(c+d x)\right ) \, dx \\ & = \frac {(4 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {1}{20} \int \cos ^3(c+d x) (a+b \sec (c+d x))^2 \left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)+\left (28 a A b+15 a^2 B+20 b^2 B+40 a b C\right ) \sec (c+d x)+20 b^2 C \sec ^2(c+d x)\right ) \, dx \\ & = \frac {\left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{60 d}+\frac {(4 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{5 d}+\frac {1}{60} \int \cos ^2(c+d x) (a+b \sec (c+d x)) \left (24 A b^3+45 a^3 B+130 a b^2 B+4 a^2 b (29 A+40 C)+\left (115 a^2 b B+60 b^3 B+36 a b^2 (3 A+5 C)+8 a^3 (4 A+5 C)\right ) \sec (c+d x)+60 b^3 C \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a \left (24 A b^3+45 a^3 B+130 a b^2 B+4 a^2 b (29 A+40 C)\right ) \cos (c+d x) \sin (c+d x)}{120 d}+\frac {\left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{60 d}+\frac {(4 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{5 d}-\frac {1}{120} \int \cos (c+d x) \left (-4 \left (12 A b^4+80 a^3 b B+95 a b^3 B+4 a^4 (4 A+5 C)+2 a^2 b^2 (56 A+85 C)\right )-15 \left (3 a^4 B+24 a^2 b^2 B+8 b^4 B+16 a b^3 (A+2 C)+4 a^3 b (3 A+4 C)\right ) \sec (c+d x)-120 b^4 C \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a \left (24 A b^3+45 a^3 B+130 a b^2 B+4 a^2 b (29 A+40 C)\right ) \cos (c+d x) \sin (c+d x)}{120 d}+\frac {\left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{60 d}+\frac {(4 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{5 d}-\frac {1}{120} \int \cos (c+d x) \left (-4 \left (12 A b^4+80 a^3 b B+95 a b^3 B+4 a^4 (4 A+5 C)+2 a^2 b^2 (56 A+85 C)\right )-120 b^4 C \sec ^2(c+d x)\right ) \, dx-\frac {1}{8} \left (-3 a^4 B-24 a^2 b^2 B-8 b^4 B-16 a b^3 (A+2 C)-4 a^3 b (3 A+4 C)\right ) \int 1 \, dx \\ & = \frac {1}{8} \left (3 a^4 B+24 a^2 b^2 B+8 b^4 B+16 a b^3 (A+2 C)+4 a^3 b (3 A+4 C)\right ) x+\frac {\left (12 A b^4+80 a^3 b B+95 a b^3 B+4 a^4 (4 A+5 C)+2 a^2 b^2 (56 A+85 C)\right ) \sin (c+d x)}{30 d}+\frac {a \left (24 A b^3+45 a^3 B+130 a b^2 B+4 a^2 b (29 A+40 C)\right ) \cos (c+d x) \sin (c+d x)}{120 d}+\frac {\left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{60 d}+\frac {(4 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{5 d}+\left (b^4 C\right ) \int \sec (c+d x) \, dx \\ & = \frac {1}{8} \left (3 a^4 B+24 a^2 b^2 B+8 b^4 B+16 a b^3 (A+2 C)+4 a^3 b (3 A+4 C)\right ) x+\frac {b^4 C \text {arctanh}(\sin (c+d x))}{d}+\frac {\left (12 A b^4+80 a^3 b B+95 a b^3 B+4 a^4 (4 A+5 C)+2 a^2 b^2 (56 A+85 C)\right ) \sin (c+d x)}{30 d}+\frac {a \left (24 A b^3+45 a^3 B+130 a b^2 B+4 a^2 b (29 A+40 C)\right ) \cos (c+d x) \sin (c+d x)}{120 d}+\frac {\left (12 A b^2+35 a b B+4 a^2 (4 A+5 C)\right ) \cos ^2(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{60 d}+\frac {(4 A b+5 a B) \cos ^3(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{20 d}+\frac {A \cos ^4(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{5 d} \\ \end{align*}
Time = 3.00 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.22 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {720 a^3 A b c+960 a A b^3 c+180 a^4 B c+1440 a^2 b^2 B c+480 b^4 B c+960 a^3 b c C+1920 a b^3 c C+720 a^3 A b d x+960 a A b^3 d x+180 a^4 B d x+1440 a^2 b^2 B d x+480 b^4 B d x+960 a^3 b C d x+1920 a b^3 C d x-480 b^4 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+480 b^4 C \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+60 \left (8 A b^4+24 a^3 b B+32 a b^3 B+12 a^2 b^2 (3 A+4 C)+a^4 (5 A+6 C)\right ) \sin (c+d x)+120 a \left (4 A b^3+a^3 B+6 a b^2 B+4 a^2 b (A+C)\right ) \sin (2 (c+d x))+50 a^4 A \sin (3 (c+d x))+240 a^2 A b^2 \sin (3 (c+d x))+160 a^3 b B \sin (3 (c+d x))+40 a^4 C \sin (3 (c+d x))+60 a^3 A b \sin (4 (c+d x))+15 a^4 B \sin (4 (c+d x))+6 a^4 A \sin (5 (c+d x))}{480 d} \]
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Time = 1.00 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.82
method | result | size |
parallelrisch | \(\frac {-480 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b^{4}+480 C \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b^{4}+480 a \left (\frac {B \,a^{3}}{4}+b \left (A +C \right ) a^{2}+\frac {3 B a \,b^{2}}{2}+A \,b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (\left (50 A +40 C \right ) a^{4}+160 B \,a^{3} b +240 A \,a^{2} b^{2}\right ) \sin \left (3 d x +3 c \right )+\left (60 A \,a^{3} b +15 B \,a^{4}\right ) \sin \left (4 d x +4 c \right )+6 a^{4} A \sin \left (5 d x +5 c \right )+\left (\left (300 A +360 C \right ) a^{4}+1440 B \,a^{3} b +2160 b^{2} \left (A +\frac {4 C}{3}\right ) a^{2}+1920 B a \,b^{3}+480 A \,b^{4}\right ) \sin \left (d x +c \right )+720 d \left (\frac {B \,a^{4}}{4}+b \left (A +\frac {4 C}{3}\right ) a^{3}+2 B \,a^{2} b^{2}+\frac {4 a \,b^{3} \left (A +2 C \right )}{3}+\frac {2 B \,b^{4}}{3}\right ) x}{480 d}\) | \(258\) |
derivativedivides | \(\frac {\frac {a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{4} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 A \,a^{3} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 B \,a^{3} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 a^{3} b C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 A \,a^{2} b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+6 B \,a^{2} b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 C \,a^{2} b^{2} \sin \left (d x +c \right )+4 a A \,b^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 B a \,b^{3} \sin \left (d x +c \right )+4 C a \,b^{3} \left (d x +c \right )+A \,b^{4} \sin \left (d x +c \right )+B \,b^{4} \left (d x +c \right )+C \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(355\) |
default | \(\frac {\frac {a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {a^{4} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 A \,a^{3} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 B \,a^{3} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 a^{3} b C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 A \,a^{2} b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+6 B \,a^{2} b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+6 C \,a^{2} b^{2} \sin \left (d x +c \right )+4 a A \,b^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 B a \,b^{3} \sin \left (d x +c \right )+4 C a \,b^{3} \left (d x +c \right )+A \,b^{4} \sin \left (d x +c \right )+B \,b^{4} \left (d x +c \right )+C \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}\) | \(355\) |
risch | \(\frac {5 a^{4} A \sin \left (3 d x +3 c \right )}{48 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{4}}{4 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} A \,b^{4}}{2 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} A \,b^{4}}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{3} b}{2 d}+\frac {3 B \,a^{4} x}{8}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} C \,a^{2} b^{2}}{d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} B a \,b^{3}}{d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} C \,a^{2} b^{2}}{d}+\frac {9 i {\mathrm e}^{-i \left (d x +c \right )} A \,a^{2} b^{2}}{4 d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} B a \,b^{3}}{d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} a^{4} C}{8 d}+\frac {3 a^{3} A b x}{2}-\frac {5 i a^{4} A \,{\mathrm e}^{i \left (d x +c \right )}}{16 d}+\frac {5 i a^{4} A \,{\mathrm e}^{-i \left (d x +c \right )}}{16 d}+\frac {\sin \left (3 d x +3 c \right ) a^{4} C}{12 d}+\frac {\sin \left (2 d x +2 c \right ) A \,a^{3} b}{d}+\frac {\sin \left (2 d x +2 c \right ) a A \,b^{3}}{d}+\frac {3 \sin \left (2 d x +2 c \right ) B \,a^{2} b^{2}}{2 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} b C}{d}+\frac {\sin \left (4 d x +4 c \right ) A \,a^{3} b}{8 d}+\frac {\sin \left (3 d x +3 c \right ) A \,a^{2} b^{2}}{2 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{3} b}{3 d}+x B \,b^{4}+\frac {a^{4} A \sin \left (5 d x +5 c \right )}{80 d}+4 x C a \,b^{3}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{4}}{32 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} a^{4} C}{8 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{3} b}{2 d}-\frac {9 i {\mathrm e}^{i \left (d x +c \right )} A \,a^{2} b^{2}}{4 d}+2 A a \,b^{3} x +3 B \,a^{2} b^{2} x +2 C \,a^{3} b x +\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C \,b^{4}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C \,b^{4}}{d}\) | \(606\) |
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Time = 0.33 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.83 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {60 \, C b^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 60 \, C b^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left (3 \, B a^{4} + 4 \, {\left (3 \, A + 4 \, C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 16 \, {\left (A + 2 \, C\right )} a b^{3} + 8 \, B b^{4}\right )} d x + {\left (24 \, A a^{4} \cos \left (d x + c\right )^{4} + 16 \, {\left (4 \, A + 5 \, C\right )} a^{4} + 320 \, B a^{3} b + 240 \, {\left (2 \, A + 3 \, C\right )} a^{2} b^{2} + 480 \, B a b^{3} + 120 \, A b^{4} + 30 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left ({\left (4 \, A + 5 \, C\right )} a^{4} + 20 \, B a^{3} b + 30 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (3 \, B a^{4} + 4 \, {\left (3 \, A + 4 \, C\right )} a^{3} b + 24 \, B a^{2} b^{2} + 16 \, A a b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \]
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Timed out. \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.24 (sec) , antiderivative size = 347, normalized size of antiderivative = 1.11 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 160 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 60 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} b - 640 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} b + 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} b - 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{2} b^{2} + 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} b^{2} + 480 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{3} + 1920 \, {\left (d x + c\right )} C a b^{3} + 480 \, {\left (d x + c\right )} B b^{4} + 240 \, C b^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 2880 \, C a^{2} b^{2} \sin \left (d x + c\right ) + 1920 \, B a b^{3} \sin \left (d x + c\right ) + 480 \, A b^{4} \sin \left (d x + c\right )}{480 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1094 vs. \(2 (302) = 604\).
Time = 0.39 (sec) , antiderivative size = 1094, normalized size of antiderivative = 3.48 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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Time = 20.42 (sec) , antiderivative size = 4118, normalized size of antiderivative = 13.11 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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