Integrand size = 41, antiderivative size = 303 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{2} a \left (8 A b^3+a^3 B+12 a b^2 B+4 a^2 b (A+2 C)\right ) x+\frac {b^2 \left (2 A b^2+8 a b B+12 a^2 C+b^2 C\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {\left (12 A b^2+15 a b B+a^2 (4 A+6 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(4 A b+3 a B) \cos (c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{3 d}-\frac {b \left (39 a^2 b B-6 b^3 B+4 a b^2 (11 A-6 C)+4 a^3 (2 A+3 C)\right ) \tan (c+d x)}{6 d}-\frac {b^2 \left (18 a b B+3 b^2 (6 A-C)+a^2 (4 A+6 C)\right ) \sec (c+d x) \tan (c+d x)}{6 d} \]
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Time = 0.94 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {4179, 4133, 3855, 3852, 8} \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {b^2 \left (12 a^2 C+8 a b B+2 A b^2+b^2 C\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {\sin (c+d x) \left (a^2 (4 A+6 C)+15 a b B+12 A b^2\right ) (a+b \sec (c+d x))^2}{6 d}-\frac {b^2 \tan (c+d x) \sec (c+d x) \left (a^2 (4 A+6 C)+18 a b B+3 b^2 (6 A-C)\right )}{6 d}-\frac {b \tan (c+d x) \left (4 a^3 (2 A+3 C)+39 a^2 b B+4 a b^2 (11 A-6 C)-6 b^3 B\right )}{6 d}+\frac {1}{2} a x \left (a^3 B+4 a^2 b (A+2 C)+12 a b^2 B+8 A b^3\right )+\frac {(3 a B+4 A b) \sin (c+d x) \cos (c+d x) (a+b \sec (c+d x))^3}{6 d}+\frac {A \sin (c+d x) \cos ^2(c+d x) (a+b \sec (c+d x))^4}{3 d} \]
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Rule 8
Rule 3852
Rule 3855
Rule 4133
Rule 4179
Rubi steps \begin{align*} \text {integral}& = \frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{3 d}+\frac {1}{3} \int \cos ^2(c+d x) (a+b \sec (c+d x))^3 \left (4 A b+3 a B+(2 a A+3 b B+3 a C) \sec (c+d x)-b (2 A-3 C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {(4 A b+3 a B) \cos (c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{3 d}+\frac {1}{6} \int \cos (c+d x) (a+b \sec (c+d x))^2 \left (12 A b^2+15 a b B+a^2 (4 A+6 C)+\left (3 a^2 B+6 b^2 B+4 a b (A+3 C)\right ) \sec (c+d x)-6 b (2 A b+a B-b C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {\left (12 A b^2+15 a b B+a^2 (4 A+6 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(4 A b+3 a B) \cos (c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{3 d}+\frac {1}{6} \int (a+b \sec (c+d x)) \left (3 \left (8 A b^3+a^3 B+12 a b^2 B+4 a^2 b (A+2 C)\right )-b \left (8 a A b+3 a^2 B-6 b^2 B-18 a b C\right ) \sec (c+d x)-2 b \left (18 a b B+3 b^2 (6 A-C)+a^2 (4 A+6 C)\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {\left (12 A b^2+15 a b B+a^2 (4 A+6 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(4 A b+3 a B) \cos (c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{3 d}-\frac {b^2 \left (18 a b B+3 b^2 (6 A-C)+a^2 (4 A+6 C)\right ) \sec (c+d x) \tan (c+d x)}{6 d}+\frac {1}{12} \int \left (6 a \left (8 A b^3+a^3 B+12 a b^2 B+4 a^2 b (A+2 C)\right )+6 b^2 \left (2 A b^2+8 a b B+12 a^2 C+b^2 C\right ) \sec (c+d x)-2 b \left (39 a^2 b B-6 b^3 B+4 a b^2 (11 A-6 C)+4 a^3 (2 A+3 C)\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {1}{2} a \left (8 A b^3+a^3 B+12 a b^2 B+4 a^2 b (A+2 C)\right ) x+\frac {\left (12 A b^2+15 a b B+a^2 (4 A+6 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(4 A b+3 a B) \cos (c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{3 d}-\frac {b^2 \left (18 a b B+3 b^2 (6 A-C)+a^2 (4 A+6 C)\right ) \sec (c+d x) \tan (c+d x)}{6 d}+\frac {1}{2} \left (b^2 \left (2 A b^2+8 a b B+12 a^2 C+b^2 C\right )\right ) \int \sec (c+d x) \, dx-\frac {1}{6} \left (b \left (39 a^2 b B-6 b^3 B+4 a b^2 (11 A-6 C)+4 a^3 (2 A+3 C)\right )\right ) \int \sec ^2(c+d x) \, dx \\ & = \frac {1}{2} a \left (8 A b^3+a^3 B+12 a b^2 B+4 a^2 b (A+2 C)\right ) x+\frac {b^2 \left (2 A b^2+8 a b B+12 a^2 C+b^2 C\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {\left (12 A b^2+15 a b B+a^2 (4 A+6 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(4 A b+3 a B) \cos (c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{3 d}-\frac {b^2 \left (18 a b B+3 b^2 (6 A-C)+a^2 (4 A+6 C)\right ) \sec (c+d x) \tan (c+d x)}{6 d}+\frac {\left (b \left (39 a^2 b B-6 b^3 B+4 a b^2 (11 A-6 C)+4 a^3 (2 A+3 C)\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{6 d} \\ & = \frac {1}{2} a \left (8 A b^3+a^3 B+12 a b^2 B+4 a^2 b (A+2 C)\right ) x+\frac {b^2 \left (2 A b^2+8 a b B+12 a^2 C+b^2 C\right ) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {\left (12 A b^2+15 a b B+a^2 (4 A+6 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(4 A b+3 a B) \cos (c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {A \cos ^2(c+d x) (a+b \sec (c+d x))^4 \sin (c+d x)}{3 d}-\frac {b \left (39 a^2 b B-6 b^3 B+4 a b^2 (11 A-6 C)+4 a^3 (2 A+3 C)\right ) \tan (c+d x)}{6 d}-\frac {b^2 \left (18 a b B+3 b^2 (6 A-C)+a^2 (4 A+6 C)\right ) \sec (c+d x) \tan (c+d x)}{6 d} \\ \end{align*}
Time = 8.26 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.22 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {6 a \left (8 A b^3+a^3 B+12 a b^2 B+4 a^2 b (A+2 C)\right ) (c+d x)-6 b^2 \left (2 A b^2+8 a b B+12 a^2 C+b^2 C\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 b^2 \left (2 A b^2+8 a b B+12 a^2 C+b^2 C\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {3 b^4 C}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {12 b^3 (b B+4 a C) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {3 b^4 C}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {12 b^3 (b B+4 a C) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+3 a^2 \left (24 A b^2+16 a b B+a^2 (3 A+4 C)\right ) \sin (c+d x)+3 a^3 (4 A b+a B) \sin (2 (c+d x))+a^4 A \sin (3 (c+d x))}{12 d} \]
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Time = 1.18 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {\frac {a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} C \sin \left (d x +c \right )+4 A \,a^{3} b \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 B \,a^{3} b \sin \left (d x +c \right )+4 a^{3} b C \left (d x +c \right )+6 A \,a^{2} b^{2} \sin \left (d x +c \right )+6 B \,a^{2} b^{2} \left (d x +c \right )+6 C \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a A \,b^{3} \left (d x +c \right )+4 B a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 C a \,b^{3} \tan \left (d x +c \right )+A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,b^{4} \tan \left (d x +c \right )+C \,b^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(284\) |
default | \(\frac {\frac {a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+B \,a^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{4} C \sin \left (d x +c \right )+4 A \,a^{3} b \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 B \,a^{3} b \sin \left (d x +c \right )+4 a^{3} b C \left (d x +c \right )+6 A \,a^{2} b^{2} \sin \left (d x +c \right )+6 B \,a^{2} b^{2} \left (d x +c \right )+6 C \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a A \,b^{3} \left (d x +c \right )+4 B a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 C a \,b^{3} \tan \left (d x +c \right )+A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,b^{4} \tan \left (d x +c \right )+C \,b^{4} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(284\) |
parallelrisch | \(\frac {-24 \left (\left (A +\frac {C}{2}\right ) b^{2}+4 B a b +6 C \,a^{2}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) b^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+24 \left (\left (A +\frac {C}{2}\right ) b^{2}+4 B a b +6 C \,a^{2}\right ) \left (1+\cos \left (2 d x +2 c \right )\right ) b^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+48 d a x \left (2 A \,b^{3}+3 B a \,b^{2}+a^{2} b \left (A +2 C \right )+\frac {B \,a^{3}}{4}\right ) \cos \left (2 d x +2 c \right )+6 \left (4 A \,a^{3} b +B \,a^{4}+4 B \,b^{4}+16 C a \,b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (72 A \,a^{2} b^{2}+48 B \,a^{3} b +11 a^{4} \left (A +\frac {12 C}{11}\right )\right ) \sin \left (3 d x +3 c \right )+3 \left (4 A \,a^{3} b +B \,a^{4}\right ) \sin \left (4 d x +4 c \right )+a^{4} A \sin \left (5 d x +5 c \right )+2 \left (12 C \,b^{4}+36 A \,a^{2} b^{2}+24 B \,a^{3} b +5 \left (A +\frac {6 C}{5}\right ) a^{4}\right ) \sin \left (d x +c \right )+48 d a x \left (2 A \,b^{3}+3 B a \,b^{2}+a^{2} b \left (A +2 C \right )+\frac {B \,a^{3}}{4}\right )}{24 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(350\) |
risch | \(-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} B \,a^{3} b}{d}+\frac {B \,a^{4} x}{2}-\frac {i a^{4} A \,{\mathrm e}^{3 i \left (d x +c \right )}}{24 d}-\frac {i {\mathrm e}^{i \left (d x +c \right )} a^{4} C}{2 d}+2 a^{3} A b x -\frac {3 i a^{4} A \,{\mathrm e}^{i \left (d x +c \right )}}{8 d}+\frac {3 i a^{4} A \,{\mathrm e}^{-i \left (d x +c \right )}}{8 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} a^{4} C}{2 d}+\frac {i a^{4} A \,{\mathrm e}^{-3 i \left (d x +c \right )}}{24 d}+\frac {3 i {\mathrm e}^{-i \left (d x +c \right )} A \,a^{2} b^{2}}{d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} B \,a^{3} b}{d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} A \,a^{3} b}{2 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} A \,a^{3} b}{2 d}-\frac {3 i {\mathrm e}^{i \left (d x +c \right )} A \,a^{2} b^{2}}{d}+4 A a \,b^{3} x +6 B \,a^{2} b^{2} x +4 C \,a^{3} b x -\frac {i b^{3} \left (C b \,{\mathrm e}^{3 i \left (d x +c \right )}-2 B b \,{\mathrm e}^{2 i \left (d x +c \right )}-8 C a \,{\mathrm e}^{2 i \left (d x +c \right )}-C b \,{\mathrm e}^{i \left (d x +c \right )}-2 B b -8 C a \right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C \,a^{2} b^{2}}{d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} B \,a^{4}}{8 d}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} B \,a^{4}}{8 d}-\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B a \,b^{3}}{d}+\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B a \,b^{3}}{d}-\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C \,a^{2} b^{2}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{4}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{4}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C \,b^{4}}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C \,b^{4}}{2 d}\) | \(584\) |
norman | \(\text {Expression too large to display}\) | \(1117\) |
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Time = 0.30 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.86 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {6 \, {\left (B a^{4} + 4 \, {\left (A + 2 \, C\right )} a^{3} b + 12 \, B a^{2} b^{2} + 8 \, A a b^{3}\right )} d x \cos \left (d x + c\right )^{2} + 3 \, {\left (12 \, C a^{2} b^{2} + 8 \, B a b^{3} + {\left (2 \, A + C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (12 \, C a^{2} b^{2} + 8 \, B a b^{3} + {\left (2 \, A + C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (2 \, A a^{4} \cos \left (d x + c\right )^{4} + 3 \, C b^{4} + 3 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left ({\left (2 \, A + 3 \, C\right )} a^{4} + 12 \, B a^{3} b + 18 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 6 \, {\left (4 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{12 \, d \cos \left (d x + c\right )^{2}} \]
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Timed out. \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.03 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {4 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} - 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 12 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} b - 48 \, {\left (d x + c\right )} C a^{3} b - 72 \, {\left (d x + c\right )} B a^{2} b^{2} - 48 \, {\left (d x + c\right )} A a b^{3} + 3 \, C b^{4} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, C a^{2} b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 24 \, B a b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 6 \, A b^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a^{4} \sin \left (d x + c\right ) - 48 \, B a^{3} b \sin \left (d x + c\right ) - 72 \, A a^{2} b^{2} \sin \left (d x + c\right ) - 48 \, C a b^{3} \tan \left (d x + c\right ) - 12 \, B b^{4} \tan \left (d x + c\right )}{12 \, d} \]
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Time = 0.39 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.79 \[ \int \cos ^3(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 {3 \, {\left (B a^{4} + 4 \, A a^{3} b + 8 \, C a^{3} b + 12 \, B a^{2} b^{2} + 8 \, A a b^{3}\right )} {\left (d x + c\right )} + 3 \, {\left (12 \, C a^{2} b^{2} + 8 \, B a b^{3} + 2 \, A b^{4} + C b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (12 \, C a^{2} b^{2} + 8 \, B a b^{3} + 2 \, A b^{4} + C b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {6 \, {\left (8 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 2 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 8 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, B b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - C b^{4} \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 )}^{2}} + \frac {2 \, {\left (6 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 3 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 6 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 12 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 24 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 36 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 12 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 48 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 6 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 12 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 36 \, A a^{2} b^{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 )}^{3}}}{6 \, d} \]
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Time = 20.95 (sec) , antiderivative size = 4839, normalized size of antiderivative = 15.97 \[ \int \cos ^3(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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