Integrand size = 41, antiderivative size = 293 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=b^3 (b B+4 a C) x+\frac {\left (8 A b^4+16 a^3 b B+32 a b^3 B+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}-\frac {b^2 \left (32 a b B+2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \sin (c+d x)}{24 d}+\frac {a \left (12 A b^3+8 a^3 B+36 a b^2 B+a^2 b (23 A+36 C)\right ) \tan (c+d x)}{12 d}+\frac {\left (4 A b^2+8 a b B+a^2 (3 A+4 C)\right ) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {(A b+a B) (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d} \]
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Time = 1.08 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {3126, 3110, 3102, 2814, 3855} \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=-\frac {b^2 \sin (c+d x) \left (3 a^2 (3 A+4 C)+32 a b B+2 b^2 (13 A-12 C)\right )}{24 d}+\frac {\tan (c+d x) \sec (c+d x) \left (a^2 (3 A+4 C)+8 a b B+4 A b^2\right ) (a+b \cos (c+d x))^2}{8 d}+\frac {a \tan (c+d x) \left (8 a^3 B+a^2 b (23 A+36 C)+36 a b^2 B+12 A b^3\right )}{12 d}+\frac {\left (a^4 (3 A+4 C)+16 a^3 b B+24 a^2 b^2 (A+2 C)+32 a b^3 B+8 A b^4\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {(a B+A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^3}{3 d}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^4}{4 d}+b^3 x (4 a C+b B) \]
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Rule 2814
Rule 3102
Rule 3110
Rule 3126
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \int (a+b \cos (c+d x))^3 \left (4 (A b+a B)+(3 a A+4 b B+4 a C) \cos (c+d x)-b (A-4 C) \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx \\ & = \frac {(A b+a B) (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{12} \int (a+b \cos (c+d x))^2 \left (3 \left (4 A b^2+8 a b B+a^2 (3 A+4 C)\right )+2 \left (4 a^2 B+6 b^2 B+a b (7 A+12 C)\right ) \cos (c+d x)-b (7 A b+4 a B-12 b C) \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx \\ & = \frac {\left (4 A b^2+8 a b B+a^2 (3 A+4 C)\right ) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {(A b+a B) (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{24} \int (a+b \cos (c+d x)) \left (2 \left (12 A b^3+8 a^3 B+36 a b^2 B+\frac {1}{2} a^2 (46 A b+72 b C)\right )+\left (32 a^2 b B+24 b^3 B+3 a^3 (3 A+4 C)+2 a b^2 (13 A+36 C)\right ) \cos (c+d x)-b \left (32 a b B+2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx \\ & = \frac {a \left (12 A b^3+8 a^3 B+36 a b^2 B+a^2 b (23 A+36 C)\right ) \tan (c+d x)}{12 d}+\frac {\left (4 A b^2+8 a b B+a^2 (3 A+4 C)\right ) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {(A b+a B) (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{24} \int \left (-3 \left (8 A b^4+16 a^3 b B+32 a b^3 B+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right )-24 b^3 (b B+4 a C) \cos (c+d x)+b^2 \left (32 a b B+2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = -\frac {b^2 \left (32 a b B+2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \sin (c+d x)}{24 d}+\frac {a \left (12 A b^3+8 a^3 B+36 a b^2 B+a^2 b (23 A+36 C)\right ) \tan (c+d x)}{12 d}+\frac {\left (4 A b^2+8 a b B+a^2 (3 A+4 C)\right ) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {(A b+a B) (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{24} \int \left (-3 \left (8 A b^4+16 a^3 b B+32 a b^3 B+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right )-24 b^3 (b B+4 a C) \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = b^3 (b B+4 a C) x-\frac {b^2 \left (32 a b B+2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \sin (c+d x)}{24 d}+\frac {a \left (12 A b^3+8 a^3 B+36 a b^2 B+a^2 b (23 A+36 C)\right ) \tan (c+d x)}{12 d}+\frac {\left (4 A b^2+8 a b B+a^2 (3 A+4 C)\right ) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {(A b+a B) (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{8} \left (-8 A b^4-16 a^3 b B-32 a b^3 B-24 a^2 b^2 (A+2 C)-a^4 (3 A+4 C)\right ) \int \sec (c+d x) \, dx \\ & = b^3 (b B+4 a C) x+\frac {\left (8 A b^4+16 a^3 b B+32 a b^3 B+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}-\frac {b^2 \left (32 a b B+2 b^2 (13 A-12 C)+3 a^2 (3 A+4 C)\right ) \sin (c+d x)}{24 d}+\frac {a \left (12 A b^3+8 a^3 B+36 a b^2 B+a^2 b (23 A+36 C)\right ) \tan (c+d x)}{12 d}+\frac {\left (4 A b^2+8 a b B+a^2 (3 A+4 C)\right ) (a+b \cos (c+d x))^2 \sec (c+d x) \tan (c+d x)}{8 d}+\frac {(A b+a B) (a+b \cos (c+d x))^3 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^3(c+d x) \tan (c+d x)}{4 d} \\ \end{align*}
Time = 6.53 (sec) , antiderivative size = 462, normalized size of antiderivative = 1.58 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {-12 \left (8 A b^4+16 a^3 b B+32 a b^3 B+24 a^2 b^2 (A+2 C)+a^4 (3 A+4 C)\right ) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )+\sec ^4(c+d x) \left (36 b^4 B c+144 a b^3 c C+36 b^4 B d x+144 a b^3 C d x+48 b^3 (b B+4 a C) (c+d x) \cos (2 (c+d x))+12 b^3 (b B+4 a C) (c+d x) \cos (4 (c+d x))+9 a^4 A \sin (3 (c+d x))+72 a^2 A b^2 \sin (3 (c+d x))+48 a^3 b B \sin (3 (c+d x))+12 a^4 C \sin (3 (c+d x))+18 b^4 C \sin (3 (c+d x))+32 a^3 A b \sin (4 (c+d x))+48 a A b^3 \sin (4 (c+d x))+8 a^4 B \sin (4 (c+d x))+72 a^2 b^2 B \sin (4 (c+d x))+48 a^3 b C \sin (4 (c+d x))+6 b^4 C \sin (5 (c+d x))\right )+32 a \left (6 A b^3+2 a^3 B+9 a b^2 B+a^2 (8 A b+6 b C)\right ) \sec ^2(c+d x) \tan (c+d x)+3 \left (24 a^2 A b^2+16 a^3 b B+4 b^4 C+a^4 (11 A+4 C)\right ) \sec ^3(c+d x) \tan (c+d x)}{96 d} \]
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Time = 0.85 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.88
method | result | size |
parts | \(\frac {A \,a^{4} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (B \,b^{4}+4 C a \,b^{3}\right ) \left (d x +c \right )}{d}+\frac {\left (A \,b^{4}+4 B a \,b^{3}+6 C \,a^{2} b^{2}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (4 a A \,b^{3}+6 B \,a^{2} b^{2}+4 a^{3} b C \right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b +a^{4} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {\sin \left (d x +c \right ) C \,b^{4}}{d}\) | \(257\) |
derivativedivides | \(\frac {A \,a^{4} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 A \,a^{3} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 B \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{3} b C \tan \left (d x +c \right )+6 A \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 B \,a^{2} b^{2} \tan \left (d x +c \right )+6 C \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a A \,b^{3} \tan \left (d x +c \right )+4 B a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 C a \,b^{3} \left (d x +c \right )+A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,b^{4} \left (d x +c \right )+C \sin \left (d x +c \right ) b^{4}}{d}\) | \(355\) |
default | \(\frac {A \,a^{4} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+a^{4} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-4 A \,a^{3} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 B \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 a^{3} b C \tan \left (d x +c \right )+6 A \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 B \,a^{2} b^{2} \tan \left (d x +c \right )+6 C \,a^{2} b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 a A \,b^{3} \tan \left (d x +c \right )+4 B a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 C a \,b^{3} \left (d x +c \right )+A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,b^{4} \left (d x +c \right )+C \sin \left (d x +c \right ) b^{4}}{d}\) | \(355\) |
parallelrisch | \(\frac {-9 \left (\left (A +\frac {4 C}{3}\right ) a^{4}+\frac {16 B \,a^{3} b}{3}+8 a^{2} b^{2} \left (A +2 C \right )+\frac {32 B a \,b^{3}}{3}+\frac {8 A \,b^{4}}{3}\right ) \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+9 \left (\left (A +\frac {4 C}{3}\right ) a^{4}+\frac {16 B \,a^{3} b}{3}+8 a^{2} b^{2} \left (A +2 C \right )+\frac {32 B a \,b^{3}}{3}+\frac {8 A \,b^{4}}{3}\right ) \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+96 b^{3} d x \left (B b +4 C a \right ) \cos \left (2 d x +2 c \right )+24 b^{3} d x \left (B b +4 C a \right ) \cos \left (4 d x +4 c \right )+256 \left (\frac {B \,a^{3}}{4}+a^{2} \left (A +\frac {3 C}{4}\right ) b +\frac {9 B a \,b^{2}}{8}+\frac {3 A \,b^{3}}{4}\right ) a \sin \left (2 d x +2 c \right )+\left (\left (18 A +24 C \right ) a^{4}+96 B \,a^{3} b +144 A \,a^{2} b^{2}+36 C \,b^{4}\right ) \sin \left (3 d x +3 c \right )+64 a \left (\frac {B \,a^{3}}{4}+\left (A +\frac {3 C}{2}\right ) b \,a^{2}+\frac {9 B a \,b^{2}}{4}+\frac {3 A \,b^{3}}{2}\right ) \sin \left (4 d x +4 c \right )+12 C \sin \left (5 d x +5 c \right ) b^{4}+\left (\left (66 A +24 C \right ) a^{4}+96 B \,a^{3} b +144 A \,a^{2} b^{2}+24 C \,b^{4}\right ) \sin \left (d x +c \right )+72 b^{3} d x \left (B b +4 C a \right )}{24 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(436\) |
risch | \(x B \,b^{4}+4 a \,b^{3} C x +\frac {i a \left (64 A \,a^{2} b +16 B \,a^{3}+96 A \,b^{3}+144 B a \,b^{2}+96 a^{2} b C +288 C \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+12 C \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+288 A \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}-33 A \,a^{3} {\mathrm e}^{5 i \left (d x +c \right )}-12 C \,a^{3} {\mathrm e}^{5 i \left (d x +c \right )}+288 A \,b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+48 B \,a^{3} {\mathrm e}^{4 i \left (d x +c \right )}+33 A \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+64 B \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+9 A \,a^{3} {\mathrm e}^{i \left (d x +c \right )}+12 C \,a^{3} {\mathrm e}^{i \left (d x +c \right )}-9 A \,a^{3} {\mathrm e}^{7 i \left (d x +c \right )}-12 C \,a^{3} {\mathrm e}^{7 i \left (d x +c \right )}+96 A \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+96 C \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-72 A a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-48 B \,a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}-72 A a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-48 B \,a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+144 B a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+432 B a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+288 C \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+72 A a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+48 B \,a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}+256 A \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+432 B a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+192 A \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+72 A a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}+48 B \,a^{2} b \,{\mathrm e}^{i \left (d x +c \right )}\right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}-\frac {i C \,b^{4} {\mathrm e}^{i \left (d x +c \right )}}{2 d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} C \,b^{4}}{2 d}-\frac {3 A \,a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{2}}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{4}}{d}-\frac {2 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B b}{d}-\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B a \,b^{3}}{d}-\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 d}-\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C \,a^{2} b^{2}}{d}+\frac {3 A \,a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{2}}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{4}}{d}+\frac {2 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B b}{d}+\frac {4 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B a \,b^{3}}{d}+\frac {a^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 d}+\frac {6 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C \,a^{2} b^{2}}{d}\) | \(882\) |
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Time = 0.33 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.03 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {48 \, {\left (4 \, C a b^{3} + B b^{4}\right )} d x \cos \left (d x + c\right )^{4} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 16 \, B a^{3} b + 24 \, {\left (A + 2 \, C\right )} a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 16 \, B a^{3} b + 24 \, {\left (A + 2 \, C\right )} a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (24 \, C b^{4} \cos \left (d x + c\right )^{4} + 6 \, A a^{4} + 16 \, {\left (B a^{4} + 2 \, {\left (2 \, A + 3 \, C\right )} a^{3} b + 9 \, B a^{2} b^{2} + 6 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.47 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} + 64 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} b + 192 \, {\left (d x + c\right )} C a b^{3} + 48 \, {\left (d x + c\right )} B b^{4} - 3 \, A a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a^{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 )} - 48 \, B a^{3} b {\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 )} - 72 \, A a^{2} b^{2} {\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 )} + 144 \, 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 )} + 96 \, 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 )} + 24 \, 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 )} + 48 \, C b^{4} \sin \left (d x + c\right ) + 192 \, C a^{3} b \tan \left (d x + c\right ) + 288 \, B a^{2} b^{2} \tan \left (d x + c\right ) + 192 \, A a b^{3} \tan \left (d x + c\right )}{48 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 840 vs. \(2 (281) = 562\).
Time = 0.41 (sec) , antiderivative size = 840, normalized size of antiderivative = 2.87 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Too large to display} \]
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Time = 6.37 (sec) , antiderivative size = 4710, normalized size of antiderivative = 16.08 \[ \int (a+b \cos (c+d x))^4 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Too large to display} \]
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