Integrand size = 31, antiderivative size = 216 \[ \int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=b^4 B x+\frac {\left (3 a^4 A+24 a^2 A b^2+8 A b^4+16 a^3 b B+32 a b^3 B\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a \left (16 a^2 A b+19 A b^3+4 a^3 B+34 a b^2 B\right ) \tan (c+d x)}{6 d}+\frac {a^2 \left (9 a^2 A+26 A b^2+32 a b B\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {a (7 A b+4 a B) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac {a A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d} \]
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Time = 0.92 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3068, 3126, 3110, 3100, 2814, 3855} \[ \int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\frac {a^2 \left (9 a^2 A+32 a b B+26 A b^2\right ) \tan (c+d x) \sec (c+d x)}{24 d}+\frac {a \left (4 a^3 B+16 a^2 A b+34 a b^2 B+19 A b^3\right ) \tan (c+d x)}{6 d}+\frac {\left (3 a^4 A+16 a^3 b B+24 a^2 A b^2+32 a b^3 B+8 A b^4\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a (4 a B+7 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac {a A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}+b^4 B x \]
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Rule 2814
Rule 3068
Rule 3100
Rule 3110
Rule 3126
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {a A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{4} \int (a+b \cos (c+d x))^2 \left (a (7 A b+4 a B)+\left (3 a^2 A+4 A b^2+8 a b B\right ) \cos (c+d x)+4 b^2 B \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx \\ & = \frac {a (7 A b+4 a B) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac {a A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac {1}{12} \int (a+b \cos (c+d x)) \left (a \left (9 a^2 A+26 A b^2+32 a b B\right )+\left (23 a^2 A b+12 A b^3+8 a^3 B+36 a b^2 B\right ) \cos (c+d x)+12 b^3 B \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx \\ & = \frac {a^2 \left (9 a^2 A+26 A b^2+32 a b B\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {a (7 A b+4 a B) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac {a A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{24} \int \left (-4 a \left (16 a^2 A b+19 A b^3+4 a^3 B+34 a b^2 B\right )-3 \left (3 a^4 A+24 a^2 A b^2+8 A b^4+16 a^3 b B+32 a b^3 B\right ) \cos (c+d x)-24 b^4 B \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx \\ & = \frac {a \left (16 a^2 A b+19 A b^3+4 a^3 B+34 a b^2 B\right ) \tan (c+d x)}{6 d}+\frac {a^2 \left (9 a^2 A+26 A b^2+32 a b B\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {a (7 A b+4 a B) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac {a A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{24} \int \left (-3 \left (3 a^4 A+24 a^2 A b^2+8 A b^4+16 a^3 b B+32 a b^3 B\right )-24 b^4 B \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = b^4 B x+\frac {a \left (16 a^2 A b+19 A b^3+4 a^3 B+34 a b^2 B\right ) \tan (c+d x)}{6 d}+\frac {a^2 \left (9 a^2 A+26 A b^2+32 a b B\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {a (7 A b+4 a B) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac {a A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{8} \left (-3 a^4 A-24 a^2 A b^2-8 A b^4-16 a^3 b B-32 a b^3 B\right ) \int \sec (c+d x) \, dx \\ & = b^4 B x+\frac {\left (3 a^4 A+24 a^2 A b^2+8 A b^4+16 a^3 b B+32 a b^3 B\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {a \left (16 a^2 A b+19 A b^3+4 a^3 B+34 a b^2 B\right ) \tan (c+d x)}{6 d}+\frac {a^2 \left (9 a^2 A+26 A b^2+32 a b B\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {a (7 A b+4 a B) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac {a A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d} \\ \end{align*}
Time = 0.72 (sec) , antiderivative size = 160, normalized size of antiderivative = 0.74 \[ \int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\frac {24 b^4 B d x+3 \left (3 a^4 A+24 a^2 A b^2+8 A b^4+16 a^3 b B+32 a b^3 B\right ) \text {arctanh}(\sin (c+d x))+3 a \left (8 \left (4 a^2 A b+4 A b^3+a^3 B+6 a b^2 B\right )+a \left (3 a^2 A+24 A b^2+16 a b B\right ) \sec (c+d x)+2 a^3 A \sec ^3(c+d x)\right ) \tan (c+d x)+8 a^3 (4 A b+a B) \tan ^3(c+d x)}{24 d} \]
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Time = 5.47 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.99
method | result | size |
parts | \(\frac {a^{4} A \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}+\frac {\left (A \,b^{4}+4 B a \,b^{3}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (4 A a \,b^{3}+6 B \,a^{2} b^{2}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+4 B \,a^{3} b \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}-\frac {\left (4 A \,a^{3} b +B \,a^{4}\right ) \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {B \,b^{4} \left (d x +c \right )}{d}\) | \(213\) |
derivativedivides | \(\frac {a^{4} A \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )-4 A \,a^{3} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 B \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 A \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 B \tan \left (d x +c \right ) a^{2} b^{2}+4 A \tan \left (d x +c \right ) a \,b^{3}+4 B a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,b^{4} \left (d x +c \right )}{d}\) | \(260\) |
default | \(\frac {a^{4} A \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{4} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )-4 A \,a^{3} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+4 B \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 A \,a^{2} b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+6 B \tan \left (d x +c \right ) a^{2} b^{2}+4 A \tan \left (d x +c \right ) a \,b^{3}+4 B a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{4} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+B \,b^{4} \left (d x +c \right )}{d}\) | \(260\) |
parallelrisch | \(\frac {-36 \left (a^{4} A +8 A \,a^{2} b^{2}+\frac {8}{3} A \,b^{4}+\frac {16}{3} B \,a^{3} b +\frac {32}{3} B a \,b^{3}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+36 \left (a^{4} A +8 A \,a^{2} b^{2}+\frac {8}{3} A \,b^{4}+\frac {16}{3} B \,a^{3} b +\frac {32}{3} B a \,b^{3}\right ) \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+96 B \,b^{4} d x \cos \left (2 d x +2 c \right )+24 B \,b^{4} d x \cos \left (4 d x +4 c \right )+\left (256 A \,a^{3} b +192 A a \,b^{3}+64 B \,a^{4}+288 B \,a^{2} b^{2}\right ) \sin \left (2 d x +2 c \right )+\left (64 A \,a^{3} b +96 A a \,b^{3}+16 B \,a^{4}+144 B \,a^{2} b^{2}\right ) \sin \left (4 d x +4 c \right )+\left (18 a^{4} A +144 A \,a^{2} b^{2}+96 B \,a^{3} b \right ) \sin \left (3 d x +3 c \right )+\left (66 a^{4} A +144 A \,a^{2} b^{2}+96 B \,a^{3} b \right ) \sin \left (d x +c \right )+72 B \,b^{4} d x}{24 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(358\) |
risch | \(b^{4} B x -\frac {i a \left (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 )}-144 B a \,b^{2} {\mathrm e}^{6 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 )}+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 )}-192 A \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-16 B \,a^{3}-64 A \,a^{2} b -144 B a \,b^{2}-96 A \,b^{3}-64 B \,a^{3} {\mathrm e}^{2 i \left (d x +c \right )}+33 A \,a^{3} {\mathrm e}^{5 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}^{3 i \left (d x +c \right )}-9 A \,a^{3} {\mathrm e}^{i \left (d x +c \right )}+9 A \,a^{3} {\mathrm e}^{7 i \left (d x +c \right )}-96 A \,b^{3} {\mathrm e}^{6 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 )}-432 B a \,b^{2} {\mathrm e}^{4 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 )}-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 )}\right )}{12 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {3 a^{4} A \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 {3 a^{4} A \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}\) | \(627\) |
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Time = 0.31 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.16 \[ \int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\frac {48 \, B b^{4} d x \cos \left (d x + c\right )^{4} + 3 \, {\left (3 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left (3 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, A a^{4} + 16 \, {\left (B a^{4} + 4 \, A a^{3} b + 9 \, B a^{2} b^{2} + 6 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (3 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\text {Timed out} \]
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Time = 0.21 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.47 \[ \int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{4} + 64 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{3} b + 48 \, {\left (d x + c\right )} B b^{4} - 3 \, A a^{4} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 48 \, B a^{3} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 72 \, A a^{2} b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 96 \, B a b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, A b^{4} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 288 \, B a^{2} b^{2} \tan \left (d x + c\right ) + 192 \, A a b^{3} \tan \left (d x + c\right )}{48 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 635 vs. \(2 (206) = 412\).
Time = 0.35 (sec) , antiderivative size = 635, normalized size of antiderivative = 2.94 \[ \int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\frac {24 \, {\left (d x + c\right )} B b^{4} + 3 \, {\left (3 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, {\left (3 \, A a^{4} + 16 \, B a^{3} b + 24 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {2 \, {\left (15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 24 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 48 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 144 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 96 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 40 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 160 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 48 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 432 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 9 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 40 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 160 \, A a^{3} b \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} - 432 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 288 \, A a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 15 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 24 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 48 \, B a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 72 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 144 \, B a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 96 \, A a 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} \]
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Time = 3.17 (sec) , antiderivative size = 1969, normalized size of antiderivative = 9.12 \[ \int (a+b \cos (c+d x))^4 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx=\text {Too large to display} \]
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