Integrand size = 33, antiderivative size = 251 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {1}{2} b^2 \left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) x+\frac {2 a b \left (2 A b^2+a^2 (A+2 C)\right ) \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a b \left (b^2 (11 A-6 C)+a^2 (2 A+3 C)\right ) \sin (c+d x)}{3 d}-\frac {b^2 \left (3 b^2 (6 A-C)+a^2 (4 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {\left (6 A b^2+a^2 (2 A+3 C)\right ) (a+b \cos (c+d x))^2 \tan (c+d x)}{3 d}+\frac {2 A b (a+b \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d} \]
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Time = 0.97 (sec) , antiderivative size = 251, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3127, 3126, 3112, 3102, 2814, 3855} \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {2 a b \left (a^2 (A+2 C)+2 A b^2\right ) \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a b \left (a^2 (2 A+3 C)+b^2 (11 A-6 C)\right ) \sin (c+d x)}{3 d}-\frac {b^2 \left (a^2 (4 A+6 C)+3 b^2 (6 A-C)\right ) \sin (c+d x) \cos (c+d x)}{6 d}+\frac {\left (a^2 (2 A+3 C)+6 A b^2\right ) \tan (c+d x) (a+b \cos (c+d x))^2}{3 d}+\frac {1}{2} b^2 x \left (C \left (12 a^2+b^2\right )+2 A b^2\right )+\frac {A \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^4}{3 d}+\frac {2 A b \tan (c+d x) \sec (c+d x) (a+b \cos (c+d x))^3}{3 d} \]
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Rule 2814
Rule 3102
Rule 3112
Rule 3126
Rule 3127
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {A (a+b \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{3} \int (a+b \cos (c+d x))^3 \left (4 A b+a (2 A+3 C) \cos (c+d x)-b (2 A-3 C) \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx \\ & = \frac {2 A b (a+b \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{6} \int (a+b \cos (c+d x))^2 \left (2 \left (6 A b^2+\frac {1}{2} a^2 (4 A+6 C)\right )+4 a b (A+3 C) \cos (c+d x)-6 b^2 (2 A-C) \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx \\ & = \frac {\left (6 A b^2+a^2 (2 A+3 C)\right ) (a+b \cos (c+d x))^2 \tan (c+d x)}{3 d}+\frac {2 A b (a+b \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{6} \int (a+b \cos (c+d x)) \left (12 b \left (2 A b^2+a^2 (A+2 C)\right )-2 a b^2 (4 A-9 C) \cos (c+d x)-2 b \left (3 b^2 (6 A-C)+a^2 (4 A+6 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = -\frac {b^2 \left (3 b^2 (6 A-C)+a^2 (4 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {\left (6 A b^2+a^2 (2 A+3 C)\right ) (a+b \cos (c+d x))^2 \tan (c+d x)}{3 d}+\frac {2 A b (a+b \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{12} \int \left (24 a b \left (2 A b^2+a^2 (A+2 C)\right )+6 b^2 \left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) \cos (c+d x)-8 a b \left (b^2 (11 A-6 C)+a^2 (2 A+3 C)\right ) \cos ^2(c+d x)\right ) \sec (c+d x) \, dx \\ & = -\frac {2 a b \left (b^2 (11 A-6 C)+a^2 (2 A+3 C)\right ) \sin (c+d x)}{3 d}-\frac {b^2 \left (3 b^2 (6 A-C)+a^2 (4 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {\left (6 A b^2+a^2 (2 A+3 C)\right ) (a+b \cos (c+d x))^2 \tan (c+d x)}{3 d}+\frac {2 A b (a+b \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{12} \int \left (24 a b \left (2 A b^2+a^2 (A+2 C)\right )+6 b^2 \left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = \frac {1}{2} b^2 \left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) x-\frac {2 a b \left (b^2 (11 A-6 C)+a^2 (2 A+3 C)\right ) \sin (c+d x)}{3 d}-\frac {b^2 \left (3 b^2 (6 A-C)+a^2 (4 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {\left (6 A b^2+a^2 (2 A+3 C)\right ) (a+b \cos (c+d x))^2 \tan (c+d x)}{3 d}+\frac {2 A b (a+b \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d}+\left (2 a b \left (2 A b^2+a^2 (A+2 C)\right )\right ) \int \sec (c+d x) \, dx \\ & = \frac {1}{2} b^2 \left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) x+\frac {2 a b \left (2 A b^2+a^2 (A+2 C)\right ) \text {arctanh}(\sin (c+d x))}{d}-\frac {2 a b \left (b^2 (11 A-6 C)+a^2 (2 A+3 C)\right ) \sin (c+d x)}{3 d}-\frac {b^2 \left (3 b^2 (6 A-C)+a^2 (4 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{6 d}+\frac {\left (6 A b^2+a^2 (2 A+3 C)\right ) (a+b \cos (c+d x))^2 \tan (c+d x)}{3 d}+\frac {2 A b (a+b \cos (c+d x))^3 \sec (c+d x) \tan (c+d x)}{3 d}+\frac {A (a+b \cos (c+d x))^4 \sec ^2(c+d x) \tan (c+d x)}{3 d} \\ \end{align*}
Time = 10.29 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.64 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {6 b^2 \left (2 A b^2+\left (12 a^2+b^2\right ) C\right ) (c+d x)-24 a b \left (2 A b^2+a^2 (A+2 C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+24 a b \left (2 A b^2+a^2 (A+2 C)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {a^3 A (a+12 b)}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2 a^4 A \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {4 a^2 \left (18 A b^2+a^2 (2 A+3 C)\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}+\frac {2 a^4 A \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {a^3 A (a+12 b)}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {4 a^2 \left (18 A b^2+a^2 (2 A+3 C)\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}+48 a b^3 C \sin (c+d x)+3 b^4 C \sin (2 (c+d x))}{12 d} \]
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Time = 9.20 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.79
method | result | size |
parts | \(-\frac {a^{4} A \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (A \,b^{4}+6 C \,a^{2} b^{2}\right ) \left (d x +c \right )}{d}+\frac {\left (4 A a \,b^{3}+4 C \,a^{3} b \right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (6 A \,a^{2} b^{2}+C \,a^{4}\right ) \tan \left (d x +c \right )}{d}+\frac {C \,b^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {4 A \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {4 \sin \left (d x +c \right ) C a \,b^{3}}{d}\) | \(198\) |
derivativedivides | \(\frac {-a^{4} A \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+C \,a^{4} \tan \left (d x +c \right )+4 A \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 C \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 A \,a^{2} b^{2} \tan \left (d x +c \right )+6 C \,a^{2} b^{2} \left (d x +c \right )+4 A a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 C \sin \left (d x +c \right ) a \,b^{3}+A \,b^{4} \left (d x +c \right )+C \,b^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(200\) |
default | \(\frac {-a^{4} A \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+C \,a^{4} \tan \left (d x +c \right )+4 A \,a^{3} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+4 C \,a^{3} b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+6 A \,a^{2} b^{2} \tan \left (d x +c \right )+6 C \,a^{2} b^{2} \left (d x +c \right )+4 A a \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+4 C \sin \left (d x +c \right ) a \,b^{3}+A \,b^{4} \left (d x +c \right )+C \,b^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )}{d}\) | \(200\) |
parallelrisch | \(\frac {-144 b \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) a \left (2 A \,b^{2}+a^{2} \left (A +2 C \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+144 b \left (\frac {\cos \left (3 d x +3 c \right )}{3}+\cos \left (d x +c \right )\right ) a \left (2 A \,b^{2}+a^{2} \left (A +2 C \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+24 x \,b^{2} \left (\left (A +\frac {C}{2}\right ) b^{2}+6 a^{2} C \right ) d \cos \left (3 d x +3 c \right )+\left (9 C \,b^{4}+144 A \,a^{2} b^{2}+16 \left (A +\frac {3 C}{2}\right ) a^{4}\right ) \sin \left (3 d x +3 c \right )+96 \left (A \,a^{3} b +C a \,b^{3}\right ) \sin \left (2 d x +2 c \right )+48 C \sin \left (4 d x +4 c \right ) a \,b^{3}+3 C \sin \left (5 d x +5 c \right ) b^{4}+72 x \,b^{2} \left (\left (A +\frac {C}{2}\right ) b^{2}+6 a^{2} C \right ) d \cos \left (d x +c \right )+48 \left (\frac {C \,b^{4}}{8}+3 A \,a^{2} b^{2}+a^{4} \left (A +\frac {C}{2}\right )\right ) \sin \left (d x +c \right )}{24 d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(312\) |
risch | \(x A \,b^{4}+6 x C \,a^{2} b^{2}+\frac {b^{4} C x}{2}-\frac {i {\mathrm e}^{2 i \left (d x +c \right )} C \,b^{4}}{8 d}-\frac {2 i {\mathrm e}^{i \left (d x +c \right )} C a \,b^{3}}{d}+\frac {2 i {\mathrm e}^{-i \left (d x +c \right )} C a \,b^{3}}{d}+\frac {i {\mathrm e}^{-2 i \left (d x +c \right )} C \,b^{4}}{8 d}-\frac {2 i a^{2} \left (6 A a b \,{\mathrm e}^{5 i \left (d x +c \right )}-18 A \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-3 C \,a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-6 A \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-36 A \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 C \,a^{2} {\mathrm e}^{2 i \left (d x +c \right )}-6 A a b \,{\mathrm e}^{i \left (d x +c \right )}-2 A \,a^{2}-18 A \,b^{2}-3 a^{2} C \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}+\frac {2 A \,a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}+\frac {4 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A}{d}+\frac {4 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{d}-\frac {2 A \,a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}-\frac {4 a \,b^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A}{d}-\frac {4 a^{3} b \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{d}\) | \(385\) |
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Time = 0.29 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.83 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {3 \, {\left (12 \, C a^{2} b^{2} + {\left (2 \, A + C\right )} b^{4}\right )} d x \cos \left (d x + c\right )^{3} + 6 \, {\left ({\left (A + 2 \, C\right )} a^{3} b + 2 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 6 \, {\left ({\left (A + 2 \, C\right )} a^{3} b + 2 \, A a b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + {\left (3 \, C b^{4} \cos \left (d x + c\right )^{4} + 24 \, C a b^{3} \cos \left (d x + c\right )^{3} + 12 \, A a^{3} b \cos \left (d x + c\right ) + 2 \, A a^{4} + 2 \, {\left ({\left (2 \, A + 3 \, C\right )} a^{4} + 18 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )^{3}} \]
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Timed out. \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.88 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {4 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{4} + 72 \, {\left (d x + c\right )} C a^{2} b^{2} + 12 \, {\left (d x + c\right )} A b^{4} + 3 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{4} - 12 \, A a^{3} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, C a^{3} b {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, A a b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 48 \, C a b^{3} \sin \left (d x + c\right ) + 12 \, C a^{4} \tan \left (d x + c\right ) + 72 \, A a^{2} b^{2} \tan \left (d x + c\right )}{12 \, d} \]
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Time = 0.37 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.58 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\frac {3 \, {\left (12 \, C a^{2} b^{2} + 2 \, A b^{4} + C b^{4}\right )} {\left (d x + c\right )} + 12 \, {\left (A a^{3} b + 2 \, C a^{3} b + 2 \, A a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 12 \, {\left (A a^{3} b + 2 \, C a^{3} b + 2 \, A a b^{3}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + \frac {6 \, {\left (8 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, C a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + C b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{2}} - \frac {4 \, {\left (3 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 6 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 18 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 2 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 36 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 18 \, A a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{3}}}{6 \, d} \]
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Time = 3.91 (sec) , antiderivative size = 2662, normalized size of antiderivative = 10.61 \[ \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx=\text {Too large to display} \]
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