Integrand size = 41, antiderivative size = 223 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=b^3 C x+\frac {\left (12 a^2 b B+8 b^3 B+12 a b^2 (A+2 C)+a^3 (3 A+4 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (3 A b^3+4 a^3 B+16 a b^2 B+6 a^2 b (2 A+3 C)\right ) \tan (c+d x)}{6 d}+\frac {a \left (6 A b^2+20 a b B+3 a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(3 A b+4 a B) (a+b \cos (c+d x))^2 \sec ^2(c+d x) \tan (c+d x)}{12 d}+\frac {A (a+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d} \]
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Time = 0.72 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {3126, 3110, 3100, 2814, 3855} \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {a \tan (c+d x) \sec (c+d x) \left (3 a^2 (3 A+4 C)+20 a b B+6 A b^2\right )}{24 d}+\frac {\left (a^3 (3 A+4 C)+12 a^2 b B+12 a b^2 (A+2 C)+8 b^3 B\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\tan (c+d x) \left (4 a^3 B+6 a^2 b (2 A+3 C)+16 a b^2 B+3 A b^3\right )}{6 d}+\frac {(4 a B+3 A b) \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac {A \tan (c+d x) \sec ^3(c+d x) (a+b \cos (c+d x))^3}{4 d}+b^3 C x \]
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Rule 2814
Rule 3100
Rule 3110
Rule 3126
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {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 (3 A b+4 a B+(3 a A+4 b B+4 a C) \cos (c+d x)+4 b C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx \\ & = \frac {(3 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+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 (6 A b^2+20 a b B+3 a^2 (3 A+4 C)+\left (15 a A b+8 a^2 B+12 b^2 B+24 a b C\right ) \cos (c+d x)+12 b^2 C \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx \\ & = \frac {a \left (6 A b^2+20 a b B+3 a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(3 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+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{24} \int \left (-4 \left (3 A b^3+4 a^3 B+16 a b^2 B+6 a^2 b (2 A+3 C)\right )-3 \left (12 a^2 b B+8 b^3 B+12 a b^2 (A+2 C)+a^3 (3 A+4 C)\right ) \cos (c+d x)-24 b^3 C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx \\ & = \frac {\left (3 A b^3+4 a^3 B+16 a b^2 B+6 a^2 b (2 A+3 C)\right ) \tan (c+d x)}{6 d}+\frac {a \left (6 A b^2+20 a b B+3 a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(3 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+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{24} \int \left (-3 \left (12 a^2 b B+8 b^3 B+12 a b^2 (A+2 C)+a^3 (3 A+4 C)\right )-24 b^3 C \cos (c+d x)\right ) \sec (c+d x) \, dx \\ & = b^3 C x+\frac {\left (3 A b^3+4 a^3 B+16 a b^2 B+6 a^2 b (2 A+3 C)\right ) \tan (c+d x)}{6 d}+\frac {a \left (6 A b^2+20 a b B+3 a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(3 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+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac {1}{8} \left (-12 a^2 b B-8 b^3 B-12 a b^2 (A+2 C)-a^3 (3 A+4 C)\right ) \int \sec (c+d x) \, dx \\ & = b^3 C x+\frac {\left (12 a^2 b B+8 b^3 B+12 a b^2 (A+2 C)+a^3 (3 A+4 C)\right ) \text {arctanh}(\sin (c+d x))}{8 d}+\frac {\left (3 A b^3+4 a^3 B+16 a b^2 B+6 a^2 b (2 A+3 C)\right ) \tan (c+d x)}{6 d}+\frac {a \left (6 A b^2+20 a b B+3 a^2 (3 A+4 C)\right ) \sec (c+d x) \tan (c+d x)}{24 d}+\frac {(3 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+b \cos (c+d x))^3 \sec ^3(c+d x) \tan (c+d x)}{4 d} \\ \end{align*}
Time = 1.32 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.74 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {24 b^3 C d x+3 \left (12 a^2 b B+8 b^3 B+12 a b^2 (A+2 C)+a^3 (3 A+4 C)\right ) \text {arctanh}(\sin (c+d x))+3 \left (8 \left (A b^3+a^3 B+3 a b^2 B+3 a^2 b (A+C)\right )+a \left (12 A b^2+12 a b B+a^2 (3 A+4 C)\right ) \sec (c+d x)+2 a^3 A \sec ^3(c+d x)\right ) \tan (c+d x)+8 a^2 (3 A b+a B) \tan ^3(c+d x)}{24 d} \]
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Time = 0.62 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.98
method | result | size |
parts | \(\frac {A \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )}{d}-\frac {\left (3 A \,a^{2} b +B \,a^{3}\right ) \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}+\frac {\left (B \,b^{3}+3 C a \,b^{2}\right ) \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {\left (A \,b^{3}+3 B a \,b^{2}+3 a^{2} b C \right ) \tan \left (d x +c \right )}{d}+\frac {\left (3 a A \,b^{2}+3 B \,a^{2} b +a^{3} C \right ) \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}+\frac {C \,b^{3} \left (d x +c \right )}{d}\) | \(219\) |
derivativedivides | \(\frac {A \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+a^{3} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 A \,a^{2} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+3 B \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{2} b C \tan \left (d x +c \right )+3 a A \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 B a \,b^{2} \tan \left (d x +c \right )+3 C a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{3} \tan \left (d x +c \right )+B \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \,b^{3} \left (d x +c \right )}{d}\) | \(303\) |
default | \(\frac {A \,a^{3} \left (-\left (-\frac {\left (\sec ^{3}\left (d x +c \right )\right )}{4}-\frac {3 \sec \left (d x +c \right )}{8}\right ) \tan \left (d x +c \right )+\frac {3 \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )-B \,a^{3} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+a^{3} C \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )-3 A \,a^{2} b \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+3 B \,a^{2} b \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 a^{2} b C \tan \left (d x +c \right )+3 a A \,b^{2} \left (\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{2}+\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )+3 B a \,b^{2} \tan \left (d x +c \right )+3 C a \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+A \,b^{3} \tan \left (d x +c \right )+B \,b^{3} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+C \,b^{3} \left (d x +c \right )}{d}\) | \(303\) |
parallelrisch | \(\frac {-9 \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right ) \left (\left (A +\frac {4 C}{3}\right ) a^{3}+4 B \,a^{2} b +4 a \,b^{2} \left (A +2 C \right )+\frac {8 B \,b^{3}}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+9 \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right ) \left (\left (A +\frac {4 C}{3}\right ) a^{3}+4 B \,a^{2} b +4 a \,b^{2} \left (A +2 C \right )+\frac {8 B \,b^{3}}{3}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+96 C \,b^{3} d x \cos \left (2 d x +2 c \right )+24 C \,b^{3} d x \cos \left (4 d x +4 c \right )+\left (64 B \,a^{3}+192 a^{2} \left (A +\frac {3 C}{4}\right ) b +144 B a \,b^{2}+48 A \,b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (16 B \,a^{3}+48 \left (A +\frac {3 C}{2}\right ) b \,a^{2}+72 B a \,b^{2}+24 A \,b^{3}\right ) \sin \left (4 d x +4 c \right )+18 a \left (\left (A +\frac {4 C}{3}\right ) a^{2}+4 B a b +4 A \,b^{2}\right ) \sin \left (3 d x +3 c \right )+\left (\left (66 A +24 C \right ) a^{3}+72 B \,a^{2} b +72 a A \,b^{2}\right ) \sin \left (d x +c \right )+72 C \,b^{3} d x}{24 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(361\) |
risch | \(b^{3} C x +\frac {i \left (48 A \,a^{2} b +16 B \,a^{3}+24 A \,b^{3}+72 B a \,b^{2}+72 a^{2} b C -12 C \,a^{3} {\mathrm e}^{7 i \left (d x +c \right )}+24 A \,b^{3} {\mathrm e}^{6 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 )}+72 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 )}+12 C \,a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+72 A \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+9 A \,a^{3} {\mathrm e}^{i \left (d x +c \right )}+64 B \,a^{3} {\mathrm e}^{2 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 )}+72 C \,a^{2} b \,{\mathrm e}^{6 i \left (d x +c \right )}-36 A a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-36 B \,a^{2} b \,{\mathrm e}^{5 i \left (d x +c \right )}-36 A a \,b^{2} {\mathrm e}^{7 i \left (d x +c \right )}-36 B \,a^{2} b \,{\mathrm e}^{7 i \left (d x +c \right )}+72 B a \,b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+36 A a \,b^{2} {\mathrm e}^{3 i \left (d x +c \right )}+36 B \,a^{2} b \,{\mathrm e}^{3 i \left (d x +c \right )}+192 A \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+216 B a \,b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+216 C \,a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+144 A \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+216 B a \,b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+216 C \,a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}+36 A a \,b^{2} {\mathrm e}^{i \left (d x +c \right )}+36 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 {3 A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 d}+\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) A \,b^{2}}{2 d}+\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B b}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) B \,b^{3}}{d}+\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C}{2 d}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) C a \,b^{2}}{d}-\frac {3 A \,a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 d}-\frac {3 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) A \,b^{2}}{2 d}-\frac {3 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B b}{2 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) B \,b^{3}}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C}{2 d}-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) C a \,b^{2}}{d}\) | \(781\) |
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Time = 0.31 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.15 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {48 \, C b^{3} d x \cos \left (d x + c\right )^{4} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \, B a^{2} b + 12 \, {\left (A + 2 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \, B a^{2} b + 12 \, {\left (A + 2 \, C\right )} a b^{2} + 8 \, B b^{3}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (6 \, A a^{3} + 8 \, {\left (2 \, B a^{3} + 3 \, {\left (2 \, A + 3 \, C\right )} a^{2} b + 9 \, B a b^{2} + 3 \, A b^{3}\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left ({\left (3 \, A + 4 \, C\right )} a^{3} + 12 \, B a^{2} b + 12 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + 8 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Timed out} \]
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Time = 0.25 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.67 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\frac {16 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{3} + 48 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{2} b + 48 \, {\left (d x + c\right )} C b^{3} - 3 \, A a^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 12 \, C a^{3} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, B a^{2} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 36 \, A a b^{2} {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 72 \, C a b^{2} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 24 \, B b^{3} {\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 144 \, C a^{2} b \tan \left (d x + c\right ) + 144 \, B a b^{2} \tan \left (d x + c\right ) + 48 \, A b^{3} \tan \left (d x + c\right )}{48 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 759 vs. \(2 (213) = 426\).
Time = 0.38 (sec) , antiderivative size = 759, normalized size of antiderivative = 3.40 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Too large to display} \]
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Time = 4.96 (sec) , antiderivative size = 3210, normalized size of antiderivative = 14.39 \[ \int (a+b \cos (c+d x))^3 \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^5(c+d x) \, dx=\text {Too large to display} \]
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