Integrand size = 43, antiderivative size = 276 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx=\frac {(A+8 B-57 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^4 d}+\frac {(4 A+17 B-108 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{42 a^4 d}-\frac {(A+8 B-57 C) \sin (c+d x)}{10 a^4 d \sqrt {\cos (c+d x)}}+\frac {(13 A+29 B-141 C) \sin (c+d x)}{210 a^4 d \sqrt {\cos (c+d x)} (1+\cos (c+d x))^2}+\frac {(4 A+17 B-108 C) \sin (c+d x)}{42 a^4 d \sqrt {\cos (c+d x)} (1+\cos (c+d x))}-\frac {(A-B+C) \sin (c+d x)}{7 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4}+\frac {(3 A+4 B-11 C) \sin (c+d x)}{35 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3} \]
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Time = 0.98 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4197, 3120, 3057, 2827, 2716, 2719, 2720} \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx=\frac {(4 A+17 B-108 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{42 a^4 d}+\frac {(A+8 B-57 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^4 d}-\frac {(A+8 B-57 C) \sin (c+d x)}{10 a^4 d \sqrt {\cos (c+d x)}}+\frac {(4 A+17 B-108 C) \sin (c+d x)}{42 a^4 d \sqrt {\cos (c+d x)} (\cos (c+d x)+1)}+\frac {(13 A+29 B-141 C) \sin (c+d x)}{210 a^4 d \sqrt {\cos (c+d x)} (\cos (c+d x)+1)^2}+\frac {(3 A+4 B-11 C) \sin (c+d x)}{35 a d \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}-\frac {(A-B+C) \sin (c+d x)}{7 d \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^4} \]
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Rule 2716
Rule 2719
Rule 2720
Rule 2827
Rule 3057
Rule 3120
Rule 4197
Rubi steps \begin{align*} \text {integral}& = \int \frac {C+B \cos (c+d x)+A \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^4} \, dx \\ & = -\frac {(A-B+C) \sin (c+d x)}{7 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4}+\frac {\int \frac {\frac {1}{2} a (A-B+15 C)+\frac {7}{2} a (A+B-C) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3} \, dx}{7 a^2} \\ & = -\frac {(A-B+C) \sin (c+d x)}{7 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4}+\frac {(3 A+4 B-11 C) \sin (c+d x)}{35 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3}+\frac {\int \frac {\frac {1}{2} a^2 (2 A-9 B+86 C)+\frac {5}{2} a^2 (3 A+4 B-11 C) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2} \, dx}{35 a^4} \\ & = \frac {(13 A+29 B-141 C) \sin (c+d x)}{210 a^4 d \sqrt {\cos (c+d x)} (1+\cos (c+d x))^2}-\frac {(A-B+C) \sin (c+d x)}{7 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4}+\frac {(3 A+4 B-11 C) \sin (c+d x)}{35 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3}+\frac {\int \frac {-\frac {1}{4} a^3 (A+83 B-657 C)+\frac {3}{4} a^3 (13 A+29 B-141 C) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))} \, dx}{105 a^6} \\ & = \frac {(13 A+29 B-141 C) \sin (c+d x)}{210 a^4 d \sqrt {\cos (c+d x)} (1+\cos (c+d x))^2}-\frac {(A-B+C) \sin (c+d x)}{7 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4}+\frac {(3 A+4 B-11 C) \sin (c+d x)}{35 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3}+\frac {(4 A+17 B-108 C) \sin (c+d x)}{42 d \sqrt {\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right )}+\frac {\int \frac {-\frac {21}{4} a^4 (A+8 B-57 C)+\frac {5}{4} a^4 (4 A+17 B-108 C) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{105 a^8} \\ & = \frac {(13 A+29 B-141 C) \sin (c+d x)}{210 a^4 d \sqrt {\cos (c+d x)} (1+\cos (c+d x))^2}-\frac {(A-B+C) \sin (c+d x)}{7 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4}+\frac {(3 A+4 B-11 C) \sin (c+d x)}{35 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3}+\frac {(4 A+17 B-108 C) \sin (c+d x)}{42 d \sqrt {\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right )}+\frac {(4 A+17 B-108 C) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{84 a^4}-\frac {(A+8 B-57 C) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{20 a^4} \\ & = \frac {(4 A+17 B-108 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{42 a^4 d}-\frac {(A+8 B-57 C) \sin (c+d x)}{10 a^4 d \sqrt {\cos (c+d x)}}+\frac {(13 A+29 B-141 C) \sin (c+d x)}{210 a^4 d \sqrt {\cos (c+d x)} (1+\cos (c+d x))^2}-\frac {(A-B+C) \sin (c+d x)}{7 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4}+\frac {(3 A+4 B-11 C) \sin (c+d x)}{35 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3}+\frac {(4 A+17 B-108 C) \sin (c+d x)}{42 d \sqrt {\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right )}+\frac {(A+8 B-57 C) \int \sqrt {\cos (c+d x)} \, dx}{20 a^4} \\ & = \frac {(A+8 B-57 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^4 d}+\frac {(4 A+17 B-108 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{42 a^4 d}-\frac {(A+8 B-57 C) \sin (c+d x)}{10 a^4 d \sqrt {\cos (c+d x)}}+\frac {(13 A+29 B-141 C) \sin (c+d x)}{210 a^4 d \sqrt {\cos (c+d x)} (1+\cos (c+d x))^2}-\frac {(A-B+C) \sin (c+d x)}{7 d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^4}+\frac {(3 A+4 B-11 C) \sin (c+d x)}{35 a d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3}+\frac {(4 A+17 B-108 C) \sin (c+d x)}{42 d \sqrt {\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right )} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 16.17 (sec) , antiderivative size = 1962, normalized size of antiderivative = 7.11 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx=-\frac {32 A \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sec (d x-\arctan (\cot (c))) \sqrt {1-\sin (d x-\arctan (\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \sin (c) \sin (d x-\arctan (\cot (c)))} \sqrt {1+\sin (d x-\arctan (\cot (c)))}}{21 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sqrt {1+\cot ^2(c)} (a+a \sec (c+d x))^4}-\frac {136 B \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sec (d x-\arctan (\cot (c))) \sqrt {1-\sin (d x-\arctan (\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \sin (c) \sin (d x-\arctan (\cot (c)))} \sqrt {1+\sin (d x-\arctan (\cot (c)))}}{21 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sqrt {1+\cot ^2(c)} (a+a \sec (c+d x))^4}+\frac {288 C \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \sec (d x-\arctan (\cot (c))) \sqrt {1-\sin (d x-\arctan (\cot (c)))} \sqrt {-\sqrt {1+\cot ^2(c)} \sin (c) \sin (d x-\arctan (\cot (c)))} \sqrt {1+\sin (d x-\arctan (\cot (c)))}}{7 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) \sqrt {1+\cot ^2(c)} (a+a \sec (c+d x))^4}+\frac {\cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {8 (20 C-A \cos (c)-8 B \cos (c)+37 C \cos (c)) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec (c)}{5 d}-\frac {8 \sec \left (\frac {c}{2}\right ) \sec ^3\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A \sin \left (\frac {d x}{2}\right )+83 B \sin \left (\frac {d x}{2}\right )-237 C \sin \left (\frac {d x}{2}\right )\right )}{105 d}-\frac {16 \sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A \sin \left (\frac {d x}{2}\right )+8 B \sin \left (\frac {d x}{2}\right )-37 C \sin \left (\frac {d x}{2}\right )\right )}{5 d}+\frac {4 \sec \left (\frac {c}{2}\right ) \sec ^7\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (A \sin \left (\frac {d x}{2}\right )-B \sin \left (\frac {d x}{2}\right )+C \sin \left (\frac {d x}{2}\right )\right )}{7 d}+\frac {8 \sec \left (\frac {c}{2}\right ) \sec ^5\left (\frac {c}{2}+\frac {d x}{2}\right ) \left (2 A \sin \left (\frac {d x}{2}\right )-9 B \sin \left (\frac {d x}{2}\right )+16 C \sin \left (\frac {d x}{2}\right )\right )}{35 d}+\frac {64 C \sec (c) \sec (c+d x) \sin (d x)}{d}-\frac {8 (A+83 B-237 C) \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{105 d}+\frac {8 (2 A-9 B+16 C) \sec ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{35 d}+\frac {4 (A-B+C) \sec ^6\left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{7 d}\right )}{\cos ^{\frac {3}{2}}(c+d x) (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}-\frac {4 A \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\arctan (\tan (c)))} \sqrt {1+\cos (d x+\arctan (\tan (c)))} \sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{5 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}-\frac {32 B \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\arctan (\tan (c)))} \sqrt {1+\cos (d x+\arctan (\tan (c)))} \sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{5 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4}+\frac {228 C \cos ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \sec ^2(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {\, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x+\arctan (\tan (c)))\right ) \sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1-\cos (d x+\arctan (\tan (c)))} \sqrt {1+\cos (d x+\arctan (\tan (c)))} \sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}} \sqrt {1+\tan ^2(c)}}-\frac {\frac {\sin (d x+\arctan (\tan (c))) \tan (c)}{\sqrt {1+\tan ^2(c)}}+\frac {2 \cos ^2(c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}{\cos ^2(c)+\sin ^2(c)}}{\sqrt {\cos (c) \cos (d x+\arctan (\tan (c))) \sqrt {1+\tan ^2(c)}}}\right )}{5 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^4} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1016\) vs. \(2(304)=608\).
Time = 3.00 (sec) , antiderivative size = 1017, normalized size of antiderivative = 3.68
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 700, normalized size of antiderivative = 2.54 \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx=-\frac {2 \, {\left (21 \, {\left (A + 8 \, B - 57 \, C\right )} \cos \left (d x + c\right )^{4} + {\left (64 \, A + 587 \, B - 4248 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (53 \, A + 724 \, B - 5421 \, C\right )} \cos \left (d x + c\right )^{2} - 5 \, {\left (4 \, A - 67 \, B + 564 \, C\right )} \cos \left (d x + c\right ) - 420 \, C\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + 5 \, {\left (\sqrt {2} {\left (4 i \, A + 17 i \, B - 108 i \, C\right )} \cos \left (d x + c\right )^{5} + 4 \, \sqrt {2} {\left (4 i \, A + 17 i \, B - 108 i \, C\right )} \cos \left (d x + c\right )^{4} + 6 \, \sqrt {2} {\left (4 i \, A + 17 i \, B - 108 i \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, \sqrt {2} {\left (4 i \, A + 17 i \, B - 108 i \, C\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (4 i \, A + 17 i \, B - 108 i \, C\right )} \cos \left (d x + c\right )\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (\sqrt {2} {\left (-4 i \, A - 17 i \, B + 108 i \, C\right )} \cos \left (d x + c\right )^{5} + 4 \, \sqrt {2} {\left (-4 i \, A - 17 i \, B + 108 i \, C\right )} \cos \left (d x + c\right )^{4} + 6 \, \sqrt {2} {\left (-4 i \, A - 17 i \, B + 108 i \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, \sqrt {2} {\left (-4 i \, A - 17 i \, B + 108 i \, C\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (-4 i \, A - 17 i \, B + 108 i \, C\right )} \cos \left (d x + c\right )\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 21 \, {\left (\sqrt {2} {\left (-i \, A - 8 i \, B + 57 i \, C\right )} \cos \left (d x + c\right )^{5} + 4 \, \sqrt {2} {\left (-i \, A - 8 i \, B + 57 i \, C\right )} \cos \left (d x + c\right )^{4} + 6 \, \sqrt {2} {\left (-i \, A - 8 i \, B + 57 i \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, \sqrt {2} {\left (-i \, A - 8 i \, B + 57 i \, C\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (-i \, A - 8 i \, B + 57 i \, C\right )} \cos \left (d x + c\right )\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 21 \, {\left (\sqrt {2} {\left (i \, A + 8 i \, B - 57 i \, C\right )} \cos \left (d x + c\right )^{5} + 4 \, \sqrt {2} {\left (i \, A + 8 i \, B - 57 i \, C\right )} \cos \left (d x + c\right )^{4} + 6 \, \sqrt {2} {\left (i \, A + 8 i \, B - 57 i \, C\right )} \cos \left (d x + c\right )^{3} + 4 \, \sqrt {2} {\left (i \, A + 8 i \, B - 57 i \, C\right )} \cos \left (d x + c\right )^{2} + \sqrt {2} {\left (i \, A + 8 i \, B - 57 i \, C\right )} \cos \left (d x + c\right )\right )} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}{420 \, {\left (a^{4} d \cos \left (d x + c\right )^{5} + 4 \, a^{4} d \cos \left (d x + c\right )^{4} + 6 \, a^{4} d \cos \left (d x + c\right )^{3} + 4 \, a^{4} d \cos \left (d x + c\right )^{2} + a^{4} d \cos \left (d x + c\right )\right )}} \]
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Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx=\text {Timed out} \]
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\[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A}{{\left (a \sec \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) (a+a \sec (c+d x))^4} \, dx=\int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\cos \left (c+d\,x\right )}^{7/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^4} \,d x \]
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