Integrand size = 43, antiderivative size = 258 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=-\frac {7 (11 A-8 B+5 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d}+\frac {5 (30 A-21 B+14 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 a^2 d}+\frac {5 (30 A-21 B+14 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 a^2 d}-\frac {7 (11 A-8 B+5 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 a^2 d}+\frac {(30 A-21 B+14 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 a^2 d}-\frac {(11 A-8 B+5 C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2} \]
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Time = 0.60 (sec) , antiderivative size = 258, 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, 3056, 2827, 2715, 2719, 2720} \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {5 (30 A-21 B+14 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 a^2 d}-\frac {7 (11 A-8 B+5 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d}-\frac {(11 A-8 B+5 C) \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x)}{3 a^2 d (\cos (c+d x)+1)}+\frac {(30 A-21 B+14 C) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{7 a^2 d}-\frac {7 (11 A-8 B+5 C) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{15 a^2 d}+\frac {5 (30 A-21 B+14 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{21 a^2 d}-\frac {(A-B+C) \sin (c+d x) \cos ^{\frac {9}{2}}(c+d x)}{3 d (a \cos (c+d x)+a)^2} \]
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Rule 2715
Rule 2719
Rule 2720
Rule 2827
Rule 3056
Rule 3120
Rule 4197
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right )}{(a+a \cos (c+d x))^2} \, dx \\ & = -\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {\int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (-\frac {3}{2} a (3 A-3 B+C)+\frac {1}{2} a (13 A-7 B+7 C) \cos (c+d x)\right )}{a+a \cos (c+d x)} \, dx}{3 a^2} \\ & = -\frac {(11 A-8 B+5 C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {\int \cos ^{\frac {5}{2}}(c+d x) \left (-\frac {7}{2} a^2 (11 A-8 B+5 C)+\frac {3}{2} a^2 (30 A-21 B+14 C) \cos (c+d x)\right ) \, dx}{3 a^4} \\ & = -\frac {(11 A-8 B+5 C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac {(7 (11 A-8 B+5 C)) \int \cos ^{\frac {5}{2}}(c+d x) \, dx}{6 a^2}+\frac {(30 A-21 B+14 C) \int \cos ^{\frac {7}{2}}(c+d x) \, dx}{2 a^2} \\ & = -\frac {7 (11 A-8 B+5 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 a^2 d}+\frac {(30 A-21 B+14 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 a^2 d}-\frac {(11 A-8 B+5 C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}-\frac {(7 (11 A-8 B+5 C)) \int \sqrt {\cos (c+d x)} \, dx}{10 a^2}+\frac {(5 (30 A-21 B+14 C)) \int \cos ^{\frac {3}{2}}(c+d x) \, dx}{14 a^2} \\ & = -\frac {7 (11 A-8 B+5 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d}+\frac {5 (30 A-21 B+14 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 a^2 d}-\frac {7 (11 A-8 B+5 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 a^2 d}+\frac {(30 A-21 B+14 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 a^2 d}-\frac {(11 A-8 B+5 C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2}+\frac {(5 (30 A-21 B+14 C)) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{42 a^2} \\ & = -\frac {7 (11 A-8 B+5 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a^2 d}+\frac {5 (30 A-21 B+14 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 a^2 d}+\frac {5 (30 A-21 B+14 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{21 a^2 d}-\frac {7 (11 A-8 B+5 C) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{15 a^2 d}+\frac {(30 A-21 B+14 C) \cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{7 a^2 d}-\frac {(11 A-8 B+5 C) \cos ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{3 a^2 d (1+\cos (c+d x))}-\frac {(A-B+C) \cos ^{\frac {9}{2}}(c+d x) \sin (c+d x)}{3 d (a+a \cos (c+d x))^2} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 14.91 (sec) , antiderivative size = 1820, normalized size of antiderivative = 7.05 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=-\frac {200 A \cos ^4\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 ) \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))^2}+\frac {20 B \cos ^4\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 ) \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)))}}{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))^2}-\frac {40 C \cos ^4\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 ) \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)))}}{3 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))^2}+\frac {\cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sqrt {\cos (c+d x)} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {8 (25 A-20 B+15 C+52 A \cos (c)-36 B \cos (c)+20 C \cos (c)) \csc (c)}{5 d}+\frac {4 (107 A-56 B+28 C) \cos (d x) \sin (c)}{21 d}-\frac {8 (2 A-B) \cos (2 d x) \sin (2 c)}{5 d}+\frac {4 A \cos (3 d x) \sin (3 c)}{7 d}-\frac {4 \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 )-B \sin \left (\frac {d x}{2}\right )+C \sin \left (\frac {d x}{2}\right )\right )}{3 d}+\frac {8 \sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \left (5 A \sin \left (\frac {d x}{2}\right )-4 B \sin \left (\frac {d x}{2}\right )+3 C \sin \left (\frac {d x}{2}\right )\right )}{d}+\frac {4 (107 A-56 B+28 C) \cos (c) \sin (d x)}{21 d}-\frac {8 (2 A-B) \cos (2 c) \sin (2 d x)}{5 d}+\frac {4 A \cos (3 c) \sin (3 d x)}{7 d}-\frac {4 (A-B+C) \sec ^2\left (\frac {c}{2}+\frac {d x}{2}\right ) \tan \left (\frac {c}{2}\right )}{3 d}\right )}{(A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^2}+\frac {154 A \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \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))^2}-\frac {112 B \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \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))^2}+\frac {14 C \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \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 )}{d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)) (a+a \sec (c+d x))^2} \]
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Time = 9.92 (sec) , antiderivative size = 513, normalized size of antiderivative = 1.99
method | result | size |
default | \(-\frac {\sqrt {\left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \left (-2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \left (750 A \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+1617 A \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-525 B \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-1176 B \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+350 C \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+735 C \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, \left (750 A \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+1617 A \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-525 B \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-1176 B \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+350 C \operatorname {EllipticF}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+735 C \operatorname {EllipticE}\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+960 A \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (-2016 A -672 B \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (2608 A +896 B +560 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (-5932 A +2296 B -2660 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (6184 A -3682 B +2940 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-1839 A +1197 B -875 C \right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )}{210 a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \sqrt {-2 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\, d}\) | \(513\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.72 \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\frac {2 \, {\left (30 \, A \cos \left (d x + c\right )^{4} - 6 \, {\left (4 \, A - 7 \, B\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (61 \, A - 28 \, B + 35 \, C\right )} \cos \left (d x + c\right )^{2} + {\left (961 \, A - 658 \, B + 455 \, C\right )} \cos \left (d x + c\right ) + 750 \, A - 525 \, B + 350 \, C\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 25 \, {\left (\sqrt {2} {\left (30 i \, A - 21 i \, B + 14 i \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (30 i \, A - 21 i \, B + 14 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (30 i \, A - 21 i \, B + 14 i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 25 \, {\left (\sqrt {2} {\left (-30 i \, A + 21 i \, B - 14 i \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (-30 i \, A + 21 i \, B - 14 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-30 i \, A + 21 i \, B - 14 i \, C\right )}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 147 \, {\left (\sqrt {2} {\left (11 i \, A - 8 i \, B + 5 i \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (11 i \, A - 8 i \, B + 5 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (11 i \, A - 8 i \, B + 5 i \, 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 ) - 147 \, {\left (\sqrt {2} {\left (-11 i \, A + 8 i \, B - 5 i \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt {2} {\left (-11 i \, A + 8 i \, B - 5 i \, C\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-11 i \, A + 8 i \, B - 5 i \, 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 )}{210 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
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Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {7}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\int { \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{\frac {7}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\cos ^{\frac {7}{2}}(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^{7/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2} \,d x \]
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