Integrand size = 43, antiderivative size = 270 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {8 a^4 (19 A+24 B+21 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {8 a^4 (12 A+17 B+28 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {4 a^4 (73 A+83 B+7 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 a (A-9 C) \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {2 (5 A+3 B-21 C) \sqrt {\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 d}+\frac {4 (86 A+81 B-126 C) \sqrt {\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{315 d} \]
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Time = 1.00 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.186, Rules used = {4197, 3122, 3055, 3047, 3102, 2827, 2720, 2719} \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {8 a^4 (12 A+17 B+28 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {8 a^4 (19 A+24 B+21 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {4 a^4 (73 A+83 B+7 C) \sin (c+d x) \sqrt {\cos (c+d x)}}{105 d}+\frac {4 (86 A+81 B-126 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^4 \cos (c+d x)+a^4\right )}{315 d}+\frac {2 (5 A+3 B-21 C) \sin (c+d x) \sqrt {\cos (c+d x)} \left (a^2 \cos (c+d x)+a^2\right )^2}{21 d}+\frac {2 a (A-9 C) \sin (c+d x) \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)^3}{9 d}+\frac {2 C \sin (c+d x) (a \cos (c+d x)+a)^4}{d \sqrt {\cos (c+d x)}} \]
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Rule 2719
Rule 2720
Rule 2827
Rule 3047
Rule 3055
Rule 3102
Rule 3122
Rule 4197
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+a \cos (c+d x))^4 \left (C+B \cos (c+d x)+A \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {2 \int \frac {(a+a \cos (c+d x))^4 \left (\frac {1}{2} a (B+8 C)+\frac {1}{2} a (A-9 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{a} \\ & = \frac {2 a (A-9 C) \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {4 \int \frac {(a+a \cos (c+d x))^3 \left (\frac {1}{4} a^2 (A+9 B+63 C)+\frac {3}{4} a^2 (5 A+3 B-21 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{9 a} \\ & = \frac {2 a (A-9 C) \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {2 (5 A+3 B-21 C) \sqrt {\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 d}+\frac {8 \int \frac {(a+a \cos (c+d x))^2 \left (\frac {1}{4} a^3 (11 A+36 B+189 C)+\frac {1}{4} a^3 (86 A+81 B-126 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{63 a} \\ & = \frac {2 a (A-9 C) \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {2 (5 A+3 B-21 C) \sqrt {\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 d}+\frac {4 (86 A+81 B-126 C) \sqrt {\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{315 d}+\frac {16 \int \frac {(a+a \cos (c+d x)) \left (\frac {3}{8} a^4 (47 A+87 B+273 C)+\frac {9}{8} a^4 (73 A+83 B+7 C) \cos (c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{315 a} \\ & = \frac {2 a (A-9 C) \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {2 (5 A+3 B-21 C) \sqrt {\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 d}+\frac {4 (86 A+81 B-126 C) \sqrt {\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{315 d}+\frac {16 \int \frac {\frac {3}{8} a^5 (47 A+87 B+273 C)+\left (\frac {9}{8} a^5 (73 A+83 B+7 C)+\frac {3}{8} a^5 (47 A+87 B+273 C)\right ) \cos (c+d x)+\frac {9}{8} a^5 (73 A+83 B+7 C) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{315 a} \\ & = \frac {4 a^4 (73 A+83 B+7 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 a (A-9 C) \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {2 (5 A+3 B-21 C) \sqrt {\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 d}+\frac {4 (86 A+81 B-126 C) \sqrt {\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{315 d}+\frac {32 \int \frac {\frac {45}{8} a^5 (12 A+17 B+28 C)+\frac {63}{8} a^5 (19 A+24 B+21 C) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx}{945 a} \\ & = \frac {4 a^4 (73 A+83 B+7 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 a (A-9 C) \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {2 (5 A+3 B-21 C) \sqrt {\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 d}+\frac {4 (86 A+81 B-126 C) \sqrt {\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{315 d}+\frac {1}{15} \left (4 a^4 (19 A+24 B+21 C)\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (4 a^4 (12 A+17 B+28 C)\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx \\ & = \frac {8 a^4 (19 A+24 B+21 C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d}+\frac {8 a^4 (12 A+17 B+28 C) \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{21 d}+\frac {4 a^4 (73 A+83 B+7 C) \sqrt {\cos (c+d x)} \sin (c+d x)}{105 d}+\frac {2 a (A-9 C) \sqrt {\cos (c+d x)} (a+a \cos (c+d x))^3 \sin (c+d x)}{9 d}+\frac {2 C (a+a \cos (c+d x))^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {2 (5 A+3 B-21 C) \sqrt {\cos (c+d x)} \left (a^2+a^2 \cos (c+d x)\right )^2 \sin (c+d x)}{21 d}+\frac {4 (86 A+81 B-126 C) \sqrt {\cos (c+d x)} \left (a^4+a^4 \cos (c+d x)\right ) \sin (c+d x)}{315 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 15.24 (sec) , antiderivative size = 1742, normalized size of antiderivative = 6.45 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\cos ^{\frac {13}{2}}(c+d x) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (-\frac {(76 A+96 B+69 C+76 A \cos (2 c)+96 B \cos (2 c)+99 C \cos (2 c)) \csc (c) \sec (c)}{120 d}+\frac {(204 A+191 B+112 C) \cos (d x) \sin (c)}{336 d}+\frac {(127 A+72 B+18 C) \cos (2 d x) \sin (2 c)}{720 d}+\frac {(4 A+B) \cos (3 d x) \sin (3 c)}{112 d}+\frac {A \cos (4 d x) \sin (4 c)}{288 d}+\frac {(204 A+191 B+112 C) \cos (c) \sin (d x)}{336 d}+\frac {C \sec (c) \sec (c+d x) \sin (d x)}{4 d}+\frac {(127 A+72 B+18 C) \cos (2 c) \sin (2 d x)}{720 d}+\frac {(4 A+B) \cos (3 c) \sin (3 d x)}{112 d}+\frac {A \cos (4 c) \sin (4 d x)}{288 d}\right )}{A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x)}-\frac {4 A \cos ^6(c+d x) \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 \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)}}-\frac {17 B \cos ^6(c+d x) \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 \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)}}-\frac {4 C \cos ^6(c+d x) \csc (c) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2(d x-\arctan (\cot (c)))\right ) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 \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)}}-\frac {19 A \cos ^6(c+d x) \csc (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 \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 )}{30 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))}-\frac {4 B \cos ^6(c+d x) \csc (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 \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))}-\frac {7 C \cos ^6(c+d x) \csc (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a+a \sec (c+d x))^4 \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 )}{10 d (A+2 C+2 B \cos (c+d x)+A \cos (2 c+2 d x))} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(785\) vs. \(2(300)=600\).
Time = 429.85 (sec) , antiderivative size = 786, normalized size of antiderivative = 2.91
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.14 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.06 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 \, {\left (30 i \, \sqrt {2} {\left (12 \, A + 17 \, B + 28 \, C\right )} a^{4} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) - 30 i \, \sqrt {2} {\left (12 \, A + 17 \, B + 28 \, C\right )} a^{4} \cos \left (d x + c\right ) {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 42 i \, \sqrt {2} {\left (19 \, A + 24 \, B + 21 \, C\right )} a^{4} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 42 i \, \sqrt {2} {\left (19 \, A + 24 \, B + 21 \, C\right )} a^{4} \cos \left (d x + c\right ) {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - {\left (35 \, A a^{4} \cos \left (d x + c\right )^{4} + 45 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{3} + 7 \, {\left (61 \, A + 36 \, B + 9 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \, {\left (48 \, A + 47 \, B + 28 \, C\right )} a^{4} \cos \left (d x + c\right ) + 315 \, C a^{4}\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{315 \, d \cos \left (d x + c\right )} \]
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Timed out. \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Timed out. \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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\[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{4} \cos \left (d x + c\right )^{\frac {9}{2}} \,d x } \]
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Time = 20.70 (sec) , antiderivative size = 548, normalized size of antiderivative = 2.03 \[ \int \cos ^{\frac {9}{2}}(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {2\,\left (3\,A\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+4\,A\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+4\,A\,a^4\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{3\,d}+\frac {2\,\left (4\,B\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+3\,B\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+2\,B\,a^4\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{d}-\frac {2\,\left (\frac {66\,A\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {17\,A\,a^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )}{\sqrt {{\sin \left (c+d\,x\right )}^2}}\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {15}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{77\,d}+\frac {4\,C\,a^4\,\left (\frac {2\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3}+\frac {2\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3}\right )}{d}+\frac {12\,C\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {8\,C\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}-\frac {8\,A\,a^4\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {208\,A\,a^4\,{\cos \left (c+d\,x\right )}^{11/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {11}{4};\ \frac {19}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{385\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {8\,B\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,B\,a^4\,{\cos \left (c+d\,x\right )}^{9/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {9}{4};\ \frac {13}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{9\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {2\,C\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,C\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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