Integrand size = 41, antiderivative size = 213 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {7}{16} a^4 (7 A+8 B+10 C) x+\frac {4 a^4 (7 A+8 B+10 C) \sin (c+d x)}{5 d}+\frac {27 a^4 (7 A+8 B+10 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {a^4 (7 A+8 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {(2 A+3 B) \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}-\frac {2 a^4 (7 A+8 B+10 C) \sin ^3(c+d x)}{15 d} \]
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Time = 0.47 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4171, 4098, 3876, 2717, 2715, 8, 2713} \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {2 a^4 (7 A+8 B+10 C) \sin ^3(c+d x)}{15 d}+\frac {4 a^4 (7 A+8 B+10 C) \sin (c+d x)}{5 d}+\frac {a^4 (7 A+8 B+10 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac {27 a^4 (7 A+8 B+10 C) \sin (c+d x) \cos (c+d x)}{80 d}+\frac {7}{16} a^4 x (7 A+8 B+10 C)+\frac {(2 A+3 B) \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^4}{15 d}+\frac {A \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^4}{6 d} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2717
Rule 3876
Rule 4098
Rule 4171
Rubi steps \begin{align*} \text {integral}& = \frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac {\int \cos ^5(c+d x) (a+a \sec (c+d x))^4 (2 a (2 A+3 B)+a (A+6 C) \sec (c+d x)) \, dx}{6 a} \\ & = \frac {(2 A+3 B) \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac {1}{10} (7 A+8 B+10 C) \int \cos ^4(c+d x) (a+a \sec (c+d x))^4 \, dx \\ & = \frac {(2 A+3 B) \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac {1}{10} (7 A+8 B+10 C) \int \left (a^4+4 a^4 \cos (c+d x)+6 a^4 \cos ^2(c+d x)+4 a^4 \cos ^3(c+d x)+a^4 \cos ^4(c+d x)\right ) \, dx \\ & = \frac {1}{10} a^4 (7 A+8 B+10 C) x+\frac {(2 A+3 B) \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac {1}{10} \left (a^4 (7 A+8 B+10 C)\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{5} \left (2 a^4 (7 A+8 B+10 C)\right ) \int \cos (c+d x) \, dx+\frac {1}{5} \left (2 a^4 (7 A+8 B+10 C)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{5} \left (3 a^4 (7 A+8 B+10 C)\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {1}{10} a^4 (7 A+8 B+10 C) x+\frac {2 a^4 (7 A+8 B+10 C) \sin (c+d x)}{5 d}+\frac {3 a^4 (7 A+8 B+10 C) \cos (c+d x) \sin (c+d x)}{10 d}+\frac {a^4 (7 A+8 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {(2 A+3 B) \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}+\frac {1}{40} \left (3 a^4 (7 A+8 B+10 C)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{10} \left (3 a^4 (7 A+8 B+10 C)\right ) \int 1 \, dx-\frac {\left (2 a^4 (7 A+8 B+10 C)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d} \\ & = \frac {2}{5} a^4 (7 A+8 B+10 C) x+\frac {4 a^4 (7 A+8 B+10 C) \sin (c+d x)}{5 d}+\frac {27 a^4 (7 A+8 B+10 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {a^4 (7 A+8 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {(2 A+3 B) \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}-\frac {2 a^4 (7 A+8 B+10 C) \sin ^3(c+d x)}{15 d}+\frac {1}{80} \left (3 a^4 (7 A+8 B+10 C)\right ) \int 1 \, dx \\ & = \frac {7}{16} a^4 (7 A+8 B+10 C) x+\frac {4 a^4 (7 A+8 B+10 C) \sin (c+d x)}{5 d}+\frac {27 a^4 (7 A+8 B+10 C) \cos (c+d x) \sin (c+d x)}{80 d}+\frac {a^4 (7 A+8 B+10 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {(2 A+3 B) \cos ^4(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{15 d}+\frac {A \cos ^5(c+d x) (a+a \sec (c+d x))^4 \sin (c+d x)}{6 d}-\frac {2 a^4 (7 A+8 B+10 C) \sin ^3(c+d x)}{15 d} \\ \end{align*}
Time = 0.93 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.69 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^4 (1+\cos (c+d x))^4 \sin (c+d x) \left (-A+6 B+5 A (1+\cos (c+d x))+\frac {(7 A+8 B+10 C) \left (210 \arcsin \left (\sqrt {\sin ^2\left (\frac {1}{2} (c+d x)\right )}\right )+\left (160+81 \cos (c+d x)+32 \cos ^2(c+d x)+6 \cos ^3(c+d x)\right ) \sqrt {\sin ^2(c+d x)}\right )}{8 \sqrt {1-\cos (c+d x)} (1+\cos (c+d x))^{9/2}}\right )}{30 d} \]
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Time = 0.50 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.60
method | result | size |
parallelrisch | \(\frac {3 \left (\frac {\left (\frac {127 A}{16}+8 B +7 C \right ) \sin \left (2 d x +2 c \right )}{3}+\left (A +\frac {29 B}{36}+\frac {4 C}{9}\right ) \sin \left (3 d x +3 c \right )+\frac {\left (\frac {5 A}{8}+\frac {B}{3}+\frac {C}{12}\right ) \sin \left (4 d x +4 c \right )}{2}+\frac {\left (A +\frac {B}{4}\right ) \sin \left (5 d x +5 c \right )}{15}+\frac {A \sin \left (6 d x +6 c \right )}{144}+\frac {\left (22 A +\frac {49 B}{2}+28 C \right ) \sin \left (d x +c \right )}{3}+\frac {49 x d \left (A +\frac {8 B}{7}+\frac {10 C}{7}\right )}{12}\right ) a^{4}}{4 d}\) | \(128\) |
risch | \(\frac {49 a^{4} A x}{16}+\frac {7 a^{4} x B}{2}+\frac {35 a^{4} x C}{8}+\frac {11 \sin \left (d x +c \right ) a^{4} A}{2 d}+\frac {49 \sin \left (d x +c \right ) B \,a^{4}}{8 d}+\frac {7 \sin \left (d x +c \right ) a^{4} C}{d}+\frac {a^{4} A \sin \left (6 d x +6 c \right )}{192 d}+\frac {a^{4} A \sin \left (5 d x +5 c \right )}{20 d}+\frac {\sin \left (5 d x +5 c \right ) B \,a^{4}}{80 d}+\frac {15 a^{4} A \sin \left (4 d x +4 c \right )}{64 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{4}}{8 d}+\frac {\sin \left (4 d x +4 c \right ) a^{4} C}{32 d}+\frac {3 a^{4} A \sin \left (3 d x +3 c \right )}{4 d}+\frac {29 \sin \left (3 d x +3 c \right ) B \,a^{4}}{48 d}+\frac {\sin \left (3 d x +3 c \right ) a^{4} C}{3 d}+\frac {127 \sin \left (2 d x +2 c \right ) a^{4} A}{64 d}+\frac {2 \sin \left (2 d x +2 c \right ) B \,a^{4}}{d}+\frac {7 \sin \left (2 d x +2 c \right ) a^{4} C}{4 d}\) | \(284\) |
derivativedivides | \(\frac {a^{4} A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{4} \sin \left (d x +c \right )+a^{4} C \left (d x +c \right )+\frac {4 a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 B \,a^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} C \sin \left (d x +c \right )+6 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 B \,a^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+6 a^{4} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{4} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \,a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{4} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(416\) |
default | \(\frac {a^{4} A \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \,a^{4} \sin \left (d x +c \right )+a^{4} C \left (d x +c \right )+\frac {4 a^{4} A \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+4 B \,a^{4} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 a^{4} C \sin \left (d x +c \right )+6 a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+2 B \,a^{4} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+6 a^{4} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+\frac {4 a^{4} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+4 B \,a^{4} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 a^{4} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \,a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{4} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(416\) |
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Time = 0.26 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.68 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (7 \, A + 8 \, B + 10 \, C\right )} a^{4} d x + {\left (40 \, A a^{4} \cos \left (d x + c\right )^{5} + 48 \, {\left (4 \, A + B\right )} a^{4} \cos \left (d x + c\right )^{4} + 10 \, {\left (41 \, A + 24 \, B + 6 \, C\right )} a^{4} \cos \left (d x + c\right )^{3} + 32 \, {\left (18 \, A + 17 \, B + 10 \, C\right )} a^{4} \cos \left (d x + c\right )^{2} + 15 \, {\left (49 \, A + 56 \, B + 54 \, C\right )} a^{4} \cos \left (d x + c\right ) + 16 \, {\left (72 \, A + 83 \, B + 100 \, C\right )} a^{4}\right )} \sin \left (d x + c\right )}{240 \, d} \]
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Timed out. \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 400 vs. \(2 (199) = 398\).
Time = 0.22 (sec) , antiderivative size = 400, normalized size of antiderivative = 1.88 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {256 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{4} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a^{4} + 180 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} + 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} - 1920 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{4} + 120 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} + 960 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{4} - 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{4} + 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 1440 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{4} + 960 \, {\left (d x + c\right )} C a^{4} + 960 \, B a^{4} \sin \left (d x + c\right ) + 3840 \, C a^{4} \sin \left (d x + c\right )}{960 \, d} \]
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Time = 0.37 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.64 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {105 \, {\left (7 \, A a^{4} + 8 \, B a^{4} + 10 \, C a^{4}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (735 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 840 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1050 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 4165 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 4760 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 5950 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 9702 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 11088 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 13860 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 11802 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 13488 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 16860 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 7355 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 9320 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 10690 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3105 \, A a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3000 \, B a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2790 \, C a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]
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Time = 19.17 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.57 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (\frac {49\,A\,a^4}{8}+7\,B\,a^4+\frac {35\,C\,a^4}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {833\,A\,a^4}{24}+\frac {119\,B\,a^4}{3}+\frac {595\,C\,a^4}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {1617\,A\,a^4}{20}+\frac {462\,B\,a^4}{5}+\frac {231\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {1967\,A\,a^4}{20}+\frac {562\,B\,a^4}{5}+\frac {281\,C\,a^4}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {1471\,A\,a^4}{24}+\frac {233\,B\,a^4}{3}+\frac {1069\,C\,a^4}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {207\,A\,a^4}{8}+25\,B\,a^4+\frac {93\,C\,a^4}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {7\,a^4\,\mathrm {atan}\left (\frac {7\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (7\,A+8\,B+10\,C\right )}{8\,\left (\frac {49\,A\,a^4}{8}+7\,B\,a^4+\frac {35\,C\,a^4}{4}\right )}\right )\,\left (7\,A+8\,B+10\,C\right )}{8\,d} \]
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