Integrand size = 31, antiderivative size = 309 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {1}{16} \left (5 a^4 A+36 a^2 A b^2+8 A b^4+24 a^3 b B+32 a b^3 B\right ) x+\frac {\left (48 a^3 A b+53 a A b^3+12 a^4 B+87 a^2 b^2 B+15 b^4 B\right ) \sin (c+d x)}{15 d}+\frac {\left (5 a^4 A+36 a^2 A b^2+8 A b^4+24 a^3 b B+32 a b^3 B\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 \left (25 a^2 A+48 A b^2+72 a b B\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {a (3 A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {a A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {a \left (16 a^2 A b+13 A b^3+4 a^3 B+27 a b^2 B\right ) \sin ^3(c+d x)}{15 d} \]
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Time = 0.88 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {4110, 4179, 4159, 4132, 2715, 8, 4129, 3092} \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {a^2 \left (25 a^2 A+72 a b B+48 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{120 d}-\frac {a \left (4 a^3 B+16 a^2 A b+27 a b^2 B+13 A b^3\right ) \sin ^3(c+d x)}{15 d}+\frac {\left (12 a^4 B+48 a^3 A b+87 a^2 b^2 B+53 a A b^3+15 b^4 B\right ) \sin (c+d x)}{15 d}+\frac {\left (5 a^4 A+24 a^3 b B+36 a^2 A b^2+32 a b^3 B+8 A b^4\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} x \left (5 a^4 A+24 a^3 b B+36 a^2 A b^2+32 a b^3 B+8 A b^4\right )+\frac {a (2 a B+3 A b) \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{10 d}+\frac {a A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d} \]
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Rule 8
Rule 2715
Rule 3092
Rule 4110
Rule 4129
Rule 4132
Rule 4159
Rule 4179
Rubi steps \begin{align*} \text {integral}& = \frac {a A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {1}{6} \int \cos ^5(c+d x) (a+b \sec (c+d x))^2 \left (-3 a (3 A b+2 a B)-\left (5 a^2 A+6 A b^2+12 a b B\right ) \sec (c+d x)-2 b (a A+3 b B) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a (3 A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {a A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {1}{30} \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (-a \left (25 a^2 A+48 A b^2+72 a b B\right )-\left (71 a^2 A b+30 A b^3+24 a^3 B+90 a b^2 B\right ) \sec (c+d x)-2 b \left (14 a A b+6 a^2 B+15 b^2 B\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a^2 \left (25 a^2 A+48 A b^2+72 a b B\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {a (3 A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {a A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {1}{120} \int \cos ^3(c+d x) \left (24 a \left (16 a^2 A b+13 A b^3+4 a^3 B+27 a b^2 B\right )+15 \left (5 a^4 A+36 a^2 A b^2+8 A b^4+24 a^3 b B+32 a b^3 B\right ) \sec (c+d x)+8 b^2 \left (14 a A b+6 a^2 B+15 b^2 B\right ) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a^2 \left (25 a^2 A+48 A b^2+72 a b B\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {a (3 A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {a A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {1}{120} \int \cos ^3(c+d x) \left (24 a \left (16 a^2 A b+13 A b^3+4 a^3 B+27 a b^2 B\right )+8 b^2 \left (14 a A b+6 a^2 B+15 b^2 B\right ) \sec ^2(c+d x)\right ) \, dx+\frac {1}{8} \left (5 a^4 A+36 a^2 A b^2+8 A b^4+24 a^3 b B+32 a b^3 B\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {\left (5 a^4 A+36 a^2 A b^2+8 A b^4+24 a^3 b B+32 a b^3 B\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 \left (25 a^2 A+48 A b^2+72 a b B\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {a (3 A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {a A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}+\frac {1}{120} \int \cos (c+d x) \left (8 b^2 \left (14 a A b+6 a^2 B+15 b^2 B\right )+24 a \left (16 a^2 A b+13 A b^3+4 a^3 B+27 a b^2 B\right ) \cos ^2(c+d x)\right ) \, dx+\frac {1}{16} \left (5 a^4 A+36 a^2 A b^2+8 A b^4+24 a^3 b B+32 a b^3 B\right ) \int 1 \, dx \\ & = \frac {1}{16} \left (5 a^4 A+36 a^2 A b^2+8 A b^4+24 a^3 b B+32 a b^3 B\right ) x+\frac {\left (5 a^4 A+36 a^2 A b^2+8 A b^4+24 a^3 b B+32 a b^3 B\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 \left (25 a^2 A+48 A b^2+72 a b B\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {a (3 A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {a A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {\text {Subst}\left (\int \left (8 b^2 \left (14 a A b+6 a^2 B+15 b^2 B\right )+24 a \left (16 a^2 A b+13 A b^3+4 a^3 B+27 a b^2 B\right )-24 a \left (16 a^2 A b+13 A b^3+4 a^3 B+27 a b^2 B\right ) x^2\right ) \, dx,x,-\sin (c+d x)\right )}{120 d} \\ & = \frac {1}{16} \left (5 a^4 A+36 a^2 A b^2+8 A b^4+24 a^3 b B+32 a b^3 B\right ) x+\frac {\left (48 a^3 A b+53 a A b^3+12 a^4 B+87 a^2 b^2 B+15 b^4 B\right ) \sin (c+d x)}{15 d}+\frac {\left (5 a^4 A+36 a^2 A b^2+8 A b^4+24 a^3 b B+32 a b^3 B\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 \left (25 a^2 A+48 A b^2+72 a b B\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {a (3 A b+2 a B) \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {a A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {a \left (16 a^2 A b+13 A b^3+4 a^3 B+27 a b^2 B\right ) \sin ^3(c+d x)}{15 d} \\ \end{align*}
Time = 3.32 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.08 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {300 a^4 A c+2160 a^2 A b^2 c+480 A b^4 c+1440 a^3 b B c+1920 a b^3 B c+300 a^4 A d x+2160 a^2 A b^2 d x+480 A b^4 d x+1440 a^3 b B d x+1920 a b^3 B d x+120 \left (20 a^3 A b+24 a A b^3+5 a^4 B+36 a^2 b^2 B+8 b^4 B\right ) \sin (c+d x)+15 \left (15 a^4 A+96 a^2 A b^2+16 A b^4+64 a^3 b B+64 a b^3 B\right ) \sin (2 (c+d x))+400 a^3 A b \sin (3 (c+d x))+320 a A b^3 \sin (3 (c+d x))+100 a^4 B \sin (3 (c+d x))+480 a^2 b^2 B \sin (3 (c+d x))+45 a^4 A \sin (4 (c+d x))+180 a^2 A b^2 \sin (4 (c+d x))+120 a^3 b B \sin (4 (c+d x))+48 a^3 A b \sin (5 (c+d x))+12 a^4 B \sin (5 (c+d x))+5 a^4 A \sin (6 (c+d x))}{960 d} \]
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Time = 4.76 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.80
method | result | size |
parallelrisch | \(\frac {\left (225 a^{4} A +1440 A \,a^{2} b^{2}+240 A \,b^{4}+960 B \,a^{3} b +960 B a \,b^{3}\right ) \sin \left (2 d x +2 c \right )+\left (400 A \,a^{3} b +320 A a \,b^{3}+100 B \,a^{4}+480 B \,a^{2} b^{2}\right ) \sin \left (3 d x +3 c \right )+\left (45 a^{4} A +180 A \,a^{2} b^{2}+120 B \,a^{3} b \right ) \sin \left (4 d x +4 c \right )+\left (48 A \,a^{3} b +12 B \,a^{4}\right ) \sin \left (5 d x +5 c \right )+5 a^{4} A \sin \left (6 d x +6 c \right )+\left (2400 A \,a^{3} b +2880 A a \,b^{3}+600 B \,a^{4}+4320 B \,a^{2} b^{2}+960 B \,b^{4}\right ) \sin \left (d x +c \right )+300 \left (a^{4} A +\frac {36}{5} A \,a^{2} b^{2}+\frac {8}{5} A \,b^{4}+\frac {24}{5} B \,a^{3} b +\frac {32}{5} B a \,b^{3}\right ) d x}{960 d}\) | \(247\) |
derivativedivides | \(\frac {a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 A \,a^{3} b \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {B \,a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+6 A \,a^{2} b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 B \,a^{3} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 A a \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+2 B \,a^{2} b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+A \,b^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 B a \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \sin \left (d x +c \right ) b^{4}}{d}\) | \(316\) |
default | \(\frac {a^{4} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {4 A \,a^{3} b \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+\frac {B \,a^{4} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+6 A \,a^{2} b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+4 B \,a^{3} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {4 A a \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+2 B \,a^{2} b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+A \,b^{4} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+4 B a \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \sin \left (d x +c \right ) b^{4}}{d}\) | \(316\) |
risch | \(\frac {a^{4} A \sin \left (6 d x +6 c \right )}{192 d}+\frac {15 \sin \left (2 d x +2 c \right ) a^{4} A}{64 d}+\frac {\sin \left (3 d x +3 c \right ) A a \,b^{3}}{3 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{2} b^{2}}{2 d}+\frac {\sin \left (2 d x +2 c \right ) B a \,b^{3}}{d}+\frac {\sin \left (5 d x +5 c \right ) A \,a^{3} b}{20 d}+\frac {3 \sin \left (4 d x +4 c \right ) A \,a^{2} b^{2}}{16 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{3} b}{8 d}+\frac {5 \sin \left (d x +c \right ) A \,a^{3} b}{2 d}+\frac {3 \sin \left (d x +c \right ) A a \,b^{3}}{d}+\frac {9 \sin \left (d x +c \right ) B \,a^{2} b^{2}}{2 d}+\frac {5 \sin \left (3 d x +3 c \right ) A \,a^{3} b}{12 d}+\frac {3 \sin \left (2 d x +2 c \right ) A \,a^{2} b^{2}}{2 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{3} b}{d}+\frac {x A \,b^{4}}{2}+\frac {3 a^{4} A \sin \left (4 d x +4 c \right )}{64 d}+\frac {\sin \left (5 d x +5 c \right ) B \,a^{4}}{80 d}+\frac {9 A \,a^{2} b^{2} x}{4}+\frac {3 B \,a^{3} b x}{2}+\frac {\sin \left (d x +c \right ) B \,b^{4}}{d}+2 x B a \,b^{3}+\frac {5 \sin \left (3 d x +3 c \right ) B \,a^{4}}{48 d}+\frac {\sin \left (2 d x +2 c \right ) A \,b^{4}}{4 d}+\frac {5 \sin \left (d x +c \right ) B \,a^{4}}{8 d}+\frac {5 a^{4} A x}{16}\) | \(404\) |
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Time = 0.29 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.79 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (5 \, A a^{4} + 24 \, B a^{3} b + 36 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} d x + {\left (40 \, A a^{4} \cos \left (d x + c\right )^{5} + 128 \, B a^{4} + 512 \, A a^{3} b + 960 \, B a^{2} b^{2} + 640 \, A a b^{3} + 240 \, B b^{4} + 48 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} \cos \left (d x + c\right )^{4} + 10 \, {\left (5 \, A a^{4} + 24 \, B a^{3} b + 36 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{3} + 32 \, {\left (2 \, B a^{4} + 8 \, A a^{3} b + 15 \, B a^{2} b^{2} + 10 \, A a b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (5 \, A a^{4} + 24 \, B a^{3} b + 36 \, A a^{2} b^{2} + 32 \, B a b^{3} + 8 \, A b^{4}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]
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Timed out. \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.23 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.99 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=-\frac {5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{4} - 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{4} - 256 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{3} b - 120 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} b - 180 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b^{2} + 1920 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} b^{2} + 1280 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a b^{3} - 960 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b^{3} - 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{4} - 960 \, B b^{4} \sin \left (d x + c\right )}{960 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1127 vs. \(2 (295) = 590\).
Time = 0.36 (sec) , antiderivative size = 1127, normalized size of antiderivative = 3.65 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\text {Too large to display} \]
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Time = 15.55 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.30 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^4 (A+B \sec (c+d x)) \, dx=\frac {5\,A\,a^4\,x}{16}+\frac {A\,b^4\,x}{2}+2\,B\,a\,b^3\,x+\frac {3\,B\,a^3\,b\,x}{2}+\frac {5\,B\,a^4\,\sin \left (c+d\,x\right )}{8\,d}+\frac {B\,b^4\,\sin \left (c+d\,x\right )}{d}+\frac {9\,A\,a^2\,b^2\,x}{4}+\frac {15\,A\,a^4\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {3\,A\,a^4\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {A\,a^4\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {A\,b^4\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {5\,B\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {B\,a^4\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {A\,a\,b^3\,\sin \left (3\,c+3\,d\,x\right )}{3\,d}+\frac {5\,A\,a^3\,b\,\sin \left (3\,c+3\,d\,x\right )}{12\,d}+\frac {A\,a^3\,b\,\sin \left (5\,c+5\,d\,x\right )}{20\,d}+\frac {B\,a\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {B\,a^3\,b\,\sin \left (2\,c+2\,d\,x\right )}{d}+\frac {B\,a^3\,b\,\sin \left (4\,c+4\,d\,x\right )}{8\,d}+\frac {9\,B\,a^2\,b^2\,\sin \left (c+d\,x\right )}{2\,d}+\frac {3\,A\,a^2\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{2\,d}+\frac {3\,A\,a^2\,b^2\,\sin \left (4\,c+4\,d\,x\right )}{16\,d}+\frac {B\,a^2\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{2\,d}+\frac {3\,A\,a\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {5\,A\,a^3\,b\,\sin \left (c+d\,x\right )}{2\,d} \]
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