Integrand size = 31, antiderivative size = 221 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {1}{8} \left (9 a^2 A b+4 A b^3+3 a^3 B+12 a b^2 B\right ) x+\frac {\left (4 a^3 A+14 a A b^2+15 a^2 b B+5 b^3 B\right ) \sin (c+d x)}{5 d}+\frac {\left (9 a^2 A b+4 A b^3+3 a^3 B+12 a b^2 B\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 (7 A b+5 a B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac {a A \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}-\frac {a \left (4 a^2 A+12 A b^2+15 a b B\right ) \sin ^3(c+d x)}{15 d} \]
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Time = 0.53 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4110, 4159, 4132, 2715, 8, 4129, 3092} \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=-\frac {a \left (4 a^2 A+15 a b B+12 A b^2\right ) \sin ^3(c+d x)}{15 d}+\frac {a^2 (5 a B+7 A b) \sin (c+d x) \cos ^3(c+d x)}{20 d}+\frac {\left (4 a^3 A+15 a^2 b B+14 a A b^2+5 b^3 B\right ) \sin (c+d x)}{5 d}+\frac {\left (3 a^3 B+9 a^2 A b+12 a b^2 B+4 A b^3\right ) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} x \left (3 a^3 B+9 a^2 A b+12 a b^2 B+4 A b^3\right )+\frac {a A \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{5 d} \]
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Rule 8
Rule 2715
Rule 3092
Rule 4110
Rule 4129
Rule 4132
Rule 4159
Rubi steps \begin{align*} \text {integral}& = \frac {a A \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}-\frac {1}{5} \int \cos ^4(c+d x) (a+b \sec (c+d x)) \left (-a (7 A b+5 a B)-\left (4 a^2 A+5 A b^2+10 a b B\right ) \sec (c+d x)-b (2 a A+5 b B) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a^2 (7 A b+5 a B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac {a A \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {1}{20} \int \cos ^3(c+d x) \left (4 a \left (4 a^2 A+12 A b^2+15 a b B\right )+5 \left (9 a^2 A b+4 A b^3+3 a^3 B+12 a b^2 B\right ) \sec (c+d x)+4 b^2 (2 a A+5 b B) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a^2 (7 A b+5 a B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac {a A \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {1}{20} \int \cos ^3(c+d x) \left (4 a \left (4 a^2 A+12 A b^2+15 a b B\right )+4 b^2 (2 a A+5 b B) \sec ^2(c+d x)\right ) \, dx+\frac {1}{4} \left (9 a^2 A b+4 A b^3+3 a^3 B+12 a b^2 B\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {\left (9 a^2 A b+4 A b^3+3 a^3 B+12 a b^2 B\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 (7 A b+5 a B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac {a A \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {1}{20} \int \cos (c+d x) \left (4 b^2 (2 a A+5 b B)+4 a \left (4 a^2 A+12 A b^2+15 a b B\right ) \cos ^2(c+d x)\right ) \, dx+\frac {1}{8} \left (9 a^2 A b+4 A b^3+3 a^3 B+12 a b^2 B\right ) \int 1 \, dx \\ & = \frac {1}{8} \left (9 a^2 A b+4 A b^3+3 a^3 B+12 a b^2 B\right ) x+\frac {\left (9 a^2 A b+4 A b^3+3 a^3 B+12 a b^2 B\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 (7 A b+5 a B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac {a A \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}-\frac {\text {Subst}\left (\int \left (4 b^2 (2 a A+5 b B)+4 a \left (4 a^2 A+12 A b^2+15 a b B\right )-4 a \left (4 a^2 A+12 A b^2+15 a b B\right ) x^2\right ) \, dx,x,-\sin (c+d x)\right )}{20 d} \\ & = \frac {1}{8} \left (9 a^2 A b+4 A b^3+3 a^3 B+12 a b^2 B\right ) x+\frac {\left (4 a^3 A+14 a A b^2+15 a^2 b B+5 b^3 B\right ) \sin (c+d x)}{5 d}+\frac {\left (9 a^2 A b+4 A b^3+3 a^3 B+12 a b^2 B\right ) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 (7 A b+5 a B) \cos ^3(c+d x) \sin (c+d x)}{20 d}+\frac {a A \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{5 d}-\frac {a \left (4 a^2 A+12 A b^2+15 a b B\right ) \sin ^3(c+d x)}{15 d} \\ \end{align*}
Time = 2.54 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.80 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {60 \left (9 a^2 A b+4 A b^3+3 a^3 B+12 a b^2 B\right ) (c+d x)+60 \left (5 a^3 A+18 a A b^2+18 a^2 b B+8 b^3 B\right ) \sin (c+d x)+120 \left (3 a^2 A b+A b^3+a^3 B+3 a b^2 B\right ) \sin (2 (c+d x))+10 a \left (5 a^2 A+12 A b^2+12 a b B\right ) \sin (3 (c+d x))+15 a^2 (3 A b+a B) \sin (4 (c+d x))+6 a^3 A \sin (5 (c+d x))}{480 d} \]
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Time = 3.70 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.81
method | result | size |
parallelrisch | \(\frac {120 \left (3 A \,a^{2} b +A \,b^{3}+B \,a^{3}+3 B a \,b^{2}\right ) \sin \left (2 d x +2 c \right )+10 \left (5 a^{3} A +12 A a \,b^{2}+12 B \,a^{2} b \right ) \sin \left (3 d x +3 c \right )+15 \left (3 A \,a^{2} b +B \,a^{3}\right ) \sin \left (4 d x +4 c \right )+6 a^{3} A \sin \left (5 d x +5 c \right )+60 \left (5 a^{3} A +18 A a \,b^{2}+18 B \,a^{2} b +8 B \,b^{3}\right ) \sin \left (d x +c \right )+540 \left (A \,a^{2} b +\frac {4}{9} A \,b^{3}+\frac {1}{3} B \,a^{3}+\frac {4}{3} B a \,b^{2}\right ) d x}{480 d}\) | \(179\) |
derivativedivides | \(\frac {\frac {a^{3} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 A \,a^{2} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+A a \,b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+B \,a^{2} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+A \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B a \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \sin \left (d x +c \right ) b^{3}}{d}\) | \(227\) |
default | \(\frac {\frac {a^{3} A \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 A \,a^{2} b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+A a \,b^{2} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+B \,a^{2} b \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+A \,b^{3} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B a \,b^{2} \left (\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+B \sin \left (d x +c \right ) b^{3}}{d}\) | \(227\) |
risch | \(\frac {9 A \,a^{2} b x}{8}+\frac {x A \,b^{3}}{2}+\frac {3 a^{3} x B}{8}+\frac {3 x B a \,b^{2}}{2}+\frac {5 a^{3} A \sin \left (d x +c \right )}{8 d}+\frac {9 \sin \left (d x +c \right ) A a \,b^{2}}{4 d}+\frac {9 \sin \left (d x +c \right ) B \,a^{2} b}{4 d}+\frac {\sin \left (d x +c \right ) B \,b^{3}}{d}+\frac {a^{3} A \sin \left (5 d x +5 c \right )}{80 d}+\frac {3 \sin \left (4 d x +4 c \right ) A \,a^{2} b}{32 d}+\frac {\sin \left (4 d x +4 c \right ) B \,a^{3}}{32 d}+\frac {5 a^{3} A \sin \left (3 d x +3 c \right )}{48 d}+\frac {\sin \left (3 d x +3 c \right ) A a \,b^{2}}{4 d}+\frac {\sin \left (3 d x +3 c \right ) B \,a^{2} b}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) A \,a^{2} b}{4 d}+\frac {\sin \left (2 d x +2 c \right ) A \,b^{3}}{4 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{3}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) B a \,b^{2}}{4 d}\) | \(278\) |
norman | \(\frac {\left (-\frac {9}{8} A \,a^{2} b -\frac {1}{2} A \,b^{3}-\frac {3}{8} B \,a^{3}-\frac {3}{2} B a \,b^{2}\right ) x +\left (-\frac {27}{4} A \,a^{2} b -3 A \,b^{3}-\frac {9}{4} B \,a^{3}-9 B a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (-\frac {9}{4} A \,a^{2} b -A \,b^{3}-\frac {3}{4} B \,a^{3}-3 B a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {9}{4} A \,a^{2} b -A \,b^{3}-\frac {3}{4} B \,a^{3}-3 B a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {9}{4} A \,a^{2} b +A \,b^{3}+\frac {3}{4} B \,a^{3}+3 B a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {9}{4} A \,a^{2} b +A \,b^{3}+\frac {3}{4} B \,a^{3}+3 B a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {9}{8} A \,a^{2} b +\frac {1}{2} A \,b^{3}+\frac {3}{8} B \,a^{3}+\frac {3}{2} B a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}+\left (\frac {27}{4} A \,a^{2} b +3 A \,b^{3}+\frac {9}{4} B \,a^{3}+9 B a \,b^{2}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\frac {\left (8 a^{3} A -15 A \,a^{2} b +24 A a \,b^{2}-4 A \,b^{3}-5 B \,a^{3}+24 B \,a^{2} b -12 B a \,b^{2}+8 B \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{4 d}-\frac {\left (8 a^{3} A +15 A \,a^{2} b +24 A a \,b^{2}+4 A \,b^{3}+5 B \,a^{3}+24 B \,a^{2} b +12 B a \,b^{2}+8 B \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {\left (40 a^{3} A -117 A \,a^{2} b +24 A a \,b^{2}-12 A \,b^{3}-39 B \,a^{3}+24 B \,a^{2} b -36 B a \,b^{2}-24 B \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{12 d}+\frac {\left (40 a^{3} A +117 A \,a^{2} b +24 A a \,b^{2}+12 A \,b^{3}+39 B \,a^{3}+24 B \,a^{2} b +36 B a \,b^{2}-24 B \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}+\frac {\left (344 a^{3} A -405 A \,a^{2} b -600 A a \,b^{2}+180 A \,b^{3}-135 B \,a^{3}-600 B \,a^{2} b +540 B a \,b^{2}-360 B \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{60 d}-\frac {\left (344 a^{3} A +405 A \,a^{2} b -600 A a \,b^{2}-180 A \,b^{3}+135 B \,a^{3}-600 B \,a^{2} b -540 B a \,b^{2}-360 B \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{60 d}-\frac {\left (872 a^{3} A -45 A \,a^{2} b +120 A a \,b^{2}+180 A \,b^{3}-15 B \,a^{3}+120 B \,a^{2} b +540 B a \,b^{2}+360 B \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{60 d}+\frac {\left (872 a^{3} A +45 A \,a^{2} b +120 A a \,b^{2}-180 A \,b^{3}+15 B \,a^{3}+120 B \,a^{2} b -540 B a \,b^{2}+360 B \,b^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{60 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5} \left (-1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{3}}\) | \(890\) |
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Time = 0.30 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.79 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (3 \, B a^{3} + 9 \, A a^{2} b + 12 \, B a b^{2} + 4 \, A b^{3}\right )} d x + {\left (24 \, A a^{3} \cos \left (d x + c\right )^{4} + 64 \, A a^{3} + 240 \, B a^{2} b + 240 \, A a b^{2} + 120 \, B b^{3} + 30 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} \cos \left (d x + c\right )^{3} + 8 \, {\left (4 \, A a^{3} + 15 \, B a^{2} b + 15 \, A a b^{2}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left (3 \, B a^{3} + 9 \, A a^{2} b + 12 \, B a b^{2} + 4 \, A b^{3}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{120 \, d} \]
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Timed out. \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\text {Timed out} \]
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Time = 0.25 (sec) , antiderivative size = 217, normalized size of antiderivative = 0.98 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{3} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 45 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} b - 480 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} b - 480 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A a b^{2} + 360 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b^{2} + 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{3} + 480 \, B b^{3} \sin \left (d x + c\right )}{480 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 672 vs. \(2 (209) = 418\).
Time = 0.35 (sec) , antiderivative size = 672, normalized size of antiderivative = 3.04 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {15 \, {\left (3 \, B a^{3} + 9 \, A a^{2} b + 12 \, B a b^{2} + 4 \, A b^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (120 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 75 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 225 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 360 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 360 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 180 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 60 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 120 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 160 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 30 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 90 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 960 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 960 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 360 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 120 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 480 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 464 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1200 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1200 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 720 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 160 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 90 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 960 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 960 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 360 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 480 \, B b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, A a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 75 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 225 \, A a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 360 \, B a^{2} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 360 \, A a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 180 \, B a b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60 \, A b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 120 \, B b^{3} \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 )}^{5}}}{120 \, d} \]
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Time = 14.93 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.25 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x))^3 (A+B \sec (c+d x)) \, dx=\frac {A\,b^3\,x}{2}+\frac {3\,B\,a^3\,x}{8}+\frac {9\,A\,a^2\,b\,x}{8}+\frac {3\,B\,a\,b^2\,x}{2}+\frac {5\,A\,a^3\,\sin \left (c+d\,x\right )}{8\,d}+\frac {B\,b^3\,\sin \left (c+d\,x\right )}{d}+\frac {5\,A\,a^3\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {A\,a^3\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {A\,b^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a^3\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a^3\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,A\,a^2\,b\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {A\,a\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{4\,d}+\frac {3\,A\,a^2\,b\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {3\,B\,a\,b^2\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {B\,a^2\,b\,\sin \left (3\,c+3\,d\,x\right )}{4\,d}+\frac {9\,A\,a\,b^2\,\sin \left (c+d\,x\right )}{4\,d}+\frac {9\,B\,a^2\,b\,\sin \left (c+d\,x\right )}{4\,d} \]
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