Integrand size = 33, antiderivative size = 257 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{16} a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) x+\frac {b \left (9 a^2 (4 A+5 C)+b^2 (11 A+15 C)\right ) \sin (c+d x)}{15 d}+\frac {a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (6 A b^2+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {A b \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {b \left (A b^2+3 a^2 (4 A+5 C)\right ) \sin ^3(c+d x)}{15 d} \]
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Time = 0.81 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {4180, 4179, 4159, 4132, 2715, 8, 4129, 3092} \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {b \left (3 a^2 (4 A+5 C)+A b^2\right ) \sin ^3(c+d x)}{15 d}+\frac {b \left (9 a^2 (4 A+5 C)+b^2 (11 A+15 C)\right ) \sin (c+d x)}{15 d}+\frac {a \left (5 a^2 (5 A+6 C)+6 A b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{120 d}+\frac {a \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right ) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a x \left (a^2 (5 A+6 C)+6 b^2 (3 A+4 C)\right )+\frac {A \sin (c+d x) \cos ^5(c+d x) (a+b \sec (c+d x))^3}{6 d}+\frac {A b \sin (c+d x) \cos ^4(c+d x) (a+b \sec (c+d x))^2}{10 d} \]
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Rule 8
Rule 2715
Rule 3092
Rule 4129
Rule 4132
Rule 4159
Rule 4179
Rule 4180
Rubi steps \begin{align*} \text {integral}& = \frac {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 b+a (5 A+6 C) \sec (c+d x)+2 b (A+3 C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {A b \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {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 (6 A b^2+5 a^2 (5 A+6 C)+a b (47 A+60 C) \sec (c+d x)+2 b^2 (8 A+15 C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a \left (6 A b^2+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {A b \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {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 b \left (A b^2+3 a^2 (4 A+5 C)\right )-15 a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \sec (c+d x)-8 b^3 (8 A+15 C) \sec ^2(c+d x)\right ) \, dx \\ & = \frac {a \left (6 A b^2+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {A b \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {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 b \left (A b^2+3 a^2 (4 A+5 C)\right )-8 b^3 (8 A+15 C) \sec ^2(c+d x)\right ) \, dx+\frac {1}{8} \left (a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right )\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (6 A b^2+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {A b \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {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^3 (8 A+15 C)-24 b \left (A b^2+3 a^2 (4 A+5 C)\right ) \cos ^2(c+d x)\right ) \, dx+\frac {1}{16} \left (a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right )\right ) \int 1 \, dx \\ & = \frac {1}{16} a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) x+\frac {a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (6 A b^2+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {A b \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {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^3 (8 A+15 C)-24 b \left (A b^2+3 a^2 (4 A+5 C)\right )+24 b \left (A b^2+3 a^2 (4 A+5 C)\right ) x^2\right ) \, dx,x,-\sin (c+d x)\right )}{120 d} \\ & = \frac {1}{16} a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) x+\frac {b \left (9 a^2 (4 A+5 C)+b^2 (11 A+15 C)\right ) \sin (c+d x)}{15 d}+\frac {a \left (6 b^2 (3 A+4 C)+a^2 (5 A+6 C)\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (6 A b^2+5 a^2 (5 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)}{120 d}+\frac {A b \cos ^4(c+d x) (a+b \sec (c+d x))^2 \sin (c+d x)}{10 d}+\frac {A \cos ^5(c+d x) (a+b \sec (c+d x))^3 \sin (c+d x)}{6 d}-\frac {b \left (A b^2+3 a^2 (4 A+5 C)\right ) \sin ^3(c+d x)}{15 d} \\ \end{align*}
Time = 1.79 (sec) , antiderivative size = 253, normalized size of antiderivative = 0.98 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {300 a^3 A c+1080 a A b^2 c+360 a^3 c C+1440 a b^2 c C+300 a^3 A d x+1080 a A b^2 d x+360 a^3 C d x+1440 a b^2 C d x+120 b \left (2 b^2 (3 A+4 C)+3 a^2 (5 A+6 C)\right ) \sin (c+d x)+15 a \left (48 b^2 (A+C)+a^2 (15 A+16 C)\right ) \sin (2 (c+d x))+300 a^2 A b \sin (3 (c+d x))+80 A b^3 \sin (3 (c+d x))+240 a^2 b C \sin (3 (c+d x))+45 a^3 A \sin (4 (c+d x))+90 a A b^2 \sin (4 (c+d x))+30 a^3 C \sin (4 (c+d x))+36 a^2 A b \sin (5 (c+d x))+5 a^3 A \sin (6 (c+d x))}{960 d} \]
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Time = 0.87 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.69
method | result | size |
parallelrisch | \(\frac {225 a \left (\left (A +\frac {16 C}{15}\right ) a^{2}+\frac {16 b^{2} \left (A +C \right )}{5}\right ) \sin \left (2 d x +2 c \right )+300 \left (\left (A +\frac {4 C}{5}\right ) a^{2}+\frac {4 A \,b^{2}}{15}\right ) b \sin \left (3 d x +3 c \right )+45 \left (a^{2} \left (A +\frac {2 C}{3}\right )+2 A \,b^{2}\right ) a \sin \left (4 d x +4 c \right )+36 A \,a^{2} b \sin \left (5 d x +5 c \right )+5 a^{3} A \sin \left (6 d x +6 c \right )+1800 b \left (a^{2} \left (A +\frac {6 C}{5}\right )+\frac {2 b^{2} \left (A +\frac {4 C}{3}\right )}{5}\right ) \sin \left (d x +c \right )+300 a x d \left (a^{2} \left (A +\frac {6 C}{5}\right )+\frac {18 b^{2} \left (A +\frac {4 C}{3}\right )}{5}\right )}{960 d}\) | \(178\) |
derivativedivides | \(\frac {a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {3 A \,a^{2} b \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 a A \,b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{2} b C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 C a \,b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \sin \left (d x +c \right ) b^{3}}{d}\) | \(249\) |
default | \(\frac {a^{3} A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {3 A \,a^{2} b \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 a A \,b^{2} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+\frac {A \,b^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{2} b C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 C a \,b^{2} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+C \sin \left (d x +c \right ) b^{3}}{d}\) | \(249\) |
risch | \(\frac {5 a^{3} A x}{16}+\frac {9 A a \,b^{2} x}{8}+\frac {3 a^{3} x C}{8}+\frac {3 C a \,b^{2} x}{2}+\frac {15 \sin \left (d x +c \right ) A \,a^{2} b}{8 d}+\frac {3 \sin \left (d x +c \right ) A \,b^{3}}{4 d}+\frac {9 \sin \left (d x +c \right ) a^{2} b C}{4 d}+\frac {\sin \left (d x +c \right ) C \,b^{3}}{d}+\frac {a^{3} A \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 A \,a^{2} b \sin \left (5 d x +5 c \right )}{80 d}+\frac {3 a^{3} A \sin \left (4 d x +4 c \right )}{64 d}+\frac {3 \sin \left (4 d x +4 c \right ) a A \,b^{2}}{32 d}+\frac {\sin \left (4 d x +4 c \right ) a^{3} C}{32 d}+\frac {5 A \,a^{2} b \sin \left (3 d x +3 c \right )}{16 d}+\frac {\sin \left (3 d x +3 c \right ) A \,b^{3}}{12 d}+\frac {\sin \left (3 d x +3 c \right ) a^{2} b C}{4 d}+\frac {15 \sin \left (2 d x +2 c \right ) a^{3} A}{64 d}+\frac {3 \sin \left (2 d x +2 c \right ) a A \,b^{2}}{4 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} C}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) C a \,b^{2}}{4 d}\) | \(315\) |
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Time = 0.29 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.74 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 6 \, {\left (3 \, A + 4 \, C\right )} a b^{2}\right )} d x + {\left (40 \, A a^{3} \cos \left (d x + c\right )^{5} + 144 \, A a^{2} b \cos \left (d x + c\right )^{4} + 96 \, {\left (4 \, A + 5 \, C\right )} a^{2} b + 80 \, {\left (2 \, A + 3 \, C\right )} b^{3} + 10 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 18 \, A a b^{2}\right )} \cos \left (d x + c\right )^{3} + 16 \, {\left (3 \, {\left (4 \, A + 5 \, C\right )} a^{2} b + 5 \, A b^{3}\right )} \cos \left (d x + c\right )^{2} + 15 \, {\left ({\left (5 \, A + 6 \, C\right )} a^{3} + 6 \, {\left (3 \, A + 4 \, C\right )} a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]
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Timed out. \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.25 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.95 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=-\frac {5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} - 192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} A a^{2} b + 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{2} b - 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a b^{2} - 720 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a b^{2} + 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} A b^{3} - 960 \, C b^{3} \sin \left (d x + c\right )}{960 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 882 vs. \(2 (243) = 486\).
Time = 0.35 (sec) , antiderivative size = 882, normalized size of antiderivative = 3.43 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Too large to display} \]
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Time = 18.98 (sec) , antiderivative size = 573, normalized size of antiderivative = 2.23 \[ \int \cos ^6(c+d x) (a+b \sec (c+d x))^3 \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (2\,A\,b^3-\frac {11\,A\,a^3}{8}-\frac {5\,C\,a^3}{4}+2\,C\,b^3-\frac {15\,A\,a\,b^2}{4}+6\,A\,a^2\,b-3\,C\,a\,b^2+6\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {5\,A\,a^3}{24}+\frac {22\,A\,b^3}{3}-\frac {7\,C\,a^3}{4}+10\,C\,b^3-\frac {21\,A\,a\,b^2}{4}+14\,A\,a^2\,b-9\,C\,a\,b^2+22\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (12\,A\,b^3-\frac {15\,A\,a^3}{4}-\frac {C\,a^3}{2}+20\,C\,b^3-\frac {3\,A\,a\,b^2}{2}+\frac {156\,A\,a^2\,b}{5}-6\,C\,a\,b^2+36\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {15\,A\,a^3}{4}+12\,A\,b^3+\frac {C\,a^3}{2}+20\,C\,b^3+\frac {3\,A\,a\,b^2}{2}+\frac {156\,A\,a^2\,b}{5}+6\,C\,a\,b^2+36\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {22\,A\,b^3}{3}-\frac {5\,A\,a^3}{24}+\frac {7\,C\,a^3}{4}+10\,C\,b^3+\frac {21\,A\,a\,b^2}{4}+14\,A\,a^2\,b+9\,C\,a\,b^2+22\,C\,a^2\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {11\,A\,a^3}{8}+2\,A\,b^3+\frac {5\,C\,a^3}{4}+2\,C\,b^3+\frac {15\,A\,a\,b^2}{4}+6\,A\,a^2\,b+3\,C\,a\,b^2+6\,C\,a^2\,b\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 {a\,\mathrm {atan}\left (\frac {a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (5\,A\,a^2+18\,A\,b^2+6\,C\,a^2+24\,C\,b^2\right )}{8\,\left (\frac {5\,A\,a^3}{8}+\frac {3\,C\,a^3}{4}+\frac {9\,A\,a\,b^2}{4}+3\,C\,a\,b^2\right )}\right )\,\left (5\,A\,a^2+18\,A\,b^2+6\,C\,a^2+24\,C\,b^2\right )}{8\,d} \]
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