Integrand size = 40, antiderivative size = 201 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{16} a^3 (23 B+26 C) x+\frac {a^3 (17 B+19 C) \sin (c+d x)}{5 d}+\frac {a^3 (23 B+26 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (21 B+22 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {a B \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(4 B+3 C) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^3 (17 B+19 C) \sin ^3(c+d x)}{15 d} \]
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Time = 0.52 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {4157, 4102, 4081, 3872, 2713, 2715, 8} \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {a^3 (17 B+19 C) \sin ^3(c+d x)}{15 d}+\frac {a^3 (17 B+19 C) \sin (c+d x)}{5 d}+\frac {a^3 (21 B+22 C) \sin (c+d x) \cos ^3(c+d x)}{40 d}+\frac {a^3 (23 B+26 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {(4 B+3 C) \sin (c+d x) \cos ^4(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{15 d}+\frac {1}{16} a^3 x (23 B+26 C)+\frac {a B \sin (c+d x) \cos ^5(c+d x) (a \sec (c+d x)+a)^2}{6 d} \]
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Rule 8
Rule 2713
Rule 2715
Rule 3872
Rule 4081
Rule 4102
Rule 4157
Rubi steps \begin{align*} \text {integral}& = \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 (B+C \sec (c+d x)) \, dx \\ & = \frac {a B \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {1}{6} \int \cos ^5(c+d x) (a+a \sec (c+d x))^2 (2 a (4 B+3 C)+3 a (B+2 C) \sec (c+d x)) \, dx \\ & = \frac {a B \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(4 B+3 C) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{30} \int \cos ^4(c+d x) (a+a \sec (c+d x)) \left (3 a^2 (21 B+22 C)+3 a^2 (13 B+16 C) \sec (c+d x)\right ) \, dx \\ & = \frac {a^3 (21 B+22 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {a B \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(4 B+3 C) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {1}{120} \int \cos ^3(c+d x) \left (-24 a^3 (17 B+19 C)-15 a^3 (23 B+26 C) \sec (c+d x)\right ) \, dx \\ & = \frac {a^3 (21 B+22 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {a B \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(4 B+3 C) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{5} \left (a^3 (17 B+19 C)\right ) \int \cos ^3(c+d x) \, dx+\frac {1}{8} \left (a^3 (23 B+26 C)\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {a^3 (23 B+26 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (21 B+22 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {a B \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(4 B+3 C) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}+\frac {1}{16} \left (a^3 (23 B+26 C)\right ) \int 1 \, dx-\frac {\left (a^3 (17 B+19 C)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d} \\ & = \frac {1}{16} a^3 (23 B+26 C) x+\frac {a^3 (17 B+19 C) \sin (c+d x)}{5 d}+\frac {a^3 (23 B+26 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^3 (21 B+22 C) \cos ^3(c+d x) \sin (c+d x)}{40 d}+\frac {a B \cos ^5(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{6 d}+\frac {(4 B+3 C) \cos ^4(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{15 d}-\frac {a^3 (17 B+19 C) \sin ^3(c+d x)}{15 d} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.67 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 (1380 B c+1380 B d x+1560 C d x+120 (21 B+23 C) \sin (c+d x)+15 (63 B+64 C) \sin (2 (c+d x))+380 B \sin (3 (c+d x))+340 C \sin (3 (c+d x))+135 B \sin (4 (c+d x))+90 C \sin (4 (c+d x))+36 B \sin (5 (c+d x))+12 C \sin (5 (c+d x))+5 B \sin (6 (c+d x)))}{960 d} \]
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Time = 0.42 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.56
method | result | size |
parallelrisch | \(\frac {9 a^{3} \left (\left (7 B +\frac {64 C}{9}\right ) \sin \left (2 d x +2 c \right )+\frac {4 \left (19 B +17 C \right ) \sin \left (3 d x +3 c \right )}{27}+\left (B +\frac {2 C}{3}\right ) \sin \left (4 d x +4 c \right )+\frac {4 \left (B +\frac {C}{3}\right ) \sin \left (5 d x +5 c \right )}{15}+\frac {B \sin \left (6 d x +6 c \right )}{27}+\frac {8 \left (7 B +\frac {23 C}{3}\right ) \sin \left (d x +c \right )}{3}+\frac {92 x d \left (B +\frac {26 C}{23}\right )}{9}\right )}{64 d}\) | \(112\) |
risch | \(\frac {23 a^{3} B x}{16}+\frac {13 a^{3} x C}{8}+\frac {21 a^{3} B \sin \left (d x +c \right )}{8 d}+\frac {23 a^{3} C \sin \left (d x +c \right )}{8 d}+\frac {B \,a^{3} \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 B \,a^{3} \sin \left (5 d x +5 c \right )}{80 d}+\frac {\sin \left (5 d x +5 c \right ) a^{3} C}{80 d}+\frac {9 B \,a^{3} \sin \left (4 d x +4 c \right )}{64 d}+\frac {3 \sin \left (4 d x +4 c \right ) a^{3} C}{32 d}+\frac {19 B \,a^{3} \sin \left (3 d x +3 c \right )}{48 d}+\frac {17 \sin \left (3 d x +3 c \right ) a^{3} C}{48 d}+\frac {63 \sin \left (2 d x +2 c \right ) B \,a^{3}}{64 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} C}{d}\) | \(207\) |
derivativedivides | \(\frac {\frac {B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{3} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {3 B \,a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 a^{3} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {a^{3} C \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(266\) |
default | \(\frac {\frac {B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{3}+a^{3} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {3 B \,a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+3 a^{3} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {a^{3} C \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}}{d}\) | \(266\) |
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Time = 0.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.65 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (23 \, B + 26 \, C\right )} a^{3} d x + {\left (40 \, B a^{3} \cos \left (d x + c\right )^{5} + 48 \, {\left (3 \, B + C\right )} a^{3} \cos \left (d x + c\right )^{4} + 10 \, {\left (23 \, B + 18 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} + 16 \, {\left (17 \, B + 19 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \, {\left (23 \, B + 26 \, C\right )} a^{3} \cos \left (d x + c\right ) + 32 \, {\left (17 \, B + 19 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{240 \, d} \]
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Timed out. \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.25 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.30 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {192 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{3} - 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 320 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} + 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} C a^{3} - 960 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} + 90 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 240 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3}}{960 \, d} \]
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Time = 0.35 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.21 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (23 \, B a^{3} + 26 \, C a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (345 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 390 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1955 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 2210 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 4554 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5148 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 5814 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 5988 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 3165 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4190 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1575 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1530 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]
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Time = 19.40 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.42 \[ \int \cos ^7(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (\frac {23\,B\,a^3}{8}+\frac {13\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (\frac {391\,B\,a^3}{24}+\frac {221\,C\,a^3}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {759\,B\,a^3}{20}+\frac {429\,C\,a^3}{10}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {969\,B\,a^3}{20}+\frac {499\,C\,a^3}{10}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {211\,B\,a^3}{8}+\frac {419\,C\,a^3}{12}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {105\,B\,a^3}{8}+\frac {51\,C\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (23\,B+26\,C\right )}{8\,\left (\frac {23\,B\,a^3}{8}+\frac {13\,C\,a^3}{4}\right )}\right )\,\left (23\,B+26\,C\right )}{8\,d} \]
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