Integrand size = 40, antiderivative size = 176 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{8} a^3 (13 B+15 C) x+\frac {a^3 (38 B+45 C) \sin (c+d x)}{15 d}+\frac {a^3 (13 B+15 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^3 (43 B+45 C) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac {a B \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {(7 B+5 C) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{20 d} \]
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Time = 0.49 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {4157, 4102, 4081, 3872, 2715, 8, 2717} \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 (38 B+45 C) \sin (c+d x)}{15 d}+\frac {a^3 (43 B+45 C) \sin (c+d x) \cos ^2(c+d x)}{60 d}+\frac {a^3 (13 B+15 C) \sin (c+d x) \cos (c+d x)}{8 d}+\frac {(7 B+5 C) \sin (c+d x) \cos ^3(c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{20 d}+\frac {1}{8} a^3 x (13 B+15 C)+\frac {a B \sin (c+d x) \cos ^4(c+d x) (a \sec (c+d x)+a)^2}{5 d} \]
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Rule 8
Rule 2715
Rule 2717
Rule 3872
Rule 4081
Rule 4102
Rule 4157
Rubi steps \begin{align*} \text {integral}& = \int \cos ^5(c+d x) (a+a \sec (c+d x))^3 (B+C \sec (c+d x)) \, dx \\ & = \frac {a B \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {1}{5} \int \cos ^4(c+d x) (a+a \sec (c+d x))^2 (a (7 B+5 C)+a (2 B+5 C) \sec (c+d x)) \, dx \\ & = \frac {a B \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {(7 B+5 C) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{20 d}+\frac {1}{20} \int \cos ^3(c+d x) (a+a \sec (c+d x)) \left (a^2 (43 B+45 C)+2 a^2 (11 B+15 C) \sec (c+d x)\right ) \, dx \\ & = \frac {a^3 (43 B+45 C) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac {a B \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {(7 B+5 C) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{20 d}-\frac {1}{60} \int \cos ^2(c+d x) \left (-15 a^3 (13 B+15 C)-4 a^3 (38 B+45 C) \sec (c+d x)\right ) \, dx \\ & = \frac {a^3 (43 B+45 C) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac {a B \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {(7 B+5 C) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{20 d}+\frac {1}{4} \left (a^3 (13 B+15 C)\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{15} \left (a^3 (38 B+45 C)\right ) \int \cos (c+d x) \, dx \\ & = \frac {a^3 (38 B+45 C) \sin (c+d x)}{15 d}+\frac {a^3 (13 B+15 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^3 (43 B+45 C) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac {a B \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {(7 B+5 C) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{20 d}+\frac {1}{8} \left (a^3 (13 B+15 C)\right ) \int 1 \, dx \\ & = \frac {1}{8} a^3 (13 B+15 C) x+\frac {a^3 (38 B+45 C) \sin (c+d x)}{15 d}+\frac {a^3 (13 B+15 C) \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^3 (43 B+45 C) \cos ^2(c+d x) \sin (c+d x)}{60 d}+\frac {a B \cos ^4(c+d x) (a+a \sec (c+d x))^2 \sin (c+d x)}{5 d}+\frac {(7 B+5 C) \cos ^3(c+d x) \left (a^3+a^3 \sec (c+d x)\right ) \sin (c+d x)}{20 d} \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.61 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {a^3 (780 B c+780 B d x+900 C d x+60 (23 B+26 C) \sin (c+d x)+480 (B+C) \sin (2 (c+d x))+170 B \sin (3 (c+d x))+120 C \sin (3 (c+d x))+45 B \sin (4 (c+d x))+15 C \sin (4 (c+d x))+6 B \sin (5 (c+d x)))}{480 d} \]
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Time = 0.33 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.53
method | result | size |
parallelrisch | \(\frac {3 \left (\frac {32 \left (B +C \right ) \sin \left (2 d x +2 c \right )}{3}+\frac {2 \left (\frac {17 B}{3}+4 C \right ) \sin \left (3 d x +3 c \right )}{3}+\left (B +\frac {C}{3}\right ) \sin \left (4 d x +4 c \right )+\frac {2 B \sin \left (5 d x +5 c \right )}{15}+\frac {4 \left (23 B +26 C \right ) \sin \left (d x +c \right )}{3}+\frac {52 \left (B +\frac {15 C}{13}\right ) x d}{3}\right ) a^{3}}{32 d}\) | \(93\) |
risch | \(\frac {13 a^{3} B x}{8}+\frac {15 a^{3} x C}{8}+\frac {23 a^{3} B \sin \left (d x +c \right )}{8 d}+\frac {13 a^{3} C \sin \left (d x +c \right )}{4 d}+\frac {B \,a^{3} \sin \left (5 d x +5 c \right )}{80 d}+\frac {3 B \,a^{3} \sin \left (4 d x +4 c \right )}{32 d}+\frac {\sin \left (4 d x +4 c \right ) a^{3} C}{32 d}+\frac {17 B \,a^{3} \sin \left (3 d x +3 c \right )}{48 d}+\frac {\sin \left (3 d x +3 c \right ) a^{3} C}{4 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{3}}{d}+\frac {\sin \left (2 d x +2 c \right ) a^{3} C}{d}\) | \(170\) |
derivativedivides | \(\frac {B \,a^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{3} C \sin \left (d x +c \right )+B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 a^{3} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {B \,a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{3} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(223\) |
default | \(\frac {B \,a^{3} \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+a^{3} C \sin \left (d x +c \right )+B \,a^{3} \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+3 a^{3} C \left (\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+3 B \,a^{3} \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )+a^{3} C \left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )+\frac {B \,a^{3} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+a^{3} C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(223\) |
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Time = 0.27 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.62 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (13 \, B + 15 \, C\right )} a^{3} d x + {\left (24 \, B a^{3} \cos \left (d x + c\right )^{4} + 30 \, {\left (3 \, B + C\right )} a^{3} \cos \left (d x + c\right )^{3} + 8 \, {\left (19 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 15 \, {\left (13 \, B + 15 \, C\right )} a^{3} \cos \left (d x + c\right ) + 8 \, {\left (38 \, B + 45 \, C\right )} a^{3}\right )} \sin \left (d x + c\right )}{120 \, d} \]
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Timed out. \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.25 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.21 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B a^{3} - 480 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{3} + 45 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} + 120 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{3} - 480 \, {\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a^{3} + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 360 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{3} + 480 \, C a^{3} \sin \left (d x + c\right )}{480 \, d} \]
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Time = 0.33 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.19 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (13 \, B a^{3} + 15 \, C a^{3}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (195 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 225 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 910 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1050 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 1664 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1920 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 1330 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 1830 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 765 \, B a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 735 \, C a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \]
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Time = 19.19 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.40 \[ \int \cos ^6(c+d x) (a+a \sec (c+d x))^3 \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (\frac {13\,B\,a^3}{4}+\frac {15\,C\,a^3}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {91\,B\,a^3}{6}+\frac {35\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (\frac {416\,B\,a^3}{15}+32\,C\,a^3\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (\frac {133\,B\,a^3}{6}+\frac {61\,C\,a^3}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (\frac {51\,B\,a^3}{4}+\frac {49\,C\,a^3}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (13\,B+15\,C\right )}{4\,\left (\frac {13\,B\,a^3}{4}+\frac {15\,C\,a^3}{4}\right )}\right )\,\left (13\,B+15\,C\right )}{4\,d} \]
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