Integrand size = 29, antiderivative size = 132 \[ \int \cos ^6(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{16} (5 A+6 C) x+\frac {B \sin (c+d x)}{d}+\frac {(5 A+6 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {(5 A+6 C) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {A \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 B \sin ^3(c+d x)}{3 d}+\frac {B \sin ^5(c+d x)}{5 d} \]
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Time = 0.12 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {4132, 2713, 4130, 2715, 8} \[ \int \cos ^6(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {(5 A+6 C) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {(5 A+6 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {A \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {1}{16} x (5 A+6 C)+\frac {B \sin ^5(c+d x)}{5 d}-\frac {2 B \sin ^3(c+d x)}{3 d}+\frac {B \sin (c+d x)}{d} \]
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Rule 8
Rule 2713
Rule 2715
Rule 4130
Rule 4132
Rubi steps \begin{align*} \text {integral}& = B \int \cos ^5(c+d x) \, dx+\int \cos ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx \\ & = \frac {A \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac {1}{6} (5 A+6 C) \int \cos ^4(c+d x) \, dx-\frac {B \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d} \\ & = \frac {B \sin (c+d x)}{d}+\frac {(5 A+6 C) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {A \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 B \sin ^3(c+d x)}{3 d}+\frac {B \sin ^5(c+d x)}{5 d}+\frac {1}{8} (5 A+6 C) \int \cos ^2(c+d x) \, dx \\ & = \frac {B \sin (c+d x)}{d}+\frac {(5 A+6 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {(5 A+6 C) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {A \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 B \sin ^3(c+d x)}{3 d}+\frac {B \sin ^5(c+d x)}{5 d}+\frac {1}{16} (5 A+6 C) \int 1 \, dx \\ & = \frac {1}{16} (5 A+6 C) x+\frac {B \sin (c+d x)}{d}+\frac {(5 A+6 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {(5 A+6 C) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {A \cos ^5(c+d x) \sin (c+d x)}{6 d}-\frac {2 B \sin ^3(c+d x)}{3 d}+\frac {B \sin ^5(c+d x)}{5 d} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.77 \[ \int \cos ^6(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {960 B \sin (c+d x)-640 B \sin ^3(c+d x)+192 B \sin ^5(c+d x)+5 (60 A c+72 c C+60 A d x+72 C d x+(45 A+48 C) \sin (2 (c+d x))+(9 A+6 C) \sin (4 (c+d x))+A \sin (6 (c+d x)))}{960 d} \]
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Time = 0.34 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.72
method | result | size |
parallelrisch | \(\frac {\left (225 A +240 C \right ) \sin \left (2 d x +2 c \right )+\left (45 A +30 C \right ) \sin \left (4 d x +4 c \right )+100 B \sin \left (3 d x +3 c \right )+12 B \sin \left (5 d x +5 c \right )+5 A \sin \left (6 d x +6 c \right )+600 B \sin \left (d x +c \right )+300 d x \left (A +\frac {6 C}{5}\right )}{960 d}\) | \(95\) |
derivativedivides | \(\frac {A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(115\) |
default | \(\frac {A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+\frac {B \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(115\) |
risch | \(\frac {5 A x}{16}+\frac {3 C x}{8}+\frac {5 B \sin \left (d x +c \right )}{8 d}+\frac {A \sin \left (6 d x +6 c \right )}{192 d}+\frac {B \sin \left (5 d x +5 c \right )}{80 d}+\frac {3 A \sin \left (4 d x +4 c \right )}{64 d}+\frac {\sin \left (4 d x +4 c \right ) C}{32 d}+\frac {5 B \sin \left (3 d x +3 c \right )}{48 d}+\frac {15 A \sin \left (2 d x +2 c \right )}{64 d}+\frac {\sin \left (2 d x +2 c \right ) C}{4 d}\) | \(127\) |
norman | \(\frac {\left (-\frac {5 A}{16}-\frac {3 C}{8}\right ) x +\left (-\frac {45 A}{16}-\frac {27 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {25 A}{16}-\frac {15 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {25 A}{16}-\frac {15 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {5 A}{16}+\frac {3 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {25 A}{16}+\frac {15 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (\frac {25 A}{16}+\frac {15 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {45 A}{16}+\frac {27 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\frac {\left (15 A +2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{2 d}-\frac {\left (11 A -16 B +10 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{8 d}-\frac {\left (11 A +16 B +10 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {\left (19 A -32 B -6 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}+\frac {\left (19 A +32 B -6 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{12 d}-\frac {\left (475 A -688 B -150 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{120 d}-\frac {\left (475 A +688 B -150 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{120 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\) | \(359\) |
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Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.70 \[ \int \cos ^6(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (5 \, A + 6 \, C\right )} d x + {\left (40 \, A \cos \left (d x + c\right )^{5} + 48 \, B \cos \left (d x + c\right )^{4} + 10 \, {\left (5 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{3} + 64 \, B \cos \left (d x + c\right )^{2} + 15 \, {\left (5 \, A + 6 \, C\right )} \cos \left (d x + c\right ) + 128 \, B\right )} \sin \left (d x + c\right )}{240 \, d} \]
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Timed out. \[ \int \cos ^6(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.22 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.87 \[ \int \cos ^6(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=-\frac {5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A - 64 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} B - 30 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C}{960 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 284 vs. \(2 (120) = 240\).
Time = 0.30 (sec) , antiderivative size = 284, normalized size of antiderivative = 2.15 \[ \int \cos ^6(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {15 \, {\left (d x + c\right )} {\left (5 \, A + 6 \, C\right )} - \frac {2 \, {\left (165 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 240 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 150 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 25 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 560 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 210 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 450 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 1248 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 60 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 450 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 1248 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 60 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 25 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 560 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 210 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 165 \, A \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 240 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 150 \, C \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{6}}}{240 \, d} \]
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Time = 15.01 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.95 \[ \int \cos ^6(c+d x) \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {5\,A\,x}{16}+\frac {3\,C\,x}{8}+\frac {15\,A\,\sin \left (2\,c+2\,d\,x\right )}{64\,d}+\frac {3\,A\,\sin \left (4\,c+4\,d\,x\right )}{64\,d}+\frac {A\,\sin \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {5\,B\,\sin \left (3\,c+3\,d\,x\right )}{48\,d}+\frac {B\,\sin \left (5\,c+5\,d\,x\right )}{80\,d}+\frac {C\,\sin \left (2\,c+2\,d\,x\right )}{4\,d}+\frac {C\,\sin \left (4\,c+4\,d\,x\right )}{32\,d}+\frac {5\,B\,\sin \left (c+d\,x\right )}{8\,d} \]
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