Integrand size = 21, antiderivative size = 89 \[ \int \cos ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {1}{16} (5 A+6 C) x+\frac {(5 A+6 C) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {(5 A+6 C) \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {A \cos ^5(c+d x) \sin (c+d x)}{6 d} \]
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Time = 0.06 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4130, 2715, 8} \[ \int \cos ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {(5 A+6 C) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {(5 A+6 C) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {A \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {1}{16} x (5 A+6 C) \]
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Rule 8
Rule 2715
Rule 4130
Rubi steps \begin{align*} \text {integral}& = \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 {(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 {1}{8} (5 A+6 C) \int \cos ^2(c+d x) \, dx \\ & = \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 {1}{16} (5 A+6 C) \int 1 \, dx \\ & = \frac {1}{16} (5 A+6 C) x+\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} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int \cos ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {60 A c+72 c C+60 A d x+72 C d x+(45 A+48 C) \sin (2 (c+d x))+(9 A+6 C) \sin (4 (c+d x))+A \sin (6 (c+d x))}{192 d} \]
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Time = 0.26 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.69
method | result | size |
parallelrisch | \(\frac {\left (45 A +48 C \right ) \sin \left (2 d x +2 c \right )+\left (9 A +6 C \right ) \sin \left (4 d x +4 c \right )+A \sin \left (6 d x +6 c \right )+60 d x \left (A +\frac {6 C}{5}\right )}{192 d}\) | \(61\) |
risch | \(\frac {5 A x}{16}+\frac {3 C x}{8}+\frac {A \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 A \sin \left (4 d x +4 c \right )}{64 d}+\frac {\sin \left (4 d x +4 c \right ) C}{32 d}+\frac {15 A \sin \left (2 d x +2 c \right )}{64 d}+\frac {\sin \left (2 d x +2 c \right ) C}{4 d}\) | \(85\) |
derivativedivides | \(\frac {A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(86\) |
default | \(\frac {A \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+C \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(86\) |
norman | \(\frac {\left (-\frac {5 A}{16}-\frac {3 C}{8}\right ) x +\left (-\frac {45 A}{16}-\frac {27 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (-\frac {25 A}{16}-\frac {15 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (-\frac {25 A}{16}-\frac {15 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {5 A}{16}+\frac {3 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {25 A}{16}+\frac {15 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (\frac {25 A}{16}+\frac {15 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {45 A}{16}+\frac {27 C}{8}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}-\frac {\left (11 A +10 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {\left (11 A +10 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{8 d}+\frac {\left (15 A +2 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{2 d}+\frac {\left (19 A -6 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{12 d}-\frac {5 \left (19 A -6 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{24 d}-\frac {5 \left (19 A -6 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}+\frac {\left (19 A -6 C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{12 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}\) | \(341\) |
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Time = 0.25 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.76 \[ \int \cos ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (5 \, A + 6 \, C\right )} d x + {\left (8 \, A \cos \left (d x + c\right )^{5} + 2 \, {\left (5 \, A + 6 \, C\right )} \cos \left (d x + c\right )^{3} + 3 \, {\left (5 \, A + 6 \, C\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d} \]
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Timed out. \[ \int \cos ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
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Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.16 \[ \int \cos ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (d x + c\right )} {\left (5 \, A + 6 \, C\right )} + \frac {3 \, {\left (5 \, A + 6 \, C\right )} \tan \left (d x + c\right )^{5} + 8 \, {\left (5 \, A + 6 \, C\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (11 \, A + 10 \, C\right )} \tan \left (d x + c\right )}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1}}{48 \, d} \]
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Time = 0.28 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.08 \[ \int \cos ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {3 \, {\left (d x + c\right )} {\left (5 \, A + 6 \, C\right )} + \frac {15 \, A \tan \left (d x + c\right )^{5} + 18 \, C \tan \left (d x + c\right )^{5} + 40 \, A \tan \left (d x + c\right )^{3} + 48 \, C \tan \left (d x + c\right )^{3} + 33 \, A \tan \left (d x + c\right ) + 30 \, C \tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )}^{3}}}{48 \, d} \]
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Time = 16.14 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.02 \[ \int \cos ^6(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=x\,\left (\frac {5\,A}{16}+\frac {3\,C}{8}\right )+\frac {\left (\frac {5\,A}{16}+\frac {3\,C}{8}\right )\,{\mathrm {tan}\left (c+d\,x\right )}^5+\left (\frac {5\,A}{6}+C\right )\,{\mathrm {tan}\left (c+d\,x\right )}^3+\left (\frac {11\,A}{16}+\frac {5\,C}{8}\right )\,\mathrm {tan}\left (c+d\,x\right )}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^6+3\,{\mathrm {tan}\left (c+d\,x\right )}^4+3\,{\mathrm {tan}\left (c+d\,x\right )}^2+1\right )} \]
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