Integrand size = 19, antiderivative size = 92 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \, dx=\frac {3 b x}{8}+\frac {a \sin (c+d x)}{d}+\frac {3 b \cos (c+d x) \sin (c+d x)}{8 d}+\frac {b \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {2 a \sin ^3(c+d x)}{3 d}+\frac {a \sin ^5(c+d x)}{5 d} \]
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Time = 0.07 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3872, 2713, 2715, 8} \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \, dx=\frac {a \sin ^5(c+d x)}{5 d}-\frac {2 a \sin ^3(c+d x)}{3 d}+\frac {a \sin (c+d x)}{d}+\frac {b \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3 b \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3 b x}{8} \]
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Rule 8
Rule 2713
Rule 2715
Rule 3872
Rubi steps \begin{align*} \text {integral}& = a \int \cos ^5(c+d x) \, dx+b \int \cos ^4(c+d x) \, dx \\ & = \frac {b \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{4} (3 b) \int \cos ^2(c+d x) \, dx-\frac {a \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d} \\ & = \frac {a \sin (c+d x)}{d}+\frac {3 b \cos (c+d x) \sin (c+d x)}{8 d}+\frac {b \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {2 a \sin ^3(c+d x)}{3 d}+\frac {a \sin ^5(c+d x)}{5 d}+\frac {1}{8} (3 b) \int 1 \, dx \\ & = \frac {3 b x}{8}+\frac {a \sin (c+d x)}{d}+\frac {3 b \cos (c+d x) \sin (c+d x)}{8 d}+\frac {b \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {2 a \sin ^3(c+d x)}{3 d}+\frac {a \sin ^5(c+d x)}{5 d} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.97 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \, dx=\frac {3 b (c+d x)}{8 d}+\frac {a \sin (c+d x)}{d}-\frac {2 a \sin ^3(c+d x)}{3 d}+\frac {a \sin ^5(c+d x)}{5 d}+\frac {b \sin (2 (c+d x))}{4 d}+\frac {b \sin (4 (c+d x))}{32 d} \]
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Time = 0.83 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.75
method | result | size |
parallelrisch | \(\frac {180 b x d +6 a \sin \left (5 d x +5 c \right )+15 b \sin \left (4 d x +4 c \right )+50 a \sin \left (3 d x +3 c \right )+120 b \sin \left (2 d x +2 c \right )+300 a \sin \left (d x +c \right )}{480 d}\) | \(69\) |
derivativedivides | \(\frac {\frac {a \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(70\) |
default | \(\frac {\frac {a \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}+b \left (\frac {\left (\cos \left (d x +c \right )^{3}+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(70\) |
risch | \(\frac {3 b x}{8}+\frac {5 a \sin \left (d x +c \right )}{8 d}+\frac {a \sin \left (5 d x +5 c \right )}{80 d}+\frac {b \sin \left (4 d x +4 c \right )}{32 d}+\frac {5 a \sin \left (3 d x +3 c \right )}{48 d}+\frac {b \sin \left (2 d x +2 c \right )}{4 d}\) | \(78\) |
norman | \(\frac {\frac {3 b x}{8}+\frac {116 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d}+\frac {15 b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8}+\frac {15 b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{4}+\frac {15 b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4}+\frac {15 b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{8}+\frac {3 b x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8}+\frac {\left (8 a -5 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{4 d}+\frac {\left (8 a +5 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {\left (16 a -3 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{6 d}+\frac {\left (16 a +3 b \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}\) | \(204\) |
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Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \, dx=\frac {45 \, b d x + {\left (24 \, a \cos \left (d x + c\right )^{4} + 30 \, b \cos \left (d x + c\right )^{3} + 32 \, a \cos \left (d x + c\right )^{2} + 45 \, b \cos \left (d x + c\right ) + 64 \, a\right )} \sin \left (d x + c\right )}{120 \, d} \]
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\[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \, dx=\int \left (a + b \sec {\left (c + d x \right )}\right ) \cos ^{5}{\left (c + d x \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.75 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \, dx=\frac {32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} b}{480 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.67 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \, dx=\frac {45 \, {\left (d x + c\right )} b + \frac {2 \, {\left (120 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 75 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 160 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 30 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 464 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 160 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 30 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 120 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 75 \, b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{5}}}{120 \, d} \]
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Time = 17.84 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.23 \[ \int \cos ^5(c+d x) (a+b \sec (c+d x)) \, dx=\frac {3\,b\,x}{8}+\frac {\left (2\,a-\frac {5\,b}{4}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (\frac {8\,a}{3}-\frac {b}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\frac {116\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15}+\left (\frac {8\,a}{3}+\frac {b}{2}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,a+\frac {5\,b}{4}\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5} \]
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