Integrand size = 19, antiderivative size = 92 \[ \int \cos ^4(c+d x) (a+a \cos (c+d x)) \, dx=\frac {3 a x}{8}+\frac {a \sin (c+d x)}{d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {2 a \sin ^3(c+d x)}{3 d}+\frac {a \sin ^5(c+d x)}{5 d} \]
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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 = {2827, 2715, 8, 2713} \[ \int \cos ^4(c+d x) (a+a \cos (c+d x)) \, dx=\frac {a \sin ^5(c+d x)}{5 d}-\frac {2 a \sin ^3(c+d x)}{3 d}+\frac {a \sin (c+d x)}{d}+\frac {a \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3 a \sin (c+d x) \cos (c+d x)}{8 d}+\frac {3 a x}{8} \]
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Rule 8
Rule 2713
Rule 2715
Rule 2827
Rubi steps \begin{align*} \text {integral}& = a \int \cos ^4(c+d x) \, dx+a \int \cos ^5(c+d x) \, dx \\ & = \frac {a \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{4} (3 a) \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 a \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a \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 a) \int 1 \, dx \\ & = \frac {3 a x}{8}+\frac {a \sin (c+d x)}{d}+\frac {3 a \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a \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 = 65, normalized size of antiderivative = 0.71 \[ \int \cos ^4(c+d x) (a+a \cos (c+d x)) \, dx=\frac {a \left (480 \sin (c+d x)-320 \sin ^3(c+d x)+96 \sin ^5(c+d x)+15 (12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x)))\right )}{480 d} \]
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Time = 2.25 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70
method | result | size |
parallelrisch | \(\frac {a \left (180 d x +300 \sin \left (d x +c \right )+6 \sin \left (5 d x +5 c \right )+15 \sin \left (4 d x +4 c \right )+50 \sin \left (3 d x +3 c \right )+120 \sin \left (2 d x +2 c \right )\right )}{480 d}\) | \(64\) |
derivativedivides | \(\frac {\frac {a \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(70\) |
default | \(\frac {\frac {a \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5}+a \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(70\) |
parts | \(\frac {a \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}+\frac {a \left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{5 d}\) | \(72\) |
risch | \(\frac {3 a x}{8}+\frac {5 a \sin \left (d x +c \right )}{8 d}+\frac {a \sin \left (5 d x +5 c \right )}{80 d}+\frac {a \sin \left (4 d x +4 c \right )}{32 d}+\frac {5 a \sin \left (3 d x +3 c \right )}{48 d}+\frac {a \sin \left (2 d x +2 c \right )}{4 d}\) | \(78\) |
norman | \(\frac {\frac {3 a x}{8}+\frac {13 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {19 a \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {116 a \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {13 a \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}+\frac {3 a \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {15 a x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {15 a x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {15 a x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {15 a x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {3 a x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}\) | \(180\) |
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Time = 0.34 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.70 \[ \int \cos ^4(c+d x) (a+a \cos (c+d x)) \, dx=\frac {45 \, a d x + {\left (24 \, a \cos \left (d x + c\right )^{4} + 30 \, a \cos \left (d x + c\right )^{3} + 32 \, a \cos \left (d x + c\right )^{2} + 45 \, a \cos \left (d x + c\right ) + 64 \, a\right )} \sin \left (d x + c\right )}{120 \, d} \]
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Time = 0.23 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.83 \[ \int \cos ^4(c+d x) (a+a \cos (c+d x)) \, dx=\begin {cases} \frac {3 a x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 a x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 a x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {8 a \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 a \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {3 a \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {a \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {5 a \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + a\right ) \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.75 \[ \int \cos ^4(c+d x) (a+a \cos (c+d x)) \, dx=\frac {32 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a + 15 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a}{480 \, d} \]
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Time = 0.33 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.84 \[ \int \cos ^4(c+d x) (a+a \cos (c+d x)) \, dx=\frac {3}{8} \, a x + \frac {a \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {a \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {5 \, a \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {a \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac {5 \, a \sin \left (d x + c\right )}{8 \, d} \]
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Time = 16.63 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.01 \[ \int \cos ^4(c+d x) (a+a \cos (c+d x)) \, dx=\frac {3\,a\,x}{8}+\frac {\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{4}+\frac {13\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{6}+\frac {116\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15}+\frac {19\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6}+\frac {13\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5} \]
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