Integrand size = 21, antiderivative size = 130 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {9 a^3 x}{16}-\frac {3 a^3 \cos ^5(c+d x)}{10 d}+\frac {9 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {3 a^3 \cos ^3(c+d x) \sin (c+d x)}{8 d}-\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {3 \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{14 d} \]
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Time = 0.16 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2757, 2748, 2715, 8} \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {3 a^3 \cos ^5(c+d x)}{10 d}-\frac {3 \cos ^5(c+d x) \left (a^3 \sin (c+d x)+a^3\right )}{14 d}+\frac {3 a^3 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac {9 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {9 a^3 x}{16}-\frac {a \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d} \]
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Rule 8
Rule 2715
Rule 2748
Rule 2757
Rubi steps \begin{align*} \text {integral}& = -\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}+\frac {1}{7} (9 a) \int \cos ^4(c+d x) (a+a \sin (c+d x))^2 \, dx \\ & = -\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {3 \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{14 d}+\frac {1}{2} \left (3 a^2\right ) \int \cos ^4(c+d x) (a+a \sin (c+d x)) \, dx \\ & = -\frac {3 a^3 \cos ^5(c+d x)}{10 d}-\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {3 \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{14 d}+\frac {1}{2} \left (3 a^3\right ) \int \cos ^4(c+d x) \, dx \\ & = -\frac {3 a^3 \cos ^5(c+d x)}{10 d}+\frac {3 a^3 \cos ^3(c+d x) \sin (c+d x)}{8 d}-\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {3 \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{14 d}+\frac {1}{8} \left (9 a^3\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {3 a^3 \cos ^5(c+d x)}{10 d}+\frac {9 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {3 a^3 \cos ^3(c+d x) \sin (c+d x)}{8 d}-\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {3 \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{14 d}+\frac {1}{16} \left (9 a^3\right ) \int 1 \, dx \\ & = \frac {9 a^3 x}{16}-\frac {3 a^3 \cos ^5(c+d x)}{10 d}+\frac {9 a^3 \cos (c+d x) \sin (c+d x)}{16 d}+\frac {3 a^3 \cos ^3(c+d x) \sin (c+d x)}{8 d}-\frac {a \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {3 \cos ^5(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{14 d} \\ \end{align*}
Time = 0.52 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.24 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \cos ^5(c+d x) \left (-630 \arcsin \left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right ) \sqrt {1-\sin (c+d x)}+\sqrt {1+\sin (c+d x)} \left (-368+613 \sin (c+d x)+411 \sin ^2(c+d x)-306 \sin ^3(c+d x)-558 \sin ^4(c+d x)-72 \sin ^5(c+d x)+200 \sin ^6(c+d x)+80 \sin ^7(c+d x)\right )\right )}{560 d (-1+\sin (c+d x))^3 (1+\sin (c+d x))^{5/2}} \]
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Time = 0.75 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.65
method | result | size |
parallelrisch | \(-\frac {\left (-36 d x +\sin \left (4 d x +4 c \right )+\sin \left (6 d x +6 c \right )+27 \cos \left (d x +c \right )+13 \cos \left (3 d x +3 c \right )+\frac {11 \cos \left (5 d x +5 c \right )}{5}-\frac {\cos \left (7 d x +7 c \right )}{7}-19 \sin \left (2 d x +2 c \right )+\frac {1472}{35}\right ) a^{3}}{64 d}\) | \(85\) |
risch | \(\frac {9 a^{3} x}{16}-\frac {27 a^{3} \cos \left (d x +c \right )}{64 d}+\frac {a^{3} \cos \left (7 d x +7 c \right )}{448 d}-\frac {a^{3} \sin \left (6 d x +6 c \right )}{64 d}-\frac {11 a^{3} \cos \left (5 d x +5 c \right )}{320 d}-\frac {a^{3} \sin \left (4 d x +4 c \right )}{64 d}-\frac {13 a^{3} \cos \left (3 d x +3 c \right )}{64 d}+\frac {19 a^{3} \sin \left (2 d x +2 c \right )}{64 d}\) | \(124\) |
derivativedivides | \(\frac {a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {3 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(143\) |
default | \(\frac {a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {3 a^{3} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}\) | \(143\) |
norman | \(\frac {\frac {9 a^{3} x}{16}-\frac {46 a^{3}}{35 d}+\frac {7 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {17 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {13 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {13 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {17 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {7 a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {63 a^{3} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {189 a^{3} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {315 a^{3} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {315 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {189 a^{3} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {63 a^{3} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {9 a^{3} x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {14 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {6 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {16 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {16 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {32 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {58 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\) | \(377\) |
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Time = 0.32 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.65 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {80 \, a^{3} \cos \left (d x + c\right )^{7} - 448 \, a^{3} \cos \left (d x + c\right )^{5} + 315 \, a^{3} d x - 35 \, {\left (8 \, a^{3} \cos \left (d x + c\right )^{5} - 6 \, a^{3} \cos \left (d x + c\right )^{3} - 9 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{560 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (122) = 244\).
Time = 0.54 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.58 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {3 a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {9 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {9 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {3 a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {3 a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {5 a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {2 a^{3} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {3 a^{3} \cos ^{5}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.18 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.88 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {1344 \, a^{3} \cos \left (d x + c\right )^{5} - 64 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{3} - 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 70 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3}}{2240 \, d} \]
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Time = 0.37 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.95 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {9}{16} \, a^{3} x + \frac {a^{3} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {11 \, a^{3} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {13 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {27 \, a^{3} \cos \left (d x + c\right )}{64 \, d} - \frac {a^{3} \sin \left (6 \, d x + 6 \, c\right )}{64 \, d} - \frac {a^{3} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {19 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
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Time = 8.46 (sec) , antiderivative size = 389, normalized size of antiderivative = 2.99 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {9\,a^3\,x}{16}-\frac {\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8}-\frac {17\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{8}+\frac {17\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{2}+\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{8}+\frac {a^3\,\left (315\,c+315\,d\,x\right )}{560}-\frac {a^3\,\left (315\,c+315\,d\,x-736\right )}{560}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^3\,\left (315\,c+315\,d\,x\right )}{80}-\frac {a^3\,\left (2205\,c+2205\,d\,x-1792\right )}{560}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {a^3\,\left (315\,c+315\,d\,x\right )}{80}-\frac {a^3\,\left (2205\,c+2205\,d\,x-3360\right )}{560}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {3\,a^3\,\left (315\,c+315\,d\,x\right )}{80}-\frac {a^3\,\left (6615\,c+6615\,d\,x-6496\right )}{560}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {3\,a^3\,\left (315\,c+315\,d\,x\right )}{80}-\frac {a^3\,\left (6615\,c+6615\,d\,x-8960\right )}{560}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {a^3\,\left (315\,c+315\,d\,x\right )}{16}-\frac {a^3\,\left (11025\,c+11025\,d\,x-7840\right )}{560}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^3\,\left (315\,c+315\,d\,x\right )}{16}-\frac {a^3\,\left (11025\,c+11025\,d\,x-17920\right )}{560}\right )-\frac {7\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]
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