Integrand size = 27, antiderivative size = 129 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 x}{8}-\frac {a^2 \cos ^5(c+d x)}{15 d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{12 d}-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {\cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{21 d} \]
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Time = 0.12 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2939, 2757, 2748, 2715, 8} \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \cos ^5(c+d x)}{15 d}-\frac {\cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{21 d}+\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{12 d}+\frac {a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {a^2 x}{8}-\frac {\cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d} \]
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Rule 8
Rule 2715
Rule 2748
Rule 2757
Rule 2939
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}+\frac {2}{7} \int \cos ^4(c+d x) (a+a \sin (c+d x))^2 \, dx \\ & = -\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {\cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{21 d}+\frac {1}{3} a \int \cos ^4(c+d x) (a+a \sin (c+d x)) \, dx \\ & = -\frac {a^2 \cos ^5(c+d x)}{15 d}-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {\cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{21 d}+\frac {1}{3} a^2 \int \cos ^4(c+d x) \, dx \\ & = -\frac {a^2 \cos ^5(c+d x)}{15 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{12 d}-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {\cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{21 d}+\frac {1}{4} a^2 \int \cos ^2(c+d x) \, dx \\ & = -\frac {a^2 \cos ^5(c+d x)}{15 d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{12 d}-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {\cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{21 d}+\frac {1}{8} a^2 \int 1 \, dx \\ & = \frac {a^2 x}{8}-\frac {a^2 \cos ^5(c+d x)}{15 d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{8 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{12 d}-\frac {\cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {\cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{21 d} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.67 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 (840 c+840 d x-1155 \cos (c+d x)-525 \cos (3 (c+d x))-63 \cos (5 (c+d x))+15 \cos (7 (c+d x))+210 \sin (2 (c+d x))-210 \sin (4 (c+d x))-70 \sin (6 (c+d x)))}{6720 d} \]
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Time = 0.33 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.69
method | result | size |
parallelrisch | \(-\frac {a^{2} \left (-840 d x +1155 \cos \left (d x +c \right )+210 \sin \left (4 d x +4 c \right )+70 \sin \left (6 d x +6 c \right )-210 \sin \left (2 d x +2 c \right )+63 \cos \left (5 d x +5 c \right )+525 \cos \left (3 d x +3 c \right )-15 \cos \left (7 d x +7 c \right )+1728\right )}{6720 d}\) | \(89\) |
derivativedivides | \(\frac {a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5}}{d}\) | \(106\) |
default | \(\frac {a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )+2 a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5}}{d}\) | \(106\) |
risch | \(\frac {a^{2} x}{8}-\frac {11 a^{2} \cos \left (d x +c \right )}{64 d}+\frac {a^{2} \cos \left (7 d x +7 c \right )}{448 d}-\frac {a^{2} \sin \left (6 d x +6 c \right )}{96 d}-\frac {3 a^{2} \cos \left (5 d x +5 c \right )}{320 d}-\frac {a^{2} \sin \left (4 d x +4 c \right )}{32 d}-\frac {5 a^{2} \cos \left (3 d x +3 c \right )}{64 d}+\frac {a^{2} \sin \left (2 d x +2 c \right )}{32 d}\) | \(124\) |
norman | \(\frac {\frac {a^{2} x}{8}-\frac {18 a^{2}}{35 d}-\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}+\frac {11 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {31 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {31 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {11 a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a^{2} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {7 a^{2} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {21 a^{2} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {35 a^{2} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {35 a^{2} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {21 a^{2} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {7 a^{2} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{2} x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {2 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {14 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {2 a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {8 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {8 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {16 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\) | \(377\) |
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Time = 0.28 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.66 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {120 \, a^{2} \cos \left (d x + c\right )^{7} - 336 \, a^{2} \cos \left (d x + c\right )^{5} + 105 \, a^{2} d x - 35 \, {\left (8 \, a^{2} \cos \left (d x + c\right )^{5} - 2 \, a^{2} \cos \left (d x + c\right )^{3} - 3 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{840 \, d} \]
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Time = 0.46 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.73 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\begin {cases} \frac {a^{2} x \sin ^{6}{\left (c + d x \right )}}{8} + \frac {3 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{8} + \frac {3 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{8} + \frac {a^{2} x \cos ^{6}{\left (c + d x \right )}}{8} + \frac {a^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {a^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{8 d} - \frac {2 a^{2} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {a^{2} \cos ^{5}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{2} \sin {\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.64 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {672 \, a^{2} \cos \left (d x + c\right )^{5} - 96 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{2} - 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2}}{3360 \, d} \]
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Time = 0.71 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.95 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {1}{8} \, a^{2} x + \frac {a^{2} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {3 \, a^{2} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {5 \, a^{2} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {11 \, a^{2} \cos \left (d x + c\right )}{64 \, d} - \frac {a^{2} \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} - \frac {a^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac {a^{2} \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} \]
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Time = 11.51 (sec) , antiderivative size = 388, normalized size of antiderivative = 3.01 \[ \int \cos ^4(c+d x) \sin (c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,x}{8}-\frac {\frac {31\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}-\frac {11\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-\frac {31\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{12}+\frac {11\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{3}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{4}+a^2\,\left (\frac {c}{8}+\frac {d\,x}{8}\right )-a^2\,\left (\frac {c}{8}+\frac {d\,x}{8}-\frac {18}{35}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (7\,a^2\,\left (\frac {c}{8}+\frac {d\,x}{8}\right )-a^2\,\left (\frac {7\,c}{8}+\frac {7\,d\,x}{8}-2\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (7\,a^2\,\left (\frac {c}{8}+\frac {d\,x}{8}\right )-a^2\,\left (\frac {7\,c}{8}+\frac {7\,d\,x}{8}-\frac {8}{5}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (21\,a^2\,\left (\frac {c}{8}+\frac {d\,x}{8}\right )-a^2\,\left (\frac {21\,c}{8}+\frac {21\,d\,x}{8}-8\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (21\,a^2\,\left (\frac {c}{8}+\frac {d\,x}{8}\right )-a^2\,\left (\frac {21\,c}{8}+\frac {21\,d\,x}{8}-\frac {14}{5}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (35\,a^2\,\left (\frac {c}{8}+\frac {d\,x}{8}\right )-a^2\,\left (\frac {35\,c}{8}+\frac {35\,d\,x}{8}-2\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (35\,a^2\,\left (\frac {c}{8}+\frac {d\,x}{8}\right )-a^2\,\left (\frac {35\,c}{8}+\frac {35\,d\,x}{8}-16\right )\right )+\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]
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