Integrand size = 27, antiderivative size = 117 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {7 a^3 x}{16}-\frac {4 a^3 \cos ^3(c+d x)}{3 d}+\frac {3 a^3 \cos ^5(c+d x)}{5 d}+\frac {7 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {7 a^3 \cos ^3(c+d x) \sin (c+d x)}{8 d}-\frac {a^3 \cos ^3(c+d x) \sin ^3(c+d x)}{6 d} \]
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Time = 0.13 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.14, 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} \[ \int \cos ^2(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {7 a^3 \cos ^3(c+d x)}{24 d}-\frac {7 \cos ^3(c+d x) \left (a^3 \sin (c+d x)+a^3\right )}{40 d}+\frac {7 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {7 a^3 x}{16}-\frac {a \cos ^3(c+d x) (a \sin (c+d x)+a)^2}{10 d}-\frac {\cos ^3(c+d x) (a \sin (c+d x)+a)^3}{6 d} \]
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Rule 8
Rule 2715
Rule 2748
Rule 2757
Rule 2939
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}+\frac {1}{2} \int \cos ^2(c+d x) (a+a \sin (c+d x))^3 \, dx \\ & = -\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^2}{10 d}-\frac {\cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}+\frac {1}{10} (7 a) \int \cos ^2(c+d x) (a+a \sin (c+d x))^2 \, dx \\ & = -\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^2}{10 d}-\frac {\cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}-\frac {7 \cos ^3(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{40 d}+\frac {1}{8} \left (7 a^2\right ) \int \cos ^2(c+d x) (a+a \sin (c+d x)) \, dx \\ & = -\frac {7 a^3 \cos ^3(c+d x)}{24 d}-\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^2}{10 d}-\frac {\cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}-\frac {7 \cos ^3(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{40 d}+\frac {1}{8} \left (7 a^3\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {7 a^3 \cos ^3(c+d x)}{24 d}+\frac {7 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^2}{10 d}-\frac {\cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}-\frac {7 \cos ^3(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{40 d}+\frac {1}{16} \left (7 a^3\right ) \int 1 \, dx \\ & = \frac {7 a^3 x}{16}-\frac {7 a^3 \cos ^3(c+d x)}{24 d}+\frac {7 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {a \cos ^3(c+d x) (a+a \sin (c+d x))^2}{10 d}-\frac {\cos ^3(c+d x) (a+a \sin (c+d x))^3}{6 d}-\frac {7 \cos ^3(c+d x) \left (a^3+a^3 \sin (c+d x)\right )}{40 d} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.65 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (450 c+420 d x-600 \cos (c+d x)-140 \cos (3 (c+d x))+36 \cos (5 (c+d x))-15 \sin (2 (c+d x))-105 \sin (4 (c+d x))+5 \sin (6 (c+d x)))}{960 d} \]
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Time = 0.30 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.67
method | result | size |
parallelrisch | \(-\frac {a^{3} \left (-420 d x +600 \cos \left (d x +c \right )+105 \sin \left (4 d x +4 c \right )-36 \cos \left (5 d x +5 c \right )+140 \cos \left (3 d x +3 c \right )+15 \sin \left (2 d x +2 c \right )-5 \sin \left (6 d x +6 c \right )+704\right )}{960 d}\) | \(78\) |
risch | \(\frac {7 a^{3} x}{16}-\frac {5 a^{3} \cos \left (d x +c \right )}{8 d}+\frac {a^{3} \sin \left (6 d x +6 c \right )}{192 d}+\frac {3 a^{3} \cos \left (5 d x +5 c \right )}{80 d}-\frac {7 a^{3} \sin \left (4 d x +4 c \right )}{64 d}-\frac {7 a^{3} \cos \left (3 d x +3 c \right )}{48 d}-\frac {a^{3} \sin \left (2 d x +2 c \right )}{64 d}\) | \(107\) |
derivativedivides | \(\frac {a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+3 a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{3}}{d}\) | \(156\) |
default | \(\frac {a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+3 a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )-\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{3}}{d}\) | \(156\) |
norman | \(\frac {\frac {7 a^{3} x}{16}-\frac {22 a^{3}}{15 d}-\frac {7 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}+\frac {73 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {37 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {37 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {73 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {7 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {21 a^{3} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {105 a^{3} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a^{3} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4}+\frac {105 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {21 a^{3} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {7 a^{3} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {2 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {18 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {44 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {34 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}\) | \(341\) |
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Time = 0.28 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.73 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {144 \, a^{3} \cos \left (d x + c\right )^{5} - 320 \, a^{3} \cos \left (d x + c\right )^{3} + 105 \, a^{3} d x + 5 \, {\left (8 \, a^{3} \cos \left (d x + c\right )^{5} - 50 \, a^{3} \cos \left (d x + c\right )^{3} + 21 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{240 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 328 vs. \(2 (110) = 220\).
Time = 0.36 (sec) , antiderivative size = 328, normalized size of antiderivative = 2.80 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {3 a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{d} - \frac {a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {3 a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {2 a^{3} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {a^{3} \cos ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin {\left (c \right )} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.23 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.91 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {320 \, a^{3} \cos \left (d x + c\right )^{3} - 192 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{3} + 5 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 90 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{960 \, d} \]
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Time = 0.44 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.91 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {7}{16} \, a^{3} x + \frac {3 \, a^{3} \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {7 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {5 \, a^{3} \cos \left (d x + c\right )}{8 \, d} + \frac {a^{3} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {7 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} - \frac {a^{3} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
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Time = 11.78 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.98 \[ \int \cos ^2(c+d x) \sin (c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {7\,a^3\,x}{16}-\frac {\frac {37\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}-\frac {37\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}-\frac {73\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+\frac {73\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}-\frac {7\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+\frac {a^3\,\left (105\,c+105\,d\,x\right )}{240}-\frac {a^3\,\left (105\,c+105\,d\,x-352\right )}{240}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {a^3\,\left (105\,c+105\,d\,x\right )}{40}-\frac {a^3\,\left (630\,c+630\,d\,x-480\right )}{240}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^3\,\left (105\,c+105\,d\,x\right )}{40}-\frac {a^3\,\left (630\,c+630\,d\,x-1632\right )}{240}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {a^3\,\left (105\,c+105\,d\,x\right )}{16}-\frac {a^3\,\left (1575\,c+1575\,d\,x-960\right )}{240}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {a^3\,\left (105\,c+105\,d\,x\right )}{16}-\frac {a^3\,\left (1575\,c+1575\,d\,x-4320\right )}{240}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^3\,\left (105\,c+105\,d\,x\right )}{12}-\frac {a^3\,\left (2100\,c+2100\,d\,x-3520\right )}{240}\right )+\frac {7\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \]
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