Integrand size = 29, antiderivative size = 132 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {5 a^3 x}{16}-\frac {4 a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos ^5(c+d x)}{d}-\frac {a^3 \cos ^7(c+d x)}{7 d}+\frac {5 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{8 d}-\frac {a^3 \cos ^3(c+d x) \sin ^3(c+d x)}{2 d} \]
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Time = 0.22 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2952, 2648, 2715, 8, 2645, 14, 276} \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \cos ^7(c+d x)}{7 d}+\frac {a^3 \cos ^5(c+d x)}{d}-\frac {4 a^3 \cos ^3(c+d x)}{3 d}-\frac {a^3 \sin ^3(c+d x) \cos ^3(c+d x)}{2 d}-\frac {5 a^3 \sin (c+d x) \cos ^3(c+d x)}{8 d}+\frac {5 a^3 \sin (c+d x) \cos (c+d x)}{16 d}+\frac {5 a^3 x}{16} \]
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Rule 8
Rule 14
Rule 276
Rule 2645
Rule 2648
Rule 2715
Rule 2952
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 \cos ^2(c+d x) \sin ^2(c+d x)+3 a^3 \cos ^2(c+d x) \sin ^3(c+d x)+3 a^3 \cos ^2(c+d x) \sin ^4(c+d x)+a^3 \cos ^2(c+d x) \sin ^5(c+d x)\right ) \, dx \\ & = a^3 \int \cos ^2(c+d x) \sin ^2(c+d x) \, dx+a^3 \int \cos ^2(c+d x) \sin ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^2(c+d x) \sin ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^2(c+d x) \sin ^4(c+d x) \, dx \\ & = -\frac {a^3 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {a^3 \cos ^3(c+d x) \sin ^3(c+d x)}{2 d}+\frac {1}{4} a^3 \int \cos ^2(c+d x) \, dx+\frac {1}{2} \left (3 a^3\right ) \int \cos ^2(c+d x) \sin ^2(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {a^3 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {5 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)}{2 d}+\frac {1}{8} a^3 \int 1 \, dx+\frac {1}{8} \left (3 a^3\right ) \int \cos ^2(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {a^3 x}{8}-\frac {4 a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos ^5(c+d x)}{d}-\frac {a^3 \cos ^7(c+d x)}{7 d}+\frac {5 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {5 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)}{2 d}+\frac {1}{16} \left (3 a^3\right ) \int 1 \, dx \\ & = \frac {5 a^3 x}{16}-\frac {4 a^3 \cos ^3(c+d x)}{3 d}+\frac {a^3 \cos ^5(c+d x)}{d}-\frac {a^3 \cos ^7(c+d x)}{7 d}+\frac {5 a^3 \cos (c+d x) \sin (c+d x)}{16 d}-\frac {5 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)}{2 d} \\ \end{align*}
Time = 6.94 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.65 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (420 c+420 d x-609 \cos (c+d x)-91 \cos (3 (c+d x))+63 \cos (5 (c+d x))-3 \cos (7 (c+d x))-63 \sin (2 (c+d x))-105 \sin (4 (c+d x))+21 \sin (6 (c+d x)))}{1344 d} \]
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Time = 0.34 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.67
method | result | size |
parallelrisch | \(-\frac {a^{3} \left (-420 d x +3 \cos \left (7 d x +7 c \right )+609 \cos \left (d x +c \right )+91 \cos \left (3 d x +3 c \right )-63 \cos \left (5 d x +5 c \right )-21 \sin \left (6 d x +6 c \right )+105 \sin \left (4 d x +4 c \right )+63 \sin \left (2 d x +2 c \right )+640\right )}{1344 d}\) | \(89\) |
risch | \(\frac {5 a^{3} x}{16}-\frac {29 a^{3} \cos \left (d x +c \right )}{64 d}-\frac {a^{3} \cos \left (7 d x +7 c \right )}{448 d}+\frac {a^{3} \sin \left (6 d x +6 c \right )}{64 d}+\frac {3 a^{3} \cos \left (5 d x +5 c \right )}{64 d}-\frac {5 a^{3} \sin \left (4 d x +4 c \right )}{64 d}-\frac {13 a^{3} \cos \left (3 d x +3 c \right )}{192 d}-\frac {3 a^{3} \sin \left (2 d x +2 c \right )}{64 d}\) | \(124\) |
derivativedivides | \(\frac {a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{7}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{35}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right )}{105}\right )+3 a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+3 a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )}{d}\) | \(194\) |
default | \(\frac {a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{7}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{35}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right )}{105}\right )+3 a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+3 a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )+a^{3} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{8}+\frac {d x}{8}+\frac {c}{8}\right )}{d}\) | \(194\) |
norman | \(\frac {\frac {5 a^{3} x}{16}-\frac {20 a^{3}}{21 d}-\frac {5 a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {3 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {119 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {119 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {3 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {5 a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {35 a^{3} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {105 a^{3} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {175 a^{3} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {175 a^{3} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {105 a^{3} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {35 a^{3} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}+\frac {5 a^{3} x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16}-\frac {12 a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {8 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {8 a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {20 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {92 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\) | \(358\) |
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Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.74 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {48 \, a^{3} \cos \left (d x + c\right )^{7} - 336 \, a^{3} \cos \left (d x + c\right )^{5} + 448 \, a^{3} \cos \left (d x + c\right )^{3} - 105 \, a^{3} d x - 21 \, {\left (8 \, a^{3} \cos \left (d x + c\right )^{5} - 18 \, a^{3} \cos \left (d x + c\right )^{3} + 5 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{336 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (122) = 244\).
Time = 0.50 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.87 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {3 a^{3} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {9 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {9 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 a^{3} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} - \frac {a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} + \frac {a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {4 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{15 d} - \frac {a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{d} - \frac {3 a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {8 a^{3} \cos ^{7}{\left (c + d x \right )}}{105 d} - \frac {2 a^{3} \cos ^{5}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin ^{2}{\left (c \right )} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.19 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.98 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {64 \, {\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} a^{3} - 1344 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{3} + 105 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 210 \, {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{6720 \, d} \]
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Time = 0.51 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.93 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {5}{16} \, a^{3} x - \frac {a^{3} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {3 \, a^{3} \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac {13 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {29 \, a^{3} \cos \left (d x + c\right )}{64 \, d} + \frac {a^{3} \sin \left (6 \, d x + 6 \, c\right )}{64 \, d} - \frac {5 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} - \frac {3 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
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Time = 13.56 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.51 \[ \int \cos ^2(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {5\,a^3\,x}{16}-\frac {\frac {5\,a^3\,\left (c+d\,x\right )}{16}+\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}-\frac {119\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{8}+\frac {119\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{8}-\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{2}-\frac {5\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{8}-\frac {a^3\,\left (105\,c+105\,d\,x-320\right )}{336}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {35\,a^3\,\left (c+d\,x\right )}{16}-\frac {a^3\,\left (735\,c+735\,d\,x-2240\right )}{336}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {105\,a^3\,\left (c+d\,x\right )}{16}-\frac {a^3\,\left (2205\,c+2205\,d\,x-2688\right )}{336}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {175\,a^3\,\left (c+d\,x\right )}{16}-\frac {a^3\,\left (3675\,c+3675\,d\,x-896\right )}{336}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {105\,a^3\,\left (c+d\,x\right )}{16}-\frac {a^3\,\left (2205\,c+2205\,d\,x-4032\right )}{336}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {175\,a^3\,\left (c+d\,x\right )}{16}-\frac {a^3\,\left (3675\,c+3675\,d\,x-10304\right )}{336}\right )+\frac {5\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]
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