Integrand size = 29, antiderivative size = 135 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 x}{8}-\frac {2 a^2 \cos ^3(c+d x)}{3 d}+\frac {3 a^2 \cos ^5(c+d x)}{5 d}-\frac {a^2 \cos ^7(c+d x)}{7 d}+\frac {a^2 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {a^2 \cos ^3(c+d x) \sin ^3(c+d x)}{3 d} \]
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Time = 0.18 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2952, 2645, 14, 2648, 2715, 8, 276} \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {a^2 \cos ^7(c+d x)}{7 d}+\frac {3 a^2 \cos ^5(c+d x)}{5 d}-\frac {2 a^2 \cos ^3(c+d x)}{3 d}-\frac {a^2 \sin ^3(c+d x) \cos ^3(c+d x)}{3 d}-\frac {a^2 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {a^2 \sin (c+d x) \cos (c+d x)}{8 d}+\frac {a^2 x}{8} \]
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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^2 \cos ^2(c+d x) \sin ^3(c+d x)+2 a^2 \cos ^2(c+d x) \sin ^4(c+d x)+a^2 \cos ^2(c+d x) \sin ^5(c+d x)\right ) \, dx \\ & = a^2 \int \cos ^2(c+d x) \sin ^3(c+d x) \, dx+a^2 \int \cos ^2(c+d x) \sin ^5(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^2(c+d x) \sin ^4(c+d x) \, dx \\ & = -\frac {a^2 \cos ^3(c+d x) \sin ^3(c+d x)}{3 d}+a^2 \int \cos ^2(c+d x) \sin ^2(c+d x) \, dx-\frac {a^2 \text {Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {a^2 \text {Subst}\left (\int x^2 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{4 d}-\frac {a^2 \cos ^3(c+d x) \sin ^3(c+d x)}{3 d}+\frac {1}{4} a^2 \int \cos ^2(c+d x) \, dx-\frac {a^2 \text {Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {a^2 \text {Subst}\left (\int \left (x^2-2 x^4+x^6\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {2 a^2 \cos ^3(c+d x)}{3 d}+\frac {3 a^2 \cos ^5(c+d x)}{5 d}-\frac {a^2 \cos ^7(c+d x)}{7 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)}{4 d}-\frac {a^2 \cos ^3(c+d x) \sin ^3(c+d x)}{3 d}+\frac {1}{8} a^2 \int 1 \, dx \\ & = \frac {a^2 x}{8}-\frac {2 a^2 \cos ^3(c+d x)}{3 d}+\frac {3 a^2 \cos ^5(c+d x)}{5 d}-\frac {a^2 \cos ^7(c+d x)}{7 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)}{4 d}-\frac {a^2 \cos ^3(c+d x) \sin ^3(c+d x)}{3 d} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.64 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 (840 c+840 d x-1365 \cos (c+d x)-175 \cos (3 (c+d x))+147 \cos (5 (c+d x))-15 \cos (7 (c+d x))-210 \sin (2 (c+d x))-210 \sin (4 (c+d x))+70 \sin (6 (c+d x)))}{6720 d} \]
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Time = 0.31 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.66
method | result | size |
parallelrisch | \(-\frac {a^{2} \left (-840 d x +1365 \cos \left (d x +c \right )-70 \sin \left (6 d x +6 c \right )+210 \sin \left (4 d x +4 c \right )+210 \sin \left (2 d x +2 c \right )-147 \cos \left (5 d x +5 c \right )+175 \cos \left (3 d x +3 c \right )+15 \cos \left (7 d x +7 c \right )+1408\right )}{6720 d}\) | \(89\) |
risch | \(\frac {a^{2} x}{8}-\frac {13 a^{2} \cos \left (d x +c \right )}{64 d}-\frac {a^{2} \cos \left (7 d x +7 c \right )}{448 d}+\frac {a^{2} \sin \left (6 d x +6 c \right )}{96 d}+\frac {7 a^{2} \cos \left (5 d x +5 c \right )}{320 d}-\frac {a^{2} \sin \left (4 d x +4 c \right )}{32 d}-\frac {5 a^{2} \cos \left (3 d x +3 c \right )}{192 d}-\frac {a^{2} \sin \left (2 d x +2 c \right )}{32 d}\) | \(124\) |
derivativedivides | \(\frac {a^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{7}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{35}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right )}{105}\right )+2 a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )}{d}\) | \(151\) |
default | \(\frac {a^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{7}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{35}-\frac {8 \left (\cos ^{3}\left (d x +c \right )\right )}{105}\right )+2 a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{6}-\frac {\sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )}{8}+\frac {\cos \left (d x +c \right ) \sin \left (d x +c \right )}{16}+\frac {d x}{16}+\frac {c}{16}\right )+a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{5}-\frac {2 \left (\cos ^{3}\left (d x +c \right )\right )}{15}\right )}{d}\) | \(151\) |
norman | \(\frac {\frac {a^{2} x}{8}-\frac {44 a^{2}}{105 d}-\frac {a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {5 a^{2} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {97 a^{2} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {97 a^{2} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {5 a^{2} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {a^{2} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {7 a^{2} x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {21 a^{2} x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {35 a^{2} x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {35 a^{2} x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {21 a^{2} x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {7 a^{2} x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{2} x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}-\frac {4 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {24 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {8 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {44 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}-\frac {52 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\) | \(358\) |
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Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.73 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {120 \, a^{2} \cos \left (d x + c\right )^{7} - 504 \, a^{2} \cos \left (d x + c\right )^{5} + 560 \, a^{2} \cos \left (d x + c\right )^{3} - 105 \, a^{2} d x - 35 \, {\left (8 \, a^{2} \cos \left (d x + c\right )^{5} - 14 \, a^{2} \cos \left (d x + c\right )^{3} + 3 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{840 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (121) = 242\).
Time = 0.49 (sec) , antiderivative size = 275, normalized size of antiderivative = 2.04 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\begin {cases} \frac {a^{2} x \sin ^{6}{\left (c + d x \right )}}{8} + \frac {3 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{8} + \frac {3 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{8} + \frac {a^{2} x \cos ^{6}{\left (c + d x \right )}}{8} + \frac {a^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {a^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {4 a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{15 d} - \frac {a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {a^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{8 d} - \frac {8 a^{2} \cos ^{7}{\left (c + d x \right )}}{105 d} - \frac {2 a^{2} \cos ^{5}{\left (c + d x \right )}}{15 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{2} \sin ^{3}{\left (c \right )} \cos ^{2}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.21 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.78 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {32 \, {\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} a^{2} - 224 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{2} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 12 \, d x - 12 \, c + 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2}}{3360 \, d} \]
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Time = 0.37 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.91 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {1}{8} \, a^{2} x - \frac {a^{2} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {7 \, a^{2} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {5 \, a^{2} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {13 \, a^{2} \cos \left (d x + c\right )}{64 \, d} + \frac {a^{2} \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} - \frac {a^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} - \frac {a^{2} \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} \]
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Time = 13.96 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.45 \[ \int \cos ^2(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,x}{8}-\frac {\frac {a^2\,\left (c+d\,x\right )}{8}+\frac {5\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3}-\frac {97\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{12}+\frac {97\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{12}-\frac {5\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{3}-\frac {a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{4}-\frac {a^2\,\left (105\,c+105\,d\,x-352\right )}{840}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {7\,a^2\,\left (c+d\,x\right )}{8}-\frac {a^2\,\left (735\,c+735\,d\,x-2464\right )}{840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {21\,a^2\,\left (c+d\,x\right )}{8}-\frac {a^2\,\left (2205\,c+2205\,d\,x-3360\right )}{840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {21\,a^2\,\left (c+d\,x\right )}{8}-\frac {a^2\,\left (2205\,c+2205\,d\,x-4032\right )}{840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {35\,a^2\,\left (c+d\,x\right )}{8}-\frac {a^2\,\left (3675\,c+3675\,d\,x+2240\right )}{840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {35\,a^2\,\left (c+d\,x\right )}{8}-\frac {a^2\,\left (3675\,c+3675\,d\,x-14560\right )}{840}\right )+\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]
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