Integrand size = 32, antiderivative size = 140 \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {1}{8} a^3 A x-\frac {2 a^3 A \cos ^3(c+d x)}{3 d}+\frac {3 a^3 A \cos ^5(c+d x)}{5 d}-\frac {a^3 A \cos ^7(c+d x)}{7 d}-\frac {a^3 A \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^3 A \cos (c+d x) \sin ^3(c+d x)}{12 d}+\frac {a^3 A \cos (c+d x) \sin ^5(c+d x)}{3 d} \]
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Time = 0.14 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3045, 2713, 2715, 8} \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=-\frac {a^3 A \cos ^7(c+d x)}{7 d}+\frac {3 a^3 A \cos ^5(c+d x)}{5 d}-\frac {2 a^3 A \cos ^3(c+d x)}{3 d}+\frac {a^3 A \sin ^5(c+d x) \cos (c+d x)}{3 d}-\frac {a^3 A \sin ^3(c+d x) \cos (c+d x)}{12 d}-\frac {a^3 A \sin (c+d x) \cos (c+d x)}{8 d}+\frac {1}{8} a^3 A x \]
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Rule 8
Rule 2713
Rule 2715
Rule 3045
Rubi steps \begin{align*} \text {integral}& = \int \left (a^3 A \sin ^3(c+d x)+2 a^3 A \sin ^4(c+d x)-2 a^3 A \sin ^6(c+d x)-a^3 A \sin ^7(c+d x)\right ) \, dx \\ & = \left (a^3 A\right ) \int \sin ^3(c+d x) \, dx-\left (a^3 A\right ) \int \sin ^7(c+d x) \, dx+\left (2 a^3 A\right ) \int \sin ^4(c+d x) \, dx-\left (2 a^3 A\right ) \int \sin ^6(c+d x) \, dx \\ & = -\frac {a^3 A \cos (c+d x) \sin ^3(c+d x)}{2 d}+\frac {a^3 A \cos (c+d x) \sin ^5(c+d x)}{3 d}+\frac {1}{2} \left (3 a^3 A\right ) \int \sin ^2(c+d x) \, dx-\frac {1}{3} \left (5 a^3 A\right ) \int \sin ^4(c+d x) \, dx-\frac {\left (a^3 A\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}+\frac {\left (a^3 A\right ) \text {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {2 a^3 A \cos ^3(c+d x)}{3 d}+\frac {3 a^3 A \cos ^5(c+d x)}{5 d}-\frac {a^3 A \cos ^7(c+d x)}{7 d}-\frac {3 a^3 A \cos (c+d x) \sin (c+d x)}{4 d}-\frac {a^3 A \cos (c+d x) \sin ^3(c+d x)}{12 d}+\frac {a^3 A \cos (c+d x) \sin ^5(c+d x)}{3 d}+\frac {1}{4} \left (3 a^3 A\right ) \int 1 \, dx-\frac {1}{4} \left (5 a^3 A\right ) \int \sin ^2(c+d x) \, dx \\ & = \frac {3}{4} a^3 A x-\frac {2 a^3 A \cos ^3(c+d x)}{3 d}+\frac {3 a^3 A \cos ^5(c+d x)}{5 d}-\frac {a^3 A \cos ^7(c+d x)}{7 d}-\frac {a^3 A \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^3 A \cos (c+d x) \sin ^3(c+d x)}{12 d}+\frac {a^3 A \cos (c+d x) \sin ^5(c+d x)}{3 d}-\frac {1}{8} \left (5 a^3 A\right ) \int 1 \, dx \\ & = \frac {1}{8} a^3 A x-\frac {2 a^3 A \cos ^3(c+d x)}{3 d}+\frac {3 a^3 A \cos ^5(c+d x)}{5 d}-\frac {a^3 A \cos ^7(c+d x)}{7 d}-\frac {a^3 A \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^3 A \cos (c+d x) \sin ^3(c+d x)}{12 d}+\frac {a^3 A \cos (c+d x) \sin ^5(c+d x)}{3 d} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.62 \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {a^3 A (840 c+840 d x-1365 \cos (c+d x)-175 \cos (3 (c+d x))+147 \cos (5 (c+d x))-15 \cos (7 (c+d x))-210 \sin (2 (c+d x))-210 \sin (4 (c+d x))+70 \sin (6 (c+d x)))}{6720 d} \]
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Time = 2.19 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.64
method | result | size |
parallelrisch | \(-\frac {A \,a^{3} \left (-840 d x +1365 \cos \left (d x +c \right )+15 \cos \left (7 d x +7 c \right )-70 \sin \left (6 d x +6 c \right )-147 \cos \left (5 d x +5 c \right )+210 \sin \left (4 d x +4 c \right )+175 \cos \left (3 d x +3 c \right )+210 \sin \left (2 d x +2 c \right )+1408\right )}{6720 d}\) | \(90\) |
risch | \(\frac {a^{3} A x}{8}-\frac {13 a^{3} A \cos \left (d x +c \right )}{64 d}-\frac {A \,a^{3} \cos \left (7 d x +7 c \right )}{448 d}+\frac {A \,a^{3} \sin \left (6 d x +6 c \right )}{96 d}+\frac {7 A \,a^{3} \cos \left (5 d x +5 c \right )}{320 d}-\frac {A \,a^{3} \sin \left (4 d x +4 c \right )}{32 d}-\frac {5 A \,a^{3} \cos \left (3 d x +3 c \right )}{192 d}-\frac {A \,a^{3} \sin \left (2 d x +2 c \right )}{32 d}\) | \(132\) |
derivativedivides | \(\frac {\frac {A \,a^{3} \left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{7}-2 A \,a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+2 A \,a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-\frac {A \,a^{3} \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}}{d}\) | \(158\) |
default | \(\frac {\frac {A \,a^{3} \left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{7}-2 A \,a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )+2 A \,a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-\frac {A \,a^{3} \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3}}{d}\) | \(158\) |
parts | \(-\frac {A \,a^{3} \left (2+\sin ^{2}\left (d x +c \right )\right ) \cos \left (d x +c \right )}{3 d}+\frac {2 A \,a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )+\frac {3 \sin \left (d x +c \right )}{2}\right ) \cos \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )}{d}-\frac {2 A \,a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )+\frac {5 \left (\sin ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \sin \left (d x +c \right )}{8}\right ) \cos \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}+\frac {A \,a^{3} \left (\frac {16}{5}+\sin ^{6}\left (d x +c \right )+\frac {6 \left (\sin ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (d x +c \right )\right )}{5}\right ) \cos \left (d x +c \right )}{7 d}\) | \(166\) |
norman | \(\frac {-\frac {44 A \,a^{3}}{105 d}+\frac {a^{3} A x}{8}-\frac {4 A \,a^{3} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {24 A \,a^{3} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {8 A \,a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {44 A \,a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}-\frac {52 A \,a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {A \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 d}-\frac {5 A \,a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {97 A \,a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}-\frac {97 A \,a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12 d}+\frac {5 A \,a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {A \,a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}+\frac {7 a^{3} A x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {21 a^{3} A x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {35 a^{3} A x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {35 a^{3} A x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {21 a^{3} A x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {7 a^{3} A x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}+\frac {a^{3} A x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\) | \(378\) |
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Time = 0.25 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.75 \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=-\frac {120 \, A a^{3} \cos \left (d x + c\right )^{7} - 504 \, A a^{3} \cos \left (d x + c\right )^{5} + 560 \, A a^{3} \cos \left (d x + c\right )^{3} - 105 \, A a^{3} d x - 35 \, {\left (8 \, A a^{3} \cos \left (d x + c\right )^{5} - 14 \, A a^{3} \cos \left (d x + c\right )^{3} + 3 \, A a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{840 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 440 vs. \(2 (131) = 262\).
Time = 0.48 (sec) , antiderivative size = 440, normalized size of antiderivative = 3.14 \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\begin {cases} - \frac {5 A a^{3} x \sin ^{6}{\left (c + d x \right )}}{8} - \frac {15 A a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{8} + \frac {3 A a^{3} x \sin ^{4}{\left (c + d x \right )}}{4} - \frac {15 A a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 A a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{2} - \frac {5 A a^{3} x \cos ^{6}{\left (c + d x \right )}}{8} + \frac {3 A a^{3} x \cos ^{4}{\left (c + d x \right )}}{4} + \frac {A a^{3} \sin ^{6}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {11 A a^{3} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {2 A a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{d} + \frac {5 A a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {5 A a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{4 d} + \frac {8 A a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {A a^{3} \sin ^{2}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} + \frac {5 A a^{3} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{8 d} - \frac {3 A a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{4 d} + \frac {16 A a^{3} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {2 A a^{3} \cos ^{3}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (- A \sin {\left (c \right )} + A\right ) \left (a \sin {\left (c \right )} + a\right )^{3} \sin ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.24 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.12 \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=-\frac {96 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 21 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3} - 35 \, \cos \left (d x + c\right )\right )} A a^{3} - 1120 \, {\left (\cos \left (d x + c\right )^{3} - 3 \, \cos \left (d x + c\right )\right )} A a^{3} + 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 60 \, d x + 60 \, c + 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3} - 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) - 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{3}}{3360 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.94 \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {1}{8} \, A a^{3} x - \frac {A a^{3} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {7 \, A a^{3} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {5 \, A a^{3} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {13 \, A a^{3} \cos \left (d x + c\right )}{64 \, d} + \frac {A a^{3} \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} - \frac {A a^{3} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} - \frac {A a^{3} \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} \]
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Time = 14.60 (sec) , antiderivative size = 300, normalized size of antiderivative = 2.14 \[ \int \sin ^3(c+d x) (a+a \sin (c+d x))^3 (A-A \sin (c+d x)) \, dx=\frac {A\,a^3\,\left (105\,c-210\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-2464\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1400\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-4032\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+6790\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+2240\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-14560\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-6790\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9-3360\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+1400\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+210\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+105\,d\,x+735\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (c+d\,x\right )+2205\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (c+d\,x\right )+3675\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (c+d\,x\right )+3675\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (c+d\,x\right )+2205\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (c+d\,x\right )+735\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (c+d\,x\right )+105\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (c+d\,x\right )-352\right )}{840\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7} \]
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