Integrand size = 31, antiderivative size = 231 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {\left (a^2-b^2\right )^2 (A b-a B) (a+b \sin (c+d x))^3}{3 b^6 d}-\frac {\left (a^2-b^2\right ) \left (4 a A b-5 a^2 B+b^2 B\right ) (a+b \sin (c+d x))^4}{4 b^6 d}+\frac {2 \left (3 a^2 A b-A b^3-5 a^3 B+3 a b^2 B\right ) (a+b \sin (c+d x))^5}{5 b^6 d}-\frac {\left (2 a A b-5 a^2 B+b^2 B\right ) (a+b \sin (c+d x))^6}{3 b^6 d}+\frac {(A b-5 a B) (a+b \sin (c+d x))^7}{7 b^6 d}+\frac {B (a+b \sin (c+d x))^8}{8 b^6 d} \]
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Time = 0.19 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {2916, 786} \[ \int \cos ^5(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=-\frac {\left (-5 a^2 B+2 a A b+b^2 B\right ) (a+b \sin (c+d x))^6}{3 b^6 d}-\frac {\left (a^2-b^2\right ) \left (-5 a^2 B+4 a A b+b^2 B\right ) (a+b \sin (c+d x))^4}{4 b^6 d}+\frac {\left (a^2-b^2\right )^2 (A b-a B) (a+b \sin (c+d x))^3}{3 b^6 d}+\frac {2 \left (-5 a^3 B+3 a^2 A b+3 a b^2 B-A b^3\right ) (a+b \sin (c+d x))^5}{5 b^6 d}+\frac {(A b-5 a B) (a+b \sin (c+d x))^7}{7 b^6 d}+\frac {B (a+b \sin (c+d x))^8}{8 b^6 d} \]
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Rule 786
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+x)^2 \left (A+\frac {B x}{b}\right ) \left (b^2-x^2\right )^2 \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {\text {Subst}\left (\int \left (\frac {\left (-a^2+b^2\right )^2 (A b-a B) (a+x)^2}{b}+\frac {\left (-a^2+b^2\right ) \left (4 a A b-5 a^2 B+b^2 B\right ) (a+x)^3}{b}-\frac {2 \left (-3 a^2 A b+A b^3+5 a^3 B-3 a b^2 B\right ) (a+x)^4}{b}+\frac {2 \left (-2 a A b+5 a^2 B-b^2 B\right ) (a+x)^5}{b}+\frac {(A b-5 a B) (a+x)^6}{b}+\frac {B (a+x)^7}{b}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d} \\ & = \frac {\left (a^2-b^2\right )^2 (A b-a B) (a+b \sin (c+d x))^3}{3 b^6 d}-\frac {\left (a^2-b^2\right ) \left (4 a A b-5 a^2 B+b^2 B\right ) (a+b \sin (c+d x))^4}{4 b^6 d}+\frac {2 \left (3 a^2 A b-A b^3-5 a^3 B+3 a b^2 B\right ) (a+b \sin (c+d x))^5}{5 b^6 d}-\frac {\left (2 a A b-5 a^2 B+b^2 B\right ) (a+b \sin (c+d x))^6}{3 b^6 d}+\frac {(A b-5 a B) (a+b \sin (c+d x))^7}{7 b^6 d}+\frac {B (a+b \sin (c+d x))^8}{8 b^6 d} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.98 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {a^4 \left (3 a^4-28 a^2 b^2+210 b^4\right ) B+840 a^2 A b^6 \sin (c+d x)+420 a b^6 (2 A b+a B) \sin ^2(c+d x)+280 b^6 \left (-2 a^2 A+A b^2+2 a b B\right ) \sin ^3(c+d x)+210 b^6 \left (-4 a A b-2 a^2 B+b^2 B\right ) \sin ^4(c+d x)+168 b^6 \left (a^2 A-2 A b^2-4 a b B\right ) \sin ^5(c+d x)+140 b^6 \left (2 a A b+a^2 B-2 b^2 B\right ) \sin ^6(c+d x)+120 b^7 (A b+2 a B) \sin ^7(c+d x)+105 b^8 B \sin ^8(c+d x)}{840 b^6 d} \]
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Time = 1.32 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\frac {B \,b^{2} \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (A \,b^{2}+2 B a b \right ) \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\left (a^{2}-2 b^{2}\right ) B +2 A a b \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (-4 B a b +\left (a^{2}-2 b^{2}\right ) A \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\left (-2 a^{2}+b^{2}\right ) B -4 A a b \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (2 B a b +\left (-2 a^{2}+b^{2}\right ) A \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (2 A a b +B \,a^{2}\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{2}+A \sin \left (d x +c \right ) a^{2}}{d}\) | \(181\) |
default | \(\frac {\frac {B \,b^{2} \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (A \,b^{2}+2 B a b \right ) \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (\left (a^{2}-2 b^{2}\right ) B +2 A a b \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (-4 B a b +\left (a^{2}-2 b^{2}\right ) A \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (\left (-2 a^{2}+b^{2}\right ) B -4 A a b \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (2 B a b +\left (-2 a^{2}+b^{2}\right ) A \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (2 A a b +B \,a^{2}\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{2}+A \sin \left (d x +c \right ) a^{2}}{d}\) | \(181\) |
parallelrisch | \(\frac {\left (-16800 A a b -8400 B \,a^{2}-2520 B \,b^{2}\right ) \cos \left (2 d x +2 c \right )+\left (-6720 A a b -3360 B \,a^{2}-420 B \,b^{2}\right ) \cos \left (4 d x +4 c \right )+\left (-1120 A a b -560 B \,a^{2}+280 B \,b^{2}\right ) \cos \left (6 d x +6 c \right )+\left (11200 A \,a^{2}-560 A \,b^{2}-1120 B a b \right ) \sin \left (3 d x +3 c \right )+\left (1344 A \,a^{2}-1008 A \,b^{2}-2016 B a b \right ) \sin \left (5 d x +5 c \right )+\left (-240 A \,b^{2}-480 B a b \right ) \sin \left (7 d x +7 c \right )+105 B \cos \left (8 d x +8 c \right ) b^{2}+\left (67200 A \,a^{2}+8400 A \,b^{2}+16800 B a b \right ) \sin \left (d x +c \right )+24640 A a b +12320 B \,a^{2}+2555 B \,b^{2}}{107520 d}\) | \(226\) |
risch | \(\frac {5 \sin \left (d x +c \right ) A \,a^{2}}{8 d}+\frac {5 \sin \left (d x +c \right ) A \,b^{2}}{64 d}+\frac {5 \sin \left (d x +c \right ) B a b}{32 d}+\frac {\cos \left (8 d x +8 c \right ) B \,b^{2}}{1024 d}-\frac {\sin \left (7 d x +7 c \right ) A \,b^{2}}{448 d}-\frac {\sin \left (7 d x +7 c \right ) B a b}{224 d}-\frac {\cos \left (6 d x +6 c \right ) A a b}{96 d}-\frac {\cos \left (6 d x +6 c \right ) B \,a^{2}}{192 d}+\frac {\cos \left (6 d x +6 c \right ) B \,b^{2}}{384 d}+\frac {\sin \left (5 d x +5 c \right ) A \,a^{2}}{80 d}-\frac {3 \sin \left (5 d x +5 c \right ) A \,b^{2}}{320 d}-\frac {3 \sin \left (5 d x +5 c \right ) B a b}{160 d}-\frac {\cos \left (4 d x +4 c \right ) A a b}{16 d}-\frac {\cos \left (4 d x +4 c \right ) B \,a^{2}}{32 d}-\frac {\cos \left (4 d x +4 c \right ) B \,b^{2}}{256 d}+\frac {5 A \,a^{2} \sin \left (3 d x +3 c \right )}{48 d}-\frac {\sin \left (3 d x +3 c \right ) A \,b^{2}}{192 d}-\frac {\sin \left (3 d x +3 c \right ) B a b}{96 d}-\frac {5 \cos \left (2 d x +2 c \right ) A a b}{32 d}-\frac {5 \cos \left (2 d x +2 c \right ) B \,a^{2}}{64 d}-\frac {3 \cos \left (2 d x +2 c \right ) B \,b^{2}}{128 d}\) | \(364\) |
norman | \(\frac {\frac {2 \left (13 A \,a^{2}+4 A \,b^{2}+8 B a b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 \left (13 A \,a^{2}+4 A \,b^{2}+8 B a b \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 \left (163 A \,a^{2}+4 A \,b^{2}+8 B a b \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {2 \left (163 A \,a^{2}+4 A \,b^{2}+8 B a b \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 d}+\frac {2 \left (1883 A \,a^{2}+344 A \,b^{2}+688 B a b \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 d}+\frac {2 \left (1883 A \,a^{2}+344 A \,b^{2}+688 B a b \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 d}+\frac {2 \left (2 A a b +B \,a^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 \left (2 A a b +B \,a^{2}\right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 \left (2 A a b +B \,a^{2}+B \,b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 \left (2 A a b +B \,a^{2}+B \,b^{2}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {10 \left (8 A a b +4 B \,a^{2}+4 B \,b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 \left (26 A a b +13 B \,a^{2}-8 B \,b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 \left (26 A a b +13 B \,a^{2}-8 B \,b^{2}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {2 A \,a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 A \,a^{2} \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}\) | \(480\) |
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Time = 0.28 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.64 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {105 \, B b^{2} \cos \left (d x + c\right )^{8} - 140 \, {\left (B a^{2} + 2 \, A a b + B b^{2}\right )} \cos \left (d x + c\right )^{6} - 8 \, {\left (15 \, {\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{6} - 3 \, {\left (7 \, A a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{4} - 56 \, A a^{2} - 16 \, B a b - 8 \, A b^{2} - 4 \, {\left (7 \, A a^{2} + 2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{840 \, d} \]
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Time = 0.67 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.34 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\begin {cases} \frac {8 A a^{2} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 A a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {A a^{2} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {A a b \cos ^{6}{\left (c + d x \right )}}{3 d} + \frac {8 A b^{2} \sin ^{7}{\left (c + d x \right )}}{105 d} + \frac {4 A b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} + \frac {A b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} - \frac {B a^{2} \cos ^{6}{\left (c + d x \right )}}{6 d} + \frac {16 B a b \sin ^{7}{\left (c + d x \right )}}{105 d} + \frac {8 B a b \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} + \frac {2 B a b \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{3 d} - \frac {B b^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{6 d} - \frac {B b^{2} \cos ^{8}{\left (c + d x \right )}}{24 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\left (c \right )}\right ) \left (a + b \sin {\left (c \right )}\right )^{2} \cos ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.80 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {105 \, B b^{2} \sin \left (d x + c\right )^{8} + 120 \, {\left (2 \, B a b + A b^{2}\right )} \sin \left (d x + c\right )^{7} + 140 \, {\left (B a^{2} + 2 \, A a b - 2 \, B b^{2}\right )} \sin \left (d x + c\right )^{6} + 168 \, {\left (A a^{2} - 4 \, B a b - 2 \, A b^{2}\right )} \sin \left (d x + c\right )^{5} - 210 \, {\left (2 \, B a^{2} + 4 \, A a b - B b^{2}\right )} \sin \left (d x + c\right )^{4} + 840 \, A a^{2} \sin \left (d x + c\right ) - 280 \, {\left (2 \, A a^{2} - 2 \, B a b - A b^{2}\right )} \sin \left (d x + c\right )^{3} + 420 \, {\left (B a^{2} + 2 \, A a b\right )} \sin \left (d x + c\right )^{2}}{840 \, d} \]
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Time = 0.57 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {B b^{2} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {{\left (2 \, B a^{2} + 4 \, A a b - B b^{2}\right )} \cos \left (6 \, d x + 6 \, c\right )}{384 \, d} - \frac {{\left (8 \, B a^{2} + 16 \, A a b + B b^{2}\right )} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {{\left (10 \, B a^{2} + 20 \, A a b + 3 \, B b^{2}\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} - \frac {{\left (2 \, B a b + A b^{2}\right )} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {{\left (4 \, A a^{2} - 6 \, B a b - 3 \, A b^{2}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (20 \, A a^{2} - 2 \, B a b - A b^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac {5 \, {\left (8 \, A a^{2} + 2 \, B a b + A b^{2}\right )} \sin \left (d x + c\right )}{64 \, d} \]
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Time = 0.12 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.78 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {{\sin \left (c+d\,x\right )}^2\,\left (\frac {B\,a^2}{2}+A\,b\,a\right )+{\sin \left (c+d\,x\right )}^7\,\left (\frac {A\,b^2}{7}+\frac {2\,B\,a\,b}{7}\right )+{\sin \left (c+d\,x\right )}^3\,\left (-\frac {2\,A\,a^2}{3}+\frac {2\,B\,a\,b}{3}+\frac {A\,b^2}{3}\right )-{\sin \left (c+d\,x\right )}^5\,\left (-\frac {A\,a^2}{5}+\frac {4\,B\,a\,b}{5}+\frac {2\,A\,b^2}{5}\right )-{\sin \left (c+d\,x\right )}^4\,\left (\frac {B\,a^2}{2}+A\,a\,b-\frac {B\,b^2}{4}\right )+{\sin \left (c+d\,x\right )}^6\,\left (\frac {B\,a^2}{6}+\frac {A\,a\,b}{3}-\frac {B\,b^2}{3}\right )+\frac {B\,b^2\,{\sin \left (c+d\,x\right )}^8}{8}+A\,a^2\,\sin \left (c+d\,x\right )}{d} \]
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