Integrand size = 29, antiderivative size = 188 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {a A \sin (c+d x)}{d}+\frac {(A b+a B) \sin ^2(c+d x)}{2 d}-\frac {(3 a A-b B) \sin ^3(c+d x)}{3 d}-\frac {3 (A b+a B) \sin ^4(c+d x)}{4 d}+\frac {3 (a A-b B) \sin ^5(c+d x)}{5 d}+\frac {(A b+a B) \sin ^6(c+d x)}{2 d}-\frac {(a A-3 b B) \sin ^7(c+d x)}{7 d}-\frac {(A b+a B) \sin ^8(c+d x)}{8 d}-\frac {b B \sin ^9(c+d x)}{9 d} \]
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Time = 0.17 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2916, 786} \[ \int \cos ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {(a B+A b) \sin ^8(c+d x)}{8 d}-\frac {(a A-3 b B) \sin ^7(c+d x)}{7 d}+\frac {(a B+A b) \sin ^6(c+d x)}{2 d}+\frac {3 (a A-b B) \sin ^5(c+d x)}{5 d}-\frac {3 (a B+A b) \sin ^4(c+d x)}{4 d}-\frac {(3 a A-b B) \sin ^3(c+d x)}{3 d}+\frac {(a B+A b) \sin ^2(c+d x)}{2 d}+\frac {a A \sin (c+d x)}{d}-\frac {b B \sin ^9(c+d x)}{9 d} \]
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Rule 786
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int (a+x) \left (A+\frac {B x}{b}\right ) \left (b^2-x^2\right )^3 \, dx,x,b \sin (c+d x)\right )}{b^7 d} \\ & = \frac {\text {Subst}\left (\int \left (a A b^6+b^5 (A b+a B) x+b^4 (-3 a A+b B) x^2-3 b^3 (A b+a B) x^3-3 b^2 (-a A+b B) x^4+3 b (A b+a B) x^5-(a A-3 b B) x^6-\frac {(A b+a B) x^7}{b}-\frac {B x^8}{b}\right ) \, dx,x,b \sin (c+d x)\right )}{b^7 d} \\ & = \frac {a A \sin (c+d x)}{d}+\frac {(A b+a B) \sin ^2(c+d x)}{2 d}-\frac {(3 a A-b B) \sin ^3(c+d x)}{3 d}-\frac {3 (A b+a B) \sin ^4(c+d x)}{4 d}+\frac {3 (a A-b B) \sin ^5(c+d x)}{5 d}+\frac {(A b+a B) \sin ^6(c+d x)}{2 d}-\frac {(a A-3 b B) \sin ^7(c+d x)}{7 d}-\frac {(A b+a B) \sin ^8(c+d x)}{8 d}-\frac {b B \sin ^9(c+d x)}{9 d} \\ \end{align*}
Time = 0.58 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.80 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\frac {\sin (c+d x) \left (2520 a A+1260 (A b+a B) \sin (c+d x)-840 (3 a A-b B) \sin ^2(c+d x)-1890 (A b+a B) \sin ^3(c+d x)+1512 (a A-b B) \sin ^4(c+d x)+1260 (A b+a B) \sin ^5(c+d x)-360 (a A-3 b B) \sin ^6(c+d x)-315 (A b+a B) \sin ^7(c+d x)-280 b B \sin ^8(c+d x)\right )}{2520 d} \]
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Time = 1.14 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(-\frac {\frac {B b \left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (A b +B a \right ) \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (a A -3 B b \right ) \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (-3 A b -3 B a \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (-3 a A +3 B b \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (3 A b +3 B a \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (3 a A -B b \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (-A b -B a \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{2}-A \sin \left (d x +c \right ) a}{d}\) | \(159\) |
default | \(-\frac {\frac {B b \left (\sin ^{9}\left (d x +c \right )\right )}{9}+\frac {\left (A b +B a \right ) \left (\sin ^{8}\left (d x +c \right )\right )}{8}+\frac {\left (a A -3 B b \right ) \left (\sin ^{7}\left (d x +c \right )\right )}{7}+\frac {\left (-3 A b -3 B a \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (-3 a A +3 B b \right ) \left (\sin ^{5}\left (d x +c \right )\right )}{5}+\frac {\left (3 A b +3 B a \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{4}+\frac {\left (3 a A -B b \right ) \left (\sin ^{3}\left (d x +c \right )\right )}{3}+\frac {\left (-A b -B a \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{2}-A \sin \left (d x +c \right ) a}{d}\) | \(159\) |
parallelrisch | \(\frac {35280 a A \sin \left (3 d x +3 c \right )-2520 A \cos \left (6 d x +6 c \right ) b -8820 A \cos \left (4 d x +4 c \right ) b -17640 A \cos \left (2 d x +2 c \right ) b -315 A \cos \left (8 d x +8 c \right ) b +720 A \sin \left (7 d x +7 c \right ) a +7056 A \sin \left (5 d x +5 c \right ) a +176400 A \sin \left (d x +c \right ) a -2520 B \cos \left (6 d x +6 c \right ) a -8820 B \cos \left (4 d x +4 c \right ) a -17640 B \cos \left (2 d x +2 c \right ) a -140 B b \sin \left (9 d x +9 c \right )-315 B \cos \left (8 d x +8 c \right ) a -900 B \sin \left (7 d x +7 c \right ) b -2016 B \sin \left (5 d x +5 c \right ) b +17640 B b \sin \left (d x +c \right )+29295 A b +29295 B a}{322560 d}\) | \(217\) |
risch | \(\frac {35 a A \sin \left (d x +c \right )}{64 d}+\frac {7 b B \sin \left (d x +c \right )}{128 d}-\frac {B b \sin \left (9 d x +9 c \right )}{2304 d}-\frac {\cos \left (8 d x +8 c \right ) A b}{1024 d}-\frac {\cos \left (8 d x +8 c \right ) B a}{1024 d}+\frac {\sin \left (7 d x +7 c \right ) a A}{448 d}-\frac {5 \sin \left (7 d x +7 c \right ) B b}{1792 d}-\frac {\cos \left (6 d x +6 c \right ) A b}{128 d}-\frac {\cos \left (6 d x +6 c \right ) B a}{128 d}+\frac {7 \sin \left (5 d x +5 c \right ) a A}{320 d}-\frac {\sin \left (5 d x +5 c \right ) B b}{160 d}-\frac {7 \cos \left (4 d x +4 c \right ) A b}{256 d}-\frac {7 \cos \left (4 d x +4 c \right ) B a}{256 d}+\frac {7 a A \sin \left (3 d x +3 c \right )}{64 d}-\frac {7 \cos \left (2 d x +2 c \right ) A b}{128 d}-\frac {7 \cos \left (2 d x +2 c \right ) B a}{128 d}\) | \(252\) |
norman | \(\frac {\frac {\left (2 A b +2 B a \right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (2 A b +2 B a \right ) \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {8 \left (3 a A +B b \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {8 \left (3 a A +B b \right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}+\frac {8 \left (17 a A -2 B b \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {8 \left (17 a A -2 B b \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {8 \left (221 a A +79 B b \right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {8 \left (221 a A +79 B b \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 d}+\frac {4 \left (4617 a A -712 B b \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 d}+\frac {14 \left (A b +B a \right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {14 \left (A b +B a \right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {14 \left (A b +B a \right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {14 \left (A b +B a \right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 \left (A b +B a \right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 \left (A b +B a \right ) \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a A \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a A \left (\tan ^{17}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}\) | \(411\) |
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Time = 0.28 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.56 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {315 \, {\left (B a + A b\right )} \cos \left (d x + c\right )^{8} + 8 \, {\left (35 \, B b \cos \left (d x + c\right )^{8} - 5 \, {\left (9 \, A a + B b\right )} \cos \left (d x + c\right )^{6} - 6 \, {\left (9 \, A a + B b\right )} \cos \left (d x + c\right )^{4} - 8 \, {\left (9 \, A a + B b\right )} \cos \left (d x + c\right )^{2} - 144 \, A a - 16 \, B b\right )} \sin \left (d x + c\right )}{2520 \, d} \]
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Time = 0.96 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.21 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx=\begin {cases} \frac {16 A a \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 A a \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 A a \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {A a \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {A b \cos ^{8}{\left (c + d x \right )}}{8 d} - \frac {B a \cos ^{8}{\left (c + d x \right )}}{8 d} + \frac {16 B b \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {8 B b \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {2 B b \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {B b \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\left (c \right )}\right ) \left (a + b \sin {\left (c \right )}\right ) \cos ^{7}{\left (c \right )} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.80 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {280 \, B b \sin \left (d x + c\right )^{9} + 315 \, {\left (B a + A b\right )} \sin \left (d x + c\right )^{8} + 360 \, {\left (A a - 3 \, B b\right )} \sin \left (d x + c\right )^{7} - 1260 \, {\left (B a + A b\right )} \sin \left (d x + c\right )^{6} - 1512 \, {\left (A a - B b\right )} \sin \left (d x + c\right )^{5} + 1890 \, {\left (B a + A b\right )} \sin \left (d x + c\right )^{4} + 840 \, {\left (3 \, A a - B b\right )} \sin \left (d x + c\right )^{3} - 2520 \, A a \sin \left (d x + c\right ) - 1260 \, {\left (B a + A b\right )} \sin \left (d x + c\right )^{2}}{2520 \, d} \]
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Time = 0.40 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.97 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {B b \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {7 \, A a \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {{\left (B a + A b\right )} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {{\left (B a + A b\right )} \cos \left (6 \, d x + 6 \, c\right )}{128 \, d} - \frac {7 \, {\left (B a + A b\right )} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {7 \, {\left (B a + A b\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {{\left (4 \, A a - 5 \, B b\right )} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {{\left (7 \, A a - 2 \, B b\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {7 \, {\left (10 \, A a + B b\right )} \sin \left (d x + c\right )}{128 \, d} \]
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Time = 0.14 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.83 \[ \int \cos ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx=-\frac {\frac {B\,b\,{\sin \left (c+d\,x\right )}^9}{9}+\left (\frac {A\,b}{8}+\frac {B\,a}{8}\right )\,{\sin \left (c+d\,x\right )}^8+\left (\frac {A\,a}{7}-\frac {3\,B\,b}{7}\right )\,{\sin \left (c+d\,x\right )}^7+\left (-\frac {A\,b}{2}-\frac {B\,a}{2}\right )\,{\sin \left (c+d\,x\right )}^6+\left (\frac {3\,B\,b}{5}-\frac {3\,A\,a}{5}\right )\,{\sin \left (c+d\,x\right )}^5+\left (\frac {3\,A\,b}{4}+\frac {3\,B\,a}{4}\right )\,{\sin \left (c+d\,x\right )}^4+\left (A\,a-\frac {B\,b}{3}\right )\,{\sin \left (c+d\,x\right )}^3+\left (-\frac {A\,b}{2}-\frac {B\,a}{2}\right )\,{\sin \left (c+d\,x\right )}^2-A\,a\,\sin \left (c+d\,x\right )}{d} \]
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