Optimal. Leaf size=188 \[ -\frac {(a B+A b) \sin ^8(c+d x)}{8 d}-\frac {(a A-3 b B) \sin ^7(c+d x)}{7 d}+\frac {(a B+A b) \sin ^6(c+d x)}{2 d}+\frac {3 (a A-b B) \sin ^5(c+d x)}{5 d}-\frac {3 (a B+A b) \sin ^4(c+d x)}{4 d}-\frac {(3 a A-b B) \sin ^3(c+d x)}{3 d}+\frac {(a B+A b) \sin ^2(c+d x)}{2 d}+\frac {a A \sin (c+d x)}{d}-\frac {b B \sin ^9(c+d x)}{9 d} \]
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Rubi [A] time = 0.24, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2837, 772} \[ -\frac {(a B+A b) \sin ^8(c+d x)}{8 d}-\frac {(a A-3 b B) \sin ^7(c+d x)}{7 d}+\frac {(a B+A b) \sin ^6(c+d x)}{2 d}+\frac {3 (a A-b B) \sin ^5(c+d x)}{5 d}-\frac {3 (a B+A b) \sin ^4(c+d x)}{4 d}-\frac {(3 a A-b B) \sin ^3(c+d x)}{3 d}+\frac {(a B+A b) \sin ^2(c+d x)}{2 d}+\frac {a A \sin (c+d x)}{d}-\frac {b B \sin ^9(c+d x)}{9 d} \]
Antiderivative was successfully verified.
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Rule 772
Rule 2837
Rubi steps
\begin {align*} \int \cos ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx &=\frac {\operatorname {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 {\operatorname {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*}
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Mathematica [A] time = 0.80, size = 151, normalized size = 0.80 \[ \frac {\sin (c+d x) \left (-315 (a B+A b) \sin ^7(c+d x)-360 (a A-3 b B) \sin ^6(c+d x)+1260 (a B+A b) \sin ^5(c+d x)+1512 (a A-b B) \sin ^4(c+d x)-1890 (a B+A b) \sin ^3(c+d x)-840 (3 a A-b B) \sin ^2(c+d x)+1260 (a B+A b) \sin (c+d x)+2520 a A-280 b B \sin ^8(c+d x)\right )}{2520 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 106, normalized size = 0.56 \[ -\frac {315 \, {\left (B a + A b\right )} \cos \left (d x + c\right )^{8} + 8 \, {\left (35 \, B b \cos \left (d x + c\right )^{8} - 5 \, {\left (9 \, A a + B b\right )} \cos \left (d x + c\right )^{6} - 6 \, {\left (9 \, A a + B b\right )} \cos \left (d x + c\right )^{4} - 8 \, {\left (9 \, A a + B b\right )} \cos \left (d x + c\right )^{2} - 144 \, A a - 16 \, B b\right )} \sin \left (d x + c\right )}{2520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 182, normalized size = 0.97 \[ -\frac {B b \sin \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {7 \, A a \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {{\left (B a + A b\right )} \cos \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {{\left (B a + A b\right )} \cos \left (6 \, d x + 6 \, c\right )}{128 \, d} - \frac {7 \, {\left (B a + A b\right )} \cos \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {7 \, {\left (B a + A b\right )} \cos \left (2 \, d x + 2 \, c\right )}{128 \, d} + \frac {{\left (4 \, A a - 5 \, B b\right )} \sin \left (7 \, d x + 7 \, c\right )}{1792 \, d} + \frac {{\left (7 \, A a - 2 \, B b\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {7 \, {\left (10 \, A a + B b\right )} \sin \left (d x + c\right )}{128 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.46, size = 128, normalized size = 0.68 \[ \frac {B b \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{8}\left (d x +c \right )\right )}{9}+\frac {\left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{63}\right )-\frac {A b \left (\cos ^{8}\left (d x +c \right )\right )}{8}-\frac {a B \left (\cos ^{8}\left (d x +c \right )\right )}{8}+\frac {a A \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 151, normalized size = 0.80 \[ -\frac {280 \, B b \sin \left (d x + c\right )^{9} + 315 \, {\left (B a + A b\right )} \sin \left (d x + c\right )^{8} + 360 \, {\left (A a - 3 \, B b\right )} \sin \left (d x + c\right )^{7} - 1260 \, {\left (B a + A b\right )} \sin \left (d x + c\right )^{6} - 1512 \, {\left (A a - B b\right )} \sin \left (d x + c\right )^{5} + 1890 \, {\left (B a + A b\right )} \sin \left (d x + c\right )^{4} + 840 \, {\left (3 \, A a - B b\right )} \sin \left (d x + c\right )^{3} - 2520 \, A a \sin \left (d x + c\right ) - 1260 \, {\left (B a + A b\right )} \sin \left (d x + c\right )^{2}}{2520 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 156, normalized size = 0.83 \[ -\frac {\frac {B\,b\,{\sin \left (c+d\,x\right )}^9}{9}+\left (\frac {A\,b}{8}+\frac {B\,a}{8}\right )\,{\sin \left (c+d\,x\right )}^8+\left (\frac {A\,a}{7}-\frac {3\,B\,b}{7}\right )\,{\sin \left (c+d\,x\right )}^7+\left (-\frac {A\,b}{2}-\frac {B\,a}{2}\right )\,{\sin \left (c+d\,x\right )}^6+\left (\frac {3\,B\,b}{5}-\frac {3\,A\,a}{5}\right )\,{\sin \left (c+d\,x\right )}^5+\left (\frac {3\,A\,b}{4}+\frac {3\,B\,a}{4}\right )\,{\sin \left (c+d\,x\right )}^4+\left (A\,a-\frac {B\,b}{3}\right )\,{\sin \left (c+d\,x\right )}^3+\left (-\frac {A\,b}{2}-\frac {B\,a}{2}\right )\,{\sin \left (c+d\,x\right )}^2-A\,a\,\sin \left (c+d\,x\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 15.63, size = 228, normalized size = 1.21 \[ \begin {cases} \frac {16 A a \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 A a \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 A a \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {A a \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {A b \cos ^{8}{\left (c + d x \right )}}{8 d} - \frac {B a \cos ^{8}{\left (c + d x \right )}}{8 d} + \frac {16 B b \sin ^{9}{\left (c + d x \right )}}{315 d} + \frac {8 B b \sin ^{7}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{35 d} + \frac {2 B b \sin ^{5}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{5 d} + \frac {B b \sin ^{3}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{3 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\relax (c )}\right ) \left (a + b \sin {\relax (c )}\right ) \cos ^{7}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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