Optimal. Leaf size=143 \[ \frac{(a B+A b) \sin ^6(c+d x)}{6 d}+\frac{(a A-2 b B) \sin ^5(c+d x)}{5 d}-\frac{(a B+A b) \sin ^4(c+d x)}{2 d}-\frac{(2 a A-b B) \sin ^3(c+d x)}{3 d}+\frac{(a B+A b) \sin ^2(c+d x)}{2 d}+\frac{a A \sin (c+d x)}{d}+\frac{b B \sin ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.170796, antiderivative size = 143, normalized size of antiderivative = 1., 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} \[ \frac{(a B+A b) \sin ^6(c+d x)}{6 d}+\frac{(a A-2 b B) \sin ^5(c+d x)}{5 d}-\frac{(a B+A b) \sin ^4(c+d x)}{2 d}-\frac{(2 a A-b B) \sin ^3(c+d x)}{3 d}+\frac{(a B+A b) \sin ^2(c+d x)}{2 d}+\frac{a A \sin (c+d x)}{d}+\frac{b B \sin ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 772
Rubi steps
\begin{align*} \int \cos ^5(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 )^2 \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{\operatorname{Subst}\left (\int \left (a A b^4+b^3 (A b+a B) x+b^2 (-2 a A+b B) x^2-2 b (A b+a B) x^3+(a A-2 b B) x^4+\frac{(A b+a B) x^5}{b}+\frac{B x^6}{b}\right ) \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac{a A \sin (c+d x)}{d}+\frac{(A b+a B) \sin ^2(c+d x)}{2 d}-\frac{(2 a A-b B) \sin ^3(c+d x)}{3 d}-\frac{(A b+a B) \sin ^4(c+d x)}{2 d}+\frac{(a A-2 b B) \sin ^5(c+d x)}{5 d}+\frac{(A b+a B) \sin ^6(c+d x)}{6 d}+\frac{b B \sin ^7(c+d x)}{7 d}\\ \end{align*}
Mathematica [A] time = 0.291036, size = 116, normalized size = 0.81 \[ \frac{\sin (c+d x) \left (35 (a B+A b) \sin ^5(c+d x)+42 (a A-2 b B) \sin ^4(c+d x)-105 (a B+A b) \sin ^3(c+d x)-70 (2 a A-b B) \sin ^2(c+d x)+105 (a B+A b) \sin (c+d x)+210 a A+30 b B \sin ^6(c+d x)\right )}{210 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.063, size = 108, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( Bb \left ( -{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{7}}+{\frac{\sin \left ( dx+c \right ) }{35} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) -{\frac{Ab \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{6}}-{\frac{aB \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{6}}+{\frac{A\sin \left ( dx+c \right ) a}{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00513, size = 157, normalized size = 1.1 \begin{align*} \frac{30 \, B b \sin \left (d x + c\right )^{7} + 35 \,{\left (B a + A b\right )} \sin \left (d x + c\right )^{6} + 42 \,{\left (A a - 2 \, B b\right )} \sin \left (d x + c\right )^{5} - 105 \,{\left (B a + A b\right )} \sin \left (d x + c\right )^{4} - 70 \,{\left (2 \, A a - B b\right )} \sin \left (d x + c\right )^{3} + 210 \, A a \sin \left (d x + c\right ) + 105 \,{\left (B a + A b\right )} \sin \left (d x + c\right )^{2}}{210 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.34837, size = 224, normalized size = 1.57 \begin{align*} -\frac{35 \,{\left (B a + A b\right )} \cos \left (d x + c\right )^{6} + 2 \,{\left (15 \, B b \cos \left (d x + c\right )^{6} - 3 \,{\left (7 \, A a + B b\right )} \cos \left (d x + c\right )^{4} - 4 \,{\left (7 \, A a + B b\right )} \cos \left (d x + c\right )^{2} - 56 \, A a - 8 \, B b\right )} \sin \left (d x + c\right )}{210 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.3374, size = 178, normalized size = 1.24 \begin{align*} \begin{cases} \frac{8 A a \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{4 A a \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{A a \sin{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac{A b \cos ^{6}{\left (c + d x \right )}}{6 d} - \frac{B a \cos ^{6}{\left (c + d x \right )}}{6 d} + \frac{8 B b \sin ^{7}{\left (c + d x \right )}}{105 d} + \frac{4 B b \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{15 d} + \frac{B b \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\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 ^{5}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23733, size = 196, normalized size = 1.37 \begin{align*} -\frac{B b \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac{{\left (B a + A b\right )} \cos \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac{{\left (B a + A b\right )} \cos \left (4 \, d x + 4 \, c\right )}{32 \, d} - \frac{5 \,{\left (B a + A b\right )} \cos \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{{\left (4 \, A a - 3 \, B b\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac{{\left (20 \, A a - B b\right )} \sin \left (3 \, d x + 3 \, c\right )}{192 \, d} + \frac{5 \,{\left (8 \, A a + B b\right )} \sin \left (d x + c\right )}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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