Optimal. Leaf size=84 \[ \frac{(3 a A+2 b B) \sin (c+d x)}{3 d}+\frac{(a B+A b) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a B+A b)+\frac{b B \sin (c+d x) \cos ^2(c+d x)}{3 d} \]
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Rubi [A] time = 0.0896968, antiderivative size = 84, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2968, 3023, 2734} \[ \frac{(3 a A+2 b B) \sin (c+d x)}{3 d}+\frac{(a B+A b) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a B+A b)+\frac{b B \sin (c+d x) \cos ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 2968
Rule 3023
Rule 2734
Rubi steps
\begin{align*} \int \cos (c+d x) (a+b \cos (c+d x)) (A+B \cos (c+d x)) \, dx &=\int \cos (c+d x) \left (a A+(A b+a B) \cos (c+d x)+b B \cos ^2(c+d x)\right ) \, dx\\ &=\frac{b B \cos ^2(c+d x) \sin (c+d x)}{3 d}+\frac{1}{3} \int \cos (c+d x) (3 a A+2 b B+3 (A b+a B) \cos (c+d x)) \, dx\\ &=\frac{1}{2} (A b+a B) x+\frac{(3 a A+2 b B) \sin (c+d x)}{3 d}+\frac{(A b+a B) \cos (c+d x) \sin (c+d x)}{2 d}+\frac{b B \cos ^2(c+d x) \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.151935, size = 75, normalized size = 0.89 \[ \frac{3 (4 a A+3 b B) \sin (c+d x)+3 (a B+A b) \sin (2 (c+d x))+6 a B c+6 a B d x+6 A b c+6 A b d x+b B \sin (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 85, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({\frac{Bb \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+Ab \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +aB \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +aA\sin \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.11735, size = 107, normalized size = 1.27 \begin{align*} \frac{3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a + 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b - 4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B b + 12 \, A a \sin \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.36506, size = 149, normalized size = 1.77 \begin{align*} \frac{3 \,{\left (B a + A b\right )} d x +{\left (2 \, B b \cos \left (d x + c\right )^{2} + 6 \, A a + 4 \, B b + 3 \,{\left (B a + A b\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.668947, size = 168, normalized size = 2. \begin{align*} \begin{cases} \frac{A a \sin{\left (c + d x \right )}}{d} + \frac{A b x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{A b x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{A b \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{B a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{B a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{B a \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{2 B b \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{B b \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (A + B \cos{\left (c \right )}\right ) \left (a + b \cos{\left (c \right )}\right ) \cos{\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 = 1.32706, size = 92, normalized size = 1.1 \begin{align*} \frac{1}{2} \,{\left (B a + A b\right )} x + \frac{B b \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac{{\left (B a + A b\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{{\left (4 \, A a + 3 \, B b\right )} \sin \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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