Optimal. Leaf size=84 \[ \frac{(3 a B+2 b C) \sin (c+d x)}{3 d}+\frac{(a C+b B) \sin (c+d x) \cos (c+d x)}{2 d}+\frac{1}{2} x (a C+b B)+\frac{b C \sin (c+d x) \cos ^2(c+d x)}{3 d} \]
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Rubi [A] time = 0.0806476, antiderivative size = 104, normalized size of antiderivative = 1.24, number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {3023, 2734} \[ \frac{\left (a^2 (-C)+3 a b B+2 b^2 C\right ) \sin (c+d x)}{3 b d}+\frac{(3 b B-a C) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{1}{2} x (a C+b B)+\frac{C \sin (c+d x) (a+b \cos (c+d x))^2}{3 b d} \]
Antiderivative was successfully verified.
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Rule 3023
Rule 2734
Rubi steps
\begin{align*} \int (a+b \cos (c+d x)) \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{C (a+b \cos (c+d x))^2 \sin (c+d x)}{3 b d}+\frac{\int (a+b \cos (c+d x)) (2 b C+(3 b B-a C) \cos (c+d x)) \, dx}{3 b}\\ &=\frac{1}{2} (b B+a C) x+\frac{\left (3 a b B-a^2 C+2 b^2 C\right ) \sin (c+d x)}{3 b d}+\frac{(3 b B-a C) \cos (c+d x) \sin (c+d x)}{6 d}+\frac{C (a+b \cos (c+d x))^2 \sin (c+d x)}{3 b d}\\ \end{align*}
Mathematica [A] time = 0.153093, size = 75, normalized size = 0.89 \[ \frac{3 (4 a B+3 b C) \sin (c+d x)+3 (a C+b B) \sin (2 (c+d x))+6 a c C+6 a C d x+6 b B c+6 b B d x+b C \sin (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 85, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({\frac{Cb \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+bB \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +aC \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +Ba\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 = 1.02136, 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 )} C a + 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B b - 4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C b + 12 \, B 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 = 1.68881, size = 149, normalized size = 1.77 \begin{align*} \frac{3 \,{\left (C a + B b\right )} d x +{\left (2 \, C b \cos \left (d x + c\right )^{2} + 6 \, B a + 4 \, C b + 3 \,{\left (C a + B 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 = 2.45925, size = 170, normalized size = 2.02 \begin{align*} \begin{cases} \frac{B a \sin{\left (c + d x \right )}}{d} + \frac{B b x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{B b x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{B b \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{C a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{C a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{C a \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{2 C b \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{C 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 (B \cos{\left (c \right )} + C \cos ^{2}{\left (c \right )}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.50569, size = 92, normalized size = 1.1 \begin{align*} \frac{1}{2} \,{\left (C a + B b\right )} x + \frac{C b \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac{{\left (C a + B b\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{{\left (4 \, B a + 3 \, C 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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