Optimal. Leaf size=170 \[ \frac{\left (4 a^2 b B+a^3 (-C)+8 a b^2 C+4 b^3 B\right ) \sin (c+d x)}{6 b d}+\frac{\left (-2 a^2 C+8 a b B+9 b^2 C\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left (4 a^2 C+8 a b B+3 b^2 C\right )+\frac{(4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d} \]
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Rubi [A] time = 0.193054, antiderivative size = 170, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.094, Rules used = {3023, 2753, 2734} \[ \frac{\left (4 a^2 b B+a^3 (-C)+8 a b^2 C+4 b^3 B\right ) \sin (c+d x)}{6 b d}+\frac{\left (-2 a^2 C+8 a b B+9 b^2 C\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left (4 a^2 C+8 a b B+3 b^2 C\right )+\frac{(4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d} \]
Antiderivative was successfully verified.
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Rule 3023
Rule 2753
Rule 2734
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^2 \left (B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx &=\frac{C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 b d}+\frac{\int (a+b \cos (c+d x))^2 (3 b C+(4 b B-a C) \cos (c+d x)) \, dx}{4 b}\\ &=\frac{(4 b B-a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 b d}+\frac{C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 b d}+\frac{\int (a+b \cos (c+d x)) \left (b (8 b B+7 a C)+\left (8 a b B-2 a^2 C+9 b^2 C\right ) \cos (c+d x)\right ) \, dx}{12 b}\\ &=\frac{1}{8} \left (8 a b B+4 a^2 C+3 b^2 C\right ) x+\frac{\left (4 a^2 b B+4 b^3 B-a^3 C+8 a b^2 C\right ) \sin (c+d x)}{6 b d}+\frac{\left (8 a b B-2 a^2 C+9 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac{(4 b B-a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 b d}+\frac{C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 b d}\\ \end{align*}
Mathematica [A] time = 0.433045, size = 118, normalized size = 0.69 \[ \frac{12 (c+d x) \left (4 a^2 C+8 a b B+3 b^2 C\right )+24 \left (4 a^2 B+6 a b C+3 b^2 B\right ) \sin (c+d x)+24 \left (a^2 C+2 a b B+b^2 C\right ) \sin (2 (c+d x))+8 b (2 a C+b B) \sin (3 (c+d x))+3 b^2 C \sin (4 (c+d x))}{96 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 152, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({b}^{2}C \left ({\frac{\sin \left ( dx+c \right ) }{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +{\frac{{b}^{2}B \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+{\frac{2\,abC \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+2\,abB \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +{a}^{2}C \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +{a}^{2}B\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 = 1.03815, size = 192, normalized size = 1.13 \begin{align*} \frac{24 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a^{2} + 48 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b - 64 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a b - 32 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B b^{2} + 3 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} C b^{2} + 96 \, B a^{2} \sin \left (d x + c\right )}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50271, size = 274, normalized size = 1.61 \begin{align*} \frac{3 \,{\left (4 \, C a^{2} + 8 \, B a b + 3 \, C b^{2}\right )} d x +{\left (6 \, C b^{2} \cos \left (d x + c\right )^{3} + 24 \, B a^{2} + 32 \, C a b + 16 \, B b^{2} + 8 \,{\left (2 \, C a b + B b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \,{\left (4 \, C a^{2} + 8 \, B a b + 3 \, C b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.94598, size = 340, normalized size = 2. \begin{align*} \begin{cases} \frac{B a^{2} \sin{\left (c + d x \right )}}{d} + B a b x \sin ^{2}{\left (c + d x \right )} + B a b x \cos ^{2}{\left (c + d x \right )} + \frac{B a b \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{d} + \frac{2 B b^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{B b^{2} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{C a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{C a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{C a^{2} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{4 C a b \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{2 C a b \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{3 C b^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{3 C b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{3 C b^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{3 C b^{2} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{5 C b^{2} \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text{for}\: d \neq 0 \\x \left (a + b \cos{\left (c \right )}\right )^{2} \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 = 1.52493, size = 167, normalized size = 0.98 \begin{align*} \frac{C b^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac{1}{8} \,{\left (4 \, C a^{2} + 8 \, B a b + 3 \, C b^{2}\right )} x + \frac{{\left (2 \, C a b + B b^{2}\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac{{\left (C a^{2} + 2 \, B a b + C b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{{\left (4 \, B a^{2} + 6 \, C a b + 3 \, B b^{2}\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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