Optimal. Leaf size=191 \[ \frac{\sin (c+d x) \left (4 a^2 b B+a^3 (-C)+4 a b^2 (3 A+2 C)+4 b^3 B\right )}{6 b d}+\frac{\sin (c+d x) \cos (c+d x) \left (-2 a^2 C+8 a b B+12 A b^2+9 b^2 C\right )}{24 d}+\frac{1}{8} x \left (4 a^2 (2 A+C)+8 a b B+b^2 (4 A+3 C)\right )+\frac{(4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d} \]
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Rubi [A] time = 0.226021, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3023, 2753, 2734} \[ \frac{\sin (c+d x) \left (4 a^2 b B+a^3 (-C)+4 a b^2 (3 A+2 C)+4 b^3 B\right )}{6 b d}+\frac{\sin (c+d x) \cos (c+d x) \left (-2 a^2 C+8 a b B+12 A b^2+9 b^2 C\right )}{24 d}+\frac{1}{8} x \left (4 a^2 (2 A+C)+8 a b B+b^2 (4 A+3 C)\right )+\frac{(4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 b d}+\frac{C \sin (c+d x) (a+b \cos (c+d x))^3}{4 b d} \]
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 (A+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 (b (4 A+3 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 (12 a A+8 b B+7 a C)+\left (12 A b^2+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 (2 A+C)+b^2 (4 A+3 C)\right ) x+\frac{\left (4 a^2 b B+4 b^3 B-a^3 C+4 a b^2 (3 A+2 C)\right ) \sin (c+d x)}{6 b d}+\frac{\left (12 A b^2+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 = 0.592331, size = 137, normalized size = 0.72 \[ \frac{12 (c+d x) \left (4 a^2 (2 A+C)+8 a b B+b^2 (4 A+3 C)\right )+24 \sin (c+d x) \left (4 a^2 B+8 a A b+6 a b C+3 b^2 B\right )+24 \sin (2 (c+d x)) \left (a^2 C+2 a b B+A b^2+b^2 C\right )+8 b (2 a C+b B) \sin (3 (c+d x))+3 b^2 C \sin (4 (c+d x))}{96 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 200, normalized size = 1.1 \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}}+A{b}^{2} \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +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 ) +2\,aAb\sin \left ( dx+c \right ) +{a}^{2}B\sin \left ( dx+c \right ) +A{a}^{2} \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.982046, size = 252, normalized size = 1.32 \begin{align*} \frac{96 \,{\left (d x + c\right )} A a^{2} + 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 + 24 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{2} - 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 ) + 192 \, A a b \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 = 1.68043, size = 320, normalized size = 1.68 \begin{align*} \frac{3 \,{\left (4 \,{\left (2 \, A + C\right )} a^{2} + 8 \, B a b +{\left (4 \, A + 3 \, C\right )} b^{2}\right )} d x +{\left (6 \, C b^{2} \cos \left (d x + c\right )^{3} + 24 \, B a^{2} + 16 \,{\left (3 \, A + 2 \, C\right )} 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 +{\left (4 \, A + 3 \, C\right )} 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 = 1.72981, size = 420, normalized size = 2.2 \begin{align*} \begin{cases} A a^{2} x + \frac{2 A a b \sin{\left (c + d x \right )}}{d} + \frac{A b^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{A b^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{A b^{2} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \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 (A + 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 = 1.18103, size = 197, normalized size = 1.03 \begin{align*} \frac{C b^{2} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac{1}{8} \,{\left (8 \, A a^{2} + 4 \, C a^{2} + 8 \, B a b + 4 \, A b^{2} + 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 + A b^{2} + C b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{{\left (4 \, B a^{2} + 8 \, A a b + 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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