Optimal. Leaf size=176 \[ \frac{b \left (16 a^2 b B-13 a^3 C+8 a b^2 C+4 b^3 B\right ) \sin (c+d x)}{6 d}+\frac{b^2 \left (-14 a^2 C+20 a b B+9 b^2 C\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left (8 a^3 b B-8 a^4 C+12 a b^3 B+3 b^4 C\right )+\frac{b (4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{b C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d} \]
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Rubi [A] time = 0.350412, antiderivative size = 176, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 48, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {3015, 2753, 2734} \[ \frac{b \left (16 a^2 b B-13 a^3 C+8 a b^2 C+4 b^3 B\right ) \sin (c+d x)}{6 d}+\frac{b^2 \left (-14 a^2 C+20 a b B+9 b^2 C\right ) \sin (c+d x) \cos (c+d x)}{24 d}+\frac{1}{8} x \left (8 a^3 b B-8 a^4 C+12 a b^3 B+3 b^4 C\right )+\frac{b (4 b B-a C) \sin (c+d x) (a+b \cos (c+d x))^2}{12 d}+\frac{b C \sin (c+d x) (a+b \cos (c+d x))^3}{4 d} \]
Antiderivative was successfully verified.
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Rule 3015
Rule 2753
Rule 2734
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^2 \left (a b B-a^2 C+b^2 B \cos (c+d x)+b^2 C \cos ^2(c+d x)\right ) \, dx &=\frac{\int (a+b \cos (c+d x))^3 \left (b^2 (b B-a C)+b^3 C \cos (c+d x)\right ) \, dx}{b^2}\\ &=\frac{b C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{\int (a+b \cos (c+d x))^2 \left (b^2 \left (3 b^2 C+4 a (b B-a C)\right )+b^3 (4 b B-a C) \cos (c+d x)\right ) \, dx}{4 b^2}\\ &=\frac{b (4 b B-a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac{b C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}+\frac{\int (a+b \cos (c+d x)) \left (b^2 \left (12 a^2 b B+8 b^3 B-12 a^3 C+7 a b^2 C\right )+b^3 \left (20 a b B-14 a^2 C+9 b^2 C\right ) \cos (c+d x)\right ) \, dx}{12 b^2}\\ &=\frac{1}{8} \left (8 a^3 b B+12 a b^3 B-8 a^4 C+3 b^4 C\right ) x+\frac{b \left (16 a^2 b B+4 b^3 B-13 a^3 C+8 a b^2 C\right ) \sin (c+d x)}{6 d}+\frac{b^2 \left (20 a b B-14 a^2 C+9 b^2 C\right ) \cos (c+d x) \sin (c+d x)}{24 d}+\frac{b (4 b B-a C) (a+b \cos (c+d x))^2 \sin (c+d x)}{12 d}+\frac{b C (a+b \cos (c+d x))^3 \sin (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.578179, size = 134, normalized size = 0.76 \[ \frac{-12 (c+d x) \left (-8 a^3 b B+8 a^4 C-12 a b^3 B-3 b^4 C\right )+24 b \left (12 a^2 b B-8 a^3 C+6 a b^2 C+3 b^3 B\right ) \sin (c+d x)+24 b^3 (3 a B+b C) \sin (2 (c+d x))+8 b^3 (2 a C+b B) \sin (3 (c+d x))+3 b^4 C \sin (4 (c+d x))}{96 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 168, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( C{b}^{4} \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}^{4}B \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+{\frac{2\,Ca{b}^{3} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+3\,a{b}^{3}B \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +3\,{a}^{2}{b}^{2}B\sin \left ( dx+c \right ) -2\,{a}^{3}bC\sin \left ( dx+c \right ) +{a}^{3}bB \left ( dx+c \right ) -{a}^{4}C \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994064, size = 219, normalized size = 1.24 \begin{align*} -\frac{96 \,{\left (d x + c\right )} C a^{4} - 96 \,{\left (d x + c\right )} B a^{3} b - 72 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b^{3} + 64 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a b^{3} + 32 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B b^{4} - 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^{4} + 192 \, C a^{3} b \sin \left (d x + c\right ) - 288 \, B a^{2} b^{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 = 1.66888, size = 311, normalized size = 1.77 \begin{align*} -\frac{3 \,{\left (8 \, C a^{4} - 8 \, B a^{3} b - 12 \, B a b^{3} - 3 \, C b^{4}\right )} d x -{\left (6 \, C b^{4} \cos \left (d x + c\right )^{3} - 48 \, C a^{3} b + 72 \, B a^{2} b^{2} + 32 \, C a b^{3} + 16 \, B b^{4} + 8 \,{\left (2 \, C a b^{3} + B b^{4}\right )} \cos \left (d x + c\right )^{2} + 9 \,{\left (4 \, B a b^{3} + C b^{4}\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 = 1.71629, size = 357, normalized size = 2.03 \begin{align*} \begin{cases} B a^{3} b x + \frac{3 B a^{2} b^{2} \sin{\left (c + d x \right )}}{d} + \frac{3 B a b^{3} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{3 B a b^{3} x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{3 B a b^{3} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{2 B b^{4} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{B b^{4} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} - C a^{4} x - \frac{2 C a^{3} b \sin{\left (c + d x \right )}}{d} + \frac{4 C a b^{3} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{2 C a b^{3} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{3 C b^{4} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{3 C b^{4} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{3 C b^{4} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{3 C b^{4} \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{5 C b^{4} \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 a b + B b^{2} \cos{\left (c \right )} - C a^{2} + C b^{2} \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 = 1.14865, size = 194, normalized size = 1.1 \begin{align*} \frac{C b^{4} \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} - \frac{1}{8} \,{\left (8 \, C a^{4} - 8 \, B a^{3} b - 12 \, B a b^{3} - 3 \, C b^{4}\right )} x + \frac{{\left (2 \, C a b^{3} + B b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac{{\left (3 \, B a b^{3} + C b^{4}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} - \frac{{\left (8 \, C a^{3} b - 12 \, B a^{2} b^{2} - 6 \, C a b^{3} - 3 \, B b^{4}\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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