Optimal. Leaf size=107 \[ \frac{2 \left (a^2 B+3 a A b+b^2 B\right ) \sin (c+d x)}{3 d}+\frac{1}{2} x \left (2 a^2 A+2 a b B+A b^2\right )+\frac{b (2 a B+3 A b) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^2}{3 d} \]
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Rubi [A] time = 0.0935754, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2753, 2734} \[ \frac{2 \left (a^2 B+3 a A b+b^2 B\right ) \sin (c+d x)}{3 d}+\frac{1}{2} x \left (2 a^2 A+2 a b B+A b^2\right )+\frac{b (2 a B+3 A b) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{B \sin (c+d x) (a+b \cos (c+d x))^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 2753
Rule 2734
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx &=\frac{B (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac{1}{3} \int (a+b \cos (c+d x)) (3 a A+2 b B+(3 A b+2 a B) \cos (c+d x)) \, dx\\ &=\frac{1}{2} \left (2 a^2 A+A b^2+2 a b B\right ) x+\frac{2 \left (3 a A b+a^2 B+b^2 B\right ) \sin (c+d x)}{3 d}+\frac{b (3 A b+2 a B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac{B (a+b \cos (c+d x))^2 \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.213408, size = 90, normalized size = 0.84 \[ \frac{6 (c+d x) \left (2 a^2 A+2 a b B+A b^2\right )+3 \left (4 a^2 B+8 a A b+3 b^2 B\right ) \sin (c+d x)+3 b (2 a B+A b) \sin (2 (c+d x))+b^2 B \sin (3 (c+d x))}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.039, size = 114, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({\frac{{b}^{2}B \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\,Bab \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +2\,Aab\sin \left ( dx+c \right ) +B{a}^{2}\sin \left ( dx+c \right ) +{a}^{2}A \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0408, size = 146, normalized size = 1.36 \begin{align*} \frac{12 \,{\left (d x + c\right )} A a^{2} + 6 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a b + 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A b^{2} - 4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B b^{2} + 12 \, B a^{2} \sin \left (d x + c\right ) + 24 \, A a b \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 = 1.33801, size = 201, normalized size = 1.88 \begin{align*} \frac{3 \,{\left (2 \, A a^{2} + 2 \, B a b + A b^{2}\right )} d x +{\left (2 \, B b^{2} \cos \left (d x + c\right )^{2} + 6 \, B a^{2} + 12 \, A a b + 4 \, B b^{2} + 3 \,{\left (2 \, B a b + A b^{2}\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 = 0.839749, size = 199, normalized size = 1.86 \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} & \text{for}\: d \neq 0 \\x \left (A + B \cos{\left (c \right )}\right ) \left (a + b \cos{\left (c \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.3853, size = 126, normalized size = 1.18 \begin{align*} \frac{B b^{2} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac{1}{2} \,{\left (2 \, A a^{2} + 2 \, B a b + A b^{2}\right )} x + \frac{{\left (2 \, B a b + A b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{{\left (4 \, B a^{2} + 8 \, A 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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