Optimal. Leaf size=94 \[ \frac{2 a^2 (3 A+2 B) \sin (c+d x)}{3 d}+\frac{a^2 (3 A+2 B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{1}{2} a^2 x (3 A+2 B)+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d} \]
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Rubi [A] time = 0.0586307, antiderivative size = 94, 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 = {2751, 2644} \[ \frac{2 a^2 (3 A+2 B) \sin (c+d x)}{3 d}+\frac{a^2 (3 A+2 B) \sin (c+d x) \cos (c+d x)}{6 d}+\frac{1}{2} a^2 x (3 A+2 B)+\frac{B \sin (c+d x) (a \cos (c+d x)+a)^2}{3 d} \]
Antiderivative was successfully verified.
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Rule 2751
Rule 2644
Rubi steps
\begin{align*} \int (a+a \cos (c+d x))^2 (A+B \cos (c+d x)) \, dx &=\frac{B (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}+\frac{1}{3} (3 A+2 B) \int (a+a \cos (c+d x))^2 \, dx\\ &=\frac{1}{2} a^2 (3 A+2 B) x+\frac{2 a^2 (3 A+2 B) \sin (c+d x)}{3 d}+\frac{a^2 (3 A+2 B) \cos (c+d x) \sin (c+d x)}{6 d}+\frac{B (a+a \cos (c+d x))^2 \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.179524, size = 61, normalized size = 0.65 \[ \frac{a^2 (3 (8 A+7 B) \sin (c+d x)+3 (A+2 B) \sin (2 (c+d x))+18 A d x+B \sin (3 (c+d x))+12 B d x)}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 116, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({\frac{B{a}^{2} \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+{a}^{2}A \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +2\,B{a}^{2} \left ( 1/2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +1/2\,dx+c/2 \right ) +2\,{a}^{2}A\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 = 1.00034, size = 149, normalized size = 1.59 \begin{align*} \frac{3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} + 12 \,{\left (d x + c\right )} A a^{2} - 4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} B a^{2} + 6 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} B a^{2} + 24 \, A a^{2} \sin \left (d x + c\right ) + 12 \, B a^{2} \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 = 1.40573, size = 165, normalized size = 1.76 \begin{align*} \frac{3 \,{\left (3 \, A + 2 \, B\right )} a^{2} d x +{\left (2 \, B a^{2} \cos \left (d x + c\right )^{2} + 3 \,{\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right ) + 2 \,{\left (6 \, A + 5 \, B\right )} a^{2}\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 = 0.792427, size = 199, normalized size = 2.12 \begin{align*} \begin{cases} \frac{A a^{2} x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{A a^{2} x \cos ^{2}{\left (c + d x \right )}}{2} + A a^{2} x + \frac{A a^{2} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{2 A a^{2} \sin{\left (c + d x \right )}}{d} + B a^{2} x \sin ^{2}{\left (c + d x \right )} + B a^{2} x \cos ^{2}{\left (c + d x \right )} + \frac{2 B a^{2} \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{B a^{2} \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{B a^{2} \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{d} + \frac{B a^{2} \sin{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (A + B \cos{\left (c \right )}\right ) \left (a \cos{\left (c \right )} + a\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 = 1.19751, size = 115, normalized size = 1.22 \begin{align*} \frac{B a^{2} \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac{1}{2} \,{\left (3 \, A a^{2} + 2 \, B a^{2}\right )} x + \frac{{\left (A a^{2} + 2 \, B a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{{\left (8 \, A a^{2} + 7 \, B a^{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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