Optimal. Leaf size=81 \[ \frac{a (3 A+C) \sin (c+d x)}{3 d}+\frac{1}{2} a x (2 A+C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 a d}-\frac{a C \sin (c+d x) \cos (c+d x)}{6 d} \]
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Rubi [A] time = 0.0635726, antiderivative size = 81, 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 = {3024, 2734} \[ \frac{a (3 A+C) \sin (c+d x)}{3 d}+\frac{1}{2} a x (2 A+C)+\frac{C \sin (c+d x) (a \cos (c+d x)+a)^2}{3 a d}-\frac{a C \sin (c+d x) \cos (c+d x)}{6 d} \]
Antiderivative was successfully verified.
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Rule 3024
Rule 2734
Rubi steps
\begin{align*} \int (a+a \cos (c+d x)) \left (A+C \cos ^2(c+d x)\right ) \, dx &=\frac{C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 a d}+\frac{\int (a+a \cos (c+d x)) (a (3 A+2 C)-a C \cos (c+d x)) \, dx}{3 a}\\ &=\frac{1}{2} a (2 A+C) x+\frac{a (3 A+C) \sin (c+d x)}{3 d}-\frac{a C \cos (c+d x) \sin (c+d x)}{6 d}+\frac{C (a+a \cos (c+d x))^2 \sin (c+d x)}{3 a d}\\ \end{align*}
Mathematica [A] time = 0.13119, size = 59, normalized size = 0.73 \[ \frac{a (3 (4 A+3 C) \sin (c+d x)+12 A d x+3 C \sin (2 (c+d x))+C \sin (3 (c+d x))+6 c C+6 C d x)}{12 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 68, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({\frac{aC \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}}+aC \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +aA\sin \left ( dx+c \right ) +aA \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12617, size = 90, normalized size = 1.11 \begin{align*} \frac{12 \,{\left (d x + c\right )} A a - 4 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} C a + 3 \,{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} C a + 12 \, A a \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 = 1.42238, size = 140, normalized size = 1.73 \begin{align*} \frac{3 \,{\left (2 \, A + C\right )} a d x +{\left (2 \, C a \cos \left (d x + c\right )^{2} + 3 \, C a \cos \left (d x + c\right ) + 2 \,{\left (3 \, A + 2 \, C\right )} a\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 = 0.632881, size = 121, normalized size = 1.49 \begin{align*} \begin{cases} A a x + \frac{A a \sin{\left (c + d x \right )}}{d} + \frac{C a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{C a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{2 C a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{C a \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{C a \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} & \text{for}\: d \neq 0 \\x \left (A + C \cos ^{2}{\left (c \right )}\right ) \left (a \cos{\left (c \right )} + a\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18375, size = 86, normalized size = 1.06 \begin{align*} \frac{1}{2} \,{\left (2 \, A a + C a\right )} x + \frac{C a \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac{C a \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{{\left (4 \, A a + 3 \, C a\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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