Optimal. Leaf size=38 \[ \frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2} \]
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Rubi [A] time = 0.0144199, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {2734} \[ \frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos (c+d x)}{2 d}+\frac{a x}{2} \]
Antiderivative was successfully verified.
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Rule 2734
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \cos (c+d x)) \, dx &=\frac{a x}{2}+\frac{a \sin (c+d x)}{d}+\frac{a \cos (c+d x) \sin (c+d x)}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0459236, size = 32, normalized size = 0.84 \[ \frac{a (2 (c+d x)+4 \sin (c+d x)+\sin (2 (c+d x)))}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 38, normalized size = 1. \begin{align*}{\frac{1}{d} \left ( a \left ({\frac{\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) }{2}}+{\frac{dx}{2}}+{\frac{c}{2}} \right ) +a\sin \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08884, size = 46, normalized size = 1.21 \begin{align*} \frac{{\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a + 4 \, a \sin \left (d x + c\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6562, size = 72, normalized size = 1.89 \begin{align*} \frac{a d x +{\left (a \cos \left (d x + c\right ) + 2 \, a\right )} \sin \left (d x + c\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.282475, size = 66, normalized size = 1.74 \begin{align*} \begin{cases} \frac{a x \sin ^{2}{\left (c + d x \right )}}{2} + \frac{a x \cos ^{2}{\left (c + d x \right )}}{2} + \frac{a \sin{\left (c + d x \right )} \cos{\left (c + d x \right )}}{2 d} + \frac{a \sin{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \cos{\left (c \right )} + a\right ) \cos{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32989, size = 42, normalized size = 1.11 \begin{align*} \frac{1}{2} \, a x + \frac{a \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{a \sin \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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