Optimal. Leaf size=29 \[ \frac{x}{a}-\frac{\sin (c+d x)}{d (a \cos (c+d x)+a)} \]
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Rubi [A] time = 0.0339496, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2735, 2648} \[ \frac{x}{a}-\frac{\sin (c+d x)}{d (a \cos (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2648
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{a+a \cos (c+d x)} \, dx &=\frac{x}{a}-\int \frac{1}{a+a \cos (c+d x)} \, dx\\ &=\frac{x}{a}-\frac{\sin (c+d x)}{d (a+a \cos (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.0734609, size = 57, normalized size = 1.97 \[ \frac{2 \cos \left (\frac{1}{2} (c+d x)\right ) \left (d x \cos \left (\frac{1}{2} (c+d x)\right )-\sec \left (\frac{c}{2}\right ) \sin \left (\frac{d x}{2}\right )\right )}{a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 37, normalized size = 1.3 \begin{align*} -{\frac{1}{da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.73606, size = 66, normalized size = 2.28 \begin{align*} \frac{\frac{2 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a} - \frac{\sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55292, size = 89, normalized size = 3.07 \begin{align*} \frac{d x \cos \left (d x + c\right ) + d x - \sin \left (d x + c\right )}{a d \cos \left (d x + c\right ) + a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.00107, size = 27, normalized size = 0.93 \begin{align*} \begin{cases} \frac{x}{a} - \frac{\tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{a d} & \text{for}\: d \neq 0 \\\frac{x \cos{\left (c \right )}}{a \cos{\left (c \right )} + a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.39856, size = 38, normalized size = 1.31 \begin{align*} \frac{\frac{d x + c}{a} - \frac{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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