Optimal. Leaf size=33 \[ -\frac{\tan ^{-1}\left (\frac{\sin (c+d x)}{3-\cos (c+d x)}\right )}{2 d}-\frac{x}{4} \]
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Rubi [A] time = 0.0127241, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {2658} \[ -\frac{\tan ^{-1}\left (\frac{\sin (c+d x)}{3-\cos (c+d x)}\right )}{2 d}-\frac{x}{4} \]
Antiderivative was successfully verified.
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Rule 2658
Rubi steps
\begin{align*} \int \frac{1}{-5+3 \cos (c+d x)} \, dx &=-\frac{x}{4}-\frac{\tan ^{-1}\left (\frac{\sin (c+d x)}{3-\cos (c+d x)}\right )}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0244086, size = 20, normalized size = 0.61 \[ -\frac{\tan ^{-1}\left (2 \tan \left (\frac{1}{2} (c+d x)\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 18, normalized size = 0.6 \begin{align*} -{\frac{1}{2\,d}\arctan \left ( 2\,\tan \left ( 1/2\,dx+c/2 \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.87304, size = 32, normalized size = 0.97 \begin{align*} -\frac{\arctan \left (\frac{2 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61104, size = 72, normalized size = 2.18 \begin{align*} \frac{\arctan \left (\frac{5 \, \cos \left (d x + c\right ) - 3}{4 \, \sin \left (d x + c\right )}\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.776051, size = 42, normalized size = 1.27 \begin{align*} \begin{cases} - \frac{\operatorname{atan}{\left (2 \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} \right )} + \pi \left \lfloor{\frac{\frac{c}{2} + \frac{d x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor }{2 d} & \text{for}\: d \neq 0 \\\frac{x}{3 \cos{\left (c \right )} - 5} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22575, size = 41, normalized size = 1.24 \begin{align*} -\frac{d x + c - 2 \, \arctan \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) - 3}\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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