Integrand size = 12, antiderivative size = 31 \[ \int \frac {1}{-5-3 \cos (c+d x)} \, dx=-\frac {x}{4}+\frac {\arctan \left (\frac {\sin (c+d x)}{3+\cos (c+d x)}\right )}{2 d} \]
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Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2737} \[ \int \frac {1}{-5-3 \cos (c+d x)} \, dx=\frac {\arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)+3}\right )}{2 d}-\frac {x}{4} \]
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Rule 2737
Rubi steps \begin{align*} \text {integral}& = -\frac {x}{4}+\frac {\arctan \left (\frac {\sin (c+d x)}{3+\cos (c+d x)}\right )}{2 d} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.65 \[ \int \frac {1}{-5-3 \cos (c+d x)} \, dx=\frac {\arctan \left (2 \cot \left (\frac {1}{2} (c+d x)\right )\right )}{2 d} \]
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Time = 0.56 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.58
method | result | size |
derivativedivides | \(-\frac {\arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{2 d}\) | \(18\) |
default | \(-\frac {\arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{2 d}\) | \(18\) |
parallelrisch | \(\frac {i \left (\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 i\right )-\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 i\right )\right )}{4 d}\) | \(36\) |
risch | \(-\frac {i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+3\right )}{4 d}+\frac {i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {1}{3}\right )}{4 d}\) | \(38\) |
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Time = 0.29 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \frac {1}{-5-3 \cos (c+d x)} \, dx=\frac {\arctan \left (\frac {5 \, \cos \left (d x + c\right ) + 3}{4 \, \sin \left (d x + c\right )}\right )}{4 \, d} \]
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Time = 0.37 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.42 \[ \int \frac {1}{-5-3 \cos (c+d x)} \, dx=\begin {cases} - \frac {\operatorname {atan}{\left (\frac {\tan {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor }{2 d} & \text {for}\: d \neq 0 \\\frac {x}{- 3 \cos {\left (c \right )} - 5} & \text {otherwise} \end {cases} \]
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Time = 0.39 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.77 \[ \int \frac {1}{-5-3 \cos (c+d x)} \, dx=-\frac {\arctan \left (\frac {\sin \left (d x + c\right )}{2 \, {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}{2 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.97 \[ \int \frac {1}{-5-3 \cos (c+d x)} \, dx=-\frac {d x + c - 2 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 3}\right )}{4 \, d} \]
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Time = 14.51 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.23 \[ \int \frac {1}{-5-3 \cos (c+d x)} \, dx=\frac {\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}}{2\,d}-\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{2\,d} \]
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