Integrand size = 21, antiderivative size = 47 \[ \int \frac {3+\cos (c+d x)}{2-\cos (c+d x)} \, dx=-x+\frac {5 x}{\sqrt {3}}+\frac {10 \arctan \left (\frac {\sin (c+d x)}{2+\sqrt {3}-\cos (c+d x)}\right )}{\sqrt {3} d} \]
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Time = 0.08 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {2814, 2736} \[ \int \frac {3+\cos (c+d x)}{2-\cos (c+d x)} \, dx=\frac {10 \arctan \left (\frac {\sin (c+d x)}{-\cos (c+d x)+\sqrt {3}+2}\right )}{\sqrt {3} d}+\frac {5 x}{\sqrt {3}}-x \]
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Rule 2736
Rule 2814
Rubi steps \begin{align*} \text {integral}& = -x+5 \int \frac {1}{2-\cos (c+d x)} \, dx \\ & = -x+\frac {5 x}{\sqrt {3}}+\frac {10 \arctan \left (\frac {\sin (c+d x)}{2+\sqrt {3}-\cos (c+d x)}\right )}{\sqrt {3} d} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.66 \[ \int \frac {3+\cos (c+d x)}{2-\cos (c+d x)} \, dx=-x+\frac {10 \arctan \left (\sqrt {3} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {3} d} \]
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Time = 1.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.79
method | result | size |
derivativedivides | \(\frac {-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {10 \sqrt {3}\, \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {3}\right )}{3}}{d}\) | \(37\) |
default | \(\frac {-2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {10 \sqrt {3}\, \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {3}\right )}{3}}{d}\) | \(37\) |
risch | \(-x +\frac {5 i \sqrt {3}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\sqrt {3}-2\right )}{3 d}-\frac {5 i \sqrt {3}\, \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\sqrt {3}-2\right )}{3 d}\) | \(55\) |
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none
Time = 0.29 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.91 \[ \int \frac {3+\cos (c+d x)}{2-\cos (c+d x)} \, dx=-\frac {3 \, d x + 5 \, \sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} \cos \left (d x + c\right ) - \sqrt {3}}{3 \, \sin \left (d x + c\right )}\right )}{3 \, d} \]
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Time = 1.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.19 \[ \int \frac {3+\cos (c+d x)}{2-\cos (c+d x)} \, dx=\begin {cases} - x + \frac {10 \sqrt {3} \left (\operatorname {atan}{\left (\sqrt {3} \tan {\left (\frac {c}{2} + \frac {d x}{2} \right )} \right )} + \pi \left \lfloor {\frac {\frac {c}{2} + \frac {d x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{3 d} & \text {for}\: d \neq 0 \\\frac {x \left (\cos {\left (c \right )} + 3\right )}{2 - \cos {\left (c \right )}} & \text {otherwise} \end {cases} \]
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none
Time = 0.29 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.11 \[ \int \frac {3+\cos (c+d x)}{2-\cos (c+d x)} \, dx=\frac {2 \, {\left (5 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right ) - 3 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )\right )}}{3 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.53 \[ \int \frac {3+\cos (c+d x)}{2-\cos (c+d x)} \, dx=-\frac {3 \, d x - 5 \, \sqrt {3} {\left (d x + c + 2 \, \arctan \left (-\frac {\sqrt {3} \sin \left (d x + c\right ) - 3 \, \sin \left (d x + c\right )}{\sqrt {3} \cos \left (d x + c\right ) + \sqrt {3} - 3 \, \cos \left (d x + c\right ) + 3}\right )\right )} + 3 \, c}{3 \, d} \]
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Time = 14.31 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.57 \[ \int \frac {3+\cos (c+d x)}{2-\cos (c+d x)} \, dx=\frac {\left (\frac {\pi -\frac {5\,\pi \,\sqrt {3}}{3}}{d}-\frac {\pi +\frac {5\,\pi \,\sqrt {3}}{3}}{d}\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{\pi }-\frac {d\,x-\frac {10\,\sqrt {3}\,\mathrm {atan}\left (\sqrt {3}\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{3}}{d} \]
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