Optimal. Leaf size=31 \[ \frac{2 x}{\sqrt{3}}-\frac{4 \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)+\sqrt{3}+2}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0114407, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2657} \[ \frac{2 x}{\sqrt{3}}-\frac{4 \tan ^{-1}\left (\frac{\sin (x)}{\cos (x)+\sqrt{3}+2}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 2657
Rubi steps
\begin{align*} \int \frac{1}{1+\frac{\cos (x)}{2}} \, dx &=\frac{2 x}{\sqrt{3}}-\frac{4 \tan ^{-1}\left (\frac{\sin (x)}{2+\sqrt{3}+\cos (x)}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0099244, size = 20, normalized size = 0.65 \[ \frac{4 \tan ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right )}{\sqrt{3}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 16, normalized size = 0.5 \begin{align*}{\frac{4\,\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3}\tan \left ({\frac{x}{2}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.41696, size = 26, normalized size = 0.84 \begin{align*} \frac{4}{3} \, \sqrt{3} \arctan \left (\frac{\sqrt{3} \sin \left (x\right )}{3 \,{\left (\cos \left (x\right ) + 1\right )}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.18087, size = 84, normalized size = 2.71 \begin{align*} -\frac{2}{3} \, \sqrt{3} \arctan \left (\frac{2 \, \sqrt{3} \cos \left (x\right ) + \sqrt{3}}{3 \, \sin \left (x\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.281181, size = 32, normalized size = 1.03 \begin{align*} \frac{4 \sqrt{3} \left (\operatorname{atan}{\left (\frac{\sqrt{3} \tan{\left (\frac{x}{2} \right )}}{3} \right )} + \pi \left \lfloor{\frac{\frac{x}{2} - \frac{\pi }{2}}{\pi }}\right \rfloor \right )}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08095, size = 54, normalized size = 1.74 \begin{align*} \frac{2}{3} \, \sqrt{3}{\left (x + 2 \, \arctan \left (-\frac{\sqrt{3} \sin \left (x\right ) - \sin \left (x\right )}{\sqrt{3} \cos \left (x\right ) + \sqrt{3} - \cos \left (x\right ) + 1}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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