Optimal. Leaf size=34 \[ \frac{4 x}{\sqrt{3}}-x+\frac{8 \tan ^{-1}\left (\frac{\cos (x)}{\sin (x)+\sqrt{3}+2}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0327097, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2735, 2657} \[ \frac{4 x}{\sqrt{3}}-x+\frac{8 \tan ^{-1}\left (\frac{\cos (x)}{\sin (x)+\sqrt{3}+2}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2657
Rubi steps
\begin{align*} \int \frac{2-\sin (x)}{2+\sin (x)} \, dx &=-x+4 \int \frac{1}{2+\sin (x)} \, dx\\ &=-x+\frac{4 x}{\sqrt{3}}+\frac{8 \tan ^{-1}\left (\frac{\cos (x)}{2+\sqrt{3}+\sin (x)}\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [A] time = 0.0282276, size = 28, normalized size = 0.82 \[ \frac{8 \tan ^{-1}\left (\frac{2 \tan \left (\frac{x}{2}\right )+1}{\sqrt{3}}\right )}{\sqrt{3}}-x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 24, normalized size = 0.7 \begin{align*}{\frac{8\,\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 1+2\,\tan \left ( x/2 \right ) \right ) } \right ) }-x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.70339, size = 49, normalized size = 1.44 \begin{align*} \frac{8}{3} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (\frac{2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + 1\right )}\right ) - 2 \, \arctan \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.32982, size = 88, normalized size = 2.59 \begin{align*} \frac{4}{3} \, \sqrt{3} \arctan \left (\frac{2 \, \sqrt{3} \sin \left (x\right ) + \sqrt{3}}{3 \, \cos \left (x\right )}\right ) - x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.08916, size = 42, normalized size = 1.24 \begin{align*} - x + \frac{8 \sqrt{3} \left (\operatorname{atan}{\left (\frac{2 \sqrt{3} \tan{\left (\frac{x}{2} \right )}}{3} + \frac{\sqrt{3}}{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 = 1.30076, size = 69, normalized size = 2.03 \begin{align*} \frac{4}{3} \, \sqrt{3}{\left (x + 2 \, \arctan \left (-\frac{\sqrt{3} \sin \left (x\right ) - \cos \left (x\right ) - 2 \, \sin \left (x\right ) - 1}{\sqrt{3} \cos \left (x\right ) + \sqrt{3} - 2 \, \cos \left (x\right ) + \sin \left (x\right ) + 2}\right )\right )} - x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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