Optimal. Leaf size=31 \[ \frac {2 x}{\sqrt {3}}-\frac {4 \tan ^{-1}\left (\frac {\sin (x)}{2+\sqrt {3}+\cos (x)}\right )}{\sqrt {3}} \]
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Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2736}
\begin {gather*} \frac {2 x}{\sqrt {3}}-\frac {4 \text {ArcTan}\left (\frac {\sin (x)}{\cos (x)+\sqrt {3}+2}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2736
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*}
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Mathematica [A]
time = 0.01, size = 20, normalized size = 0.65 \begin {gather*} \frac {4 \tan ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )}{\sqrt {3}}\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 16, normalized size = 0.52
method | result | size |
default | \(\frac {4 \sqrt {3}\, \arctan \left (\frac {\tan \left (\frac {x}{2}\right ) \sqrt {3}}{3}\right )}{3}\) | \(16\) |
risch | \(\frac {2 i \sqrt {3}\, \ln \left ({\mathrm e}^{i x}+\sqrt {3}+2\right )}{3}-\frac {2 i \sqrt {3}\, \ln \left ({\mathrm e}^{i x}-\sqrt {3}+2\right )}{3}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.62, size = 19, normalized size = 0.61 \begin {gather*} \frac {4}{3} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} \sin \left (x\right )}{3 \, {\left (\cos \left (x\right ) + 1\right )}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.61, size = 23, normalized size = 0.74 \begin {gather*} -\frac {2}{3} \, \sqrt {3} \arctan \left (\frac {2 \, \sqrt {3} \cos \left (x\right ) + \sqrt {3}}{3 \, \sin \left (x\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 32, normalized size = 1.03 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 40, normalized size = 1.29 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.22, size = 32, normalized size = 1.03 \begin {gather*} \frac {4\,\sqrt {3}\,\left (\frac {x}{2}-\mathrm {atan}\left (\mathrm {tan}\left (\frac {x}{2}\right )\right )\right )}{3}+\frac {4\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\mathrm {tan}\left (\frac {x}{2}\right )}{3}\right )}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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