Optimal. Leaf size=56 \[ -\frac {\log \left (\sqrt {3} \cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )}{\sqrt {3}}+\frac {\log \left (\sqrt {3} \cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )}{\sqrt {3}} \]
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Rubi [A]
time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2738, 212}
\begin {gather*} \frac {\log \left (\sin \left (\frac {x}{2}\right )+\sqrt {3} \cos \left (\frac {x}{2}\right )\right )}{\sqrt {3}}-\frac {\log \left (\sqrt {3} \cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )}{\sqrt {3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2738
Rubi steps
\begin {align*} \int \frac {1}{1+2 \cos (x)} \, dx &=2 \text {Subst}\left (\int \frac {1}{3-x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\frac {\log \left (\sqrt {3} \cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )}{\sqrt {3}}+\frac {\log \left (\sqrt {3} \cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )}{\sqrt {3}}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 20, normalized size = 0.36 \begin {gather*} \frac {2 \tanh ^{-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 = 0.02, size = 16, normalized size = 0.29
method | result | size |
default | \(\frac {2 \sqrt {3}\, \arctanh \left (\frac {\tan \left (\frac {x}{2}\right ) \sqrt {3}}{3}\right )}{3}\) | \(16\) |
risch | \(\frac {\sqrt {3}\, \ln \left ({\mathrm e}^{i x}+\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{3}-\frac {\sqrt {3}\, \ln \left ({\mathrm e}^{i x}+\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{3}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.40, size = 37, normalized size = 0.66 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \log \left (-\frac {\sqrt {3} - \frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}{\sqrt {3} + \frac {\sin \left (x\right )}{\cos \left (x\right ) + 1}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.64, size = 50, normalized size = 0.89 \begin {gather*} \frac {1}{6} \, \sqrt {3} \log \left (-\frac {2 \, \cos \left (x\right )^{2} - 2 \, {\left (\sqrt {3} \cos \left (x\right ) + 2 \, \sqrt {3}\right )} \sin \left (x\right ) - 4 \, \cos \left (x\right ) - 7}{4 \, \cos \left (x\right )^{2} + 4 \, \cos \left (x\right ) + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.13, size = 36, normalized size = 0.64 \begin {gather*} - \frac {\sqrt {3} \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {3} \right )}}{3} + \frac {\sqrt {3} \log {\left (\tan {\left (\frac {x}{2} \right )} + \sqrt {3} \right )}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 35, normalized size = 0.62 \begin {gather*} -\frac {1}{3} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \tan \left (\frac {1}{2} \, x\right ) \right |}}{{\left | 2 \, \sqrt {3} + 2 \, \tan \left (\frac {1}{2} \, x\right ) \right |}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.24, size = 15, normalized size = 0.27 \begin {gather*} \frac {2\,\sqrt {3}\,\mathrm {atanh}\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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