Optimal. Leaf size=45 \[ \frac {\log \left (\sin (x)+\sqrt {3} \cos (x)\right )}{2 \sqrt {3}}-\frac {\log \left (\sqrt {3} \cos (x)-\sin (x)\right )}{2 \sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {206} \[ \frac {\log \left (\sin (x)+\sqrt {3} \cos (x)\right )}{2 \sqrt {3}}-\frac {\log \left (\sqrt {3} \cos (x)-\sin (x)\right )}{2 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 206
Rubi steps
\begin {align*} \int \csc (3 x) \sin (x) \, dx &=\operatorname {Subst}\left (\int \frac {1}{3-x^2} \, dx,x,\tan (x)\right )\\ &=-\frac {\log \left (\sqrt {3} \cos (x)-\sin (x)\right )}{2 \sqrt {3}}+\frac {\log \left (\sqrt {3} \cos (x)+\sin (x)\right )}{2 \sqrt {3}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 15, normalized size = 0.33 \[ \frac {\tanh ^{-1}\left (\frac {\tan (x)}{\sqrt {3}}\right )}{\sqrt {3}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.09, size = 58, normalized size = 1.29 \[ \frac {1}{12} \, \sqrt {3} \log \left (-\frac {8 \, \cos \relax (x)^{4} - 16 \, \cos \relax (x)^{2} - 4 \, {\left (2 \, \sqrt {3} \cos \relax (x)^{3} + \sqrt {3} \cos \relax (x)\right )} \sin \relax (x) - 1}{16 \, \cos \relax (x)^{4} - 8 \, \cos \relax (x)^{2} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 31, normalized size = 0.69 \[ -\frac {1}{6} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \tan \relax (x) \right |}}{{\left | 2 \, \sqrt {3} + 2 \, \tan \relax (x) \right |}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 14, normalized size = 0.31 \[ \frac {\sqrt {3}\, \arctanh \left (\frac {\tan \relax (x ) \sqrt {3}}{3}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 125, normalized size = 2.78 \[ -\frac {1}{12} \, \sqrt {3} \log \left (\frac {4}{3} \, \cos \relax (x)^{2} + \frac {4}{3} \, \sin \relax (x)^{2} + \frac {4}{3} \, \sqrt {3} \sin \relax (x) + \frac {4}{3} \, \cos \relax (x) + \frac {4}{3}\right ) + \frac {1}{12} \, \sqrt {3} \log \left (\frac {4}{3} \, \cos \relax (x)^{2} + \frac {4}{3} \, \sin \relax (x)^{2} + \frac {4}{3} \, \sqrt {3} \sin \relax (x) - \frac {4}{3} \, \cos \relax (x) + \frac {4}{3}\right ) + \frac {1}{12} \, \sqrt {3} \log \left (\frac {4}{3} \, \cos \relax (x)^{2} + \frac {4}{3} \, \sin \relax (x)^{2} - \frac {4}{3} \, \sqrt {3} \sin \relax (x) + \frac {4}{3} \, \cos \relax (x) + \frac {4}{3}\right ) - \frac {1}{12} \, \sqrt {3} \log \left (\frac {4}{3} \, \cos \relax (x)^{2} + \frac {4}{3} \, \sin \relax (x)^{2} - \frac {4}{3} \, \sqrt {3} \sin \relax (x) - \frac {4}{3} \, \cos \relax (x) + \frac {4}{3}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.79, size = 17, normalized size = 0.38 \[ \frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {\sqrt {3}\,\sin \relax (x)}{3\,\cos \relax (x)}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.87, size = 76, normalized size = 1.69 \[ \frac {\sqrt {3} \log {\left (\tan {\left (\frac {x}{2} \right )} - \sqrt {3} \right )}}{6} - \frac {\sqrt {3} \log {\left (\tan {\left (\frac {x}{2} \right )} - \frac {\sqrt {3}}{3} \right )}}{6} + \frac {\sqrt {3} \log {\left (\tan {\left (\frac {x}{2} \right )} + \frac {\sqrt {3}}{3} \right )}}{6} - \frac {\sqrt {3} \log {\left (\tan {\left (\frac {x}{2} \right )} + \sqrt {3} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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