Optimal. Leaf size=38 \[ \cos (x)-\frac{1}{6} \tanh ^{-1}(\cos (x))-\frac{1}{6} \tanh ^{-1}(2 \cos (x))-\frac{\tanh ^{-1}\left (\frac{2 \cos (x)}{\sqrt{3}}\right )}{2 \sqrt{3}} \]
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Rubi [A] time = 0.0712837, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 7, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {12, 2073, 207} \[ \cos (x)-\frac{1}{6} \tanh ^{-1}(\cos (x))-\frac{1}{6} \tanh ^{-1}(2 \cos (x))-\frac{\tanh ^{-1}\left (\frac{2 \cos (x)}{\sqrt{3}}\right )}{2 \sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 2073
Rule 207
Rubi steps
\begin{align*} \int \cos (x) \cot (6 x) \, dx &=-\operatorname{Subst}\left (\int \frac{-1+18 x^2-48 x^4+32 x^6}{2 \left (3-19 x^2+32 x^4-16 x^6\right )} \, dx,x,\cos (x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \frac{-1+18 x^2-48 x^4+32 x^6}{3-19 x^2+32 x^4-16 x^6} \, dx,x,\cos (x)\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (-2-\frac{1}{3 \left (-1+x^2\right )}-\frac{2}{-3+4 x^2}-\frac{2}{3 \left (-1+4 x^2\right )}\right ) \, dx,x,\cos (x)\right )\right )\\ &=\cos (x)+\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\cos (x)\right )+\frac{1}{3} \operatorname{Subst}\left (\int \frac{1}{-1+4 x^2} \, dx,x,\cos (x)\right )+\operatorname{Subst}\left (\int \frac{1}{-3+4 x^2} \, dx,x,\cos (x)\right )\\ &=-\frac{1}{6} \tanh ^{-1}(\cos (x))-\frac{1}{6} \tanh ^{-1}(2 \cos (x))-\frac{\tanh ^{-1}\left (\frac{2 \cos (x)}{\sqrt{3}}\right )}{2 \sqrt{3}}+\cos (x)\\ \end{align*}
Mathematica [B] time = 0.0860789, size = 87, normalized size = 2.29 \[ \frac{1}{12} \left (12 \cos (x)+2 \log \left (\sin \left (\frac{x}{2}\right )\right )-2 \log \left (\cos \left (\frac{x}{2}\right )\right )+\log (1-2 \cos (x))-\log (2 \cos (x)+1)+2 \sqrt{3} \tanh ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right )-2}{\sqrt{3}}\right )-2 \sqrt{3} \tanh ^{-1}\left (\frac{\tan \left (\frac{x}{2}\right )+2}{\sqrt{3}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.2, size = 49, normalized size = 1.3 \begin{align*} -{\frac{\ln \left ( 1+\cos \left ( x \right ) \right ) }{12}}+{\frac{\ln \left ( -1+\cos \left ( x \right ) \right ) }{12}}-{\frac{\ln \left ( 1+2\,\cos \left ( x \right ) \right ) }{12}}+{\frac{\ln \left ( 2\,\cos \left ( x \right ) -1 \right ) }{12}}-{\frac{\sqrt{3}}{6}{\it Artanh} \left ({\frac{2\,\cos \left ( x \right ) \sqrt{3}}{3}} \right ) }+\cos \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \cos \left (x\right ) + \int \frac{{\left (\sin \left (3 \, x\right ) - \sin \left (x\right )\right )} \cos \left (4 \, x\right ) -{\left (\cos \left (3 \, x\right ) - \cos \left (x\right )\right )} \sin \left (4 \, x\right ) -{\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (3 \, x\right ) + \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) - \cos \left (x\right ) \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) \sin \left (x\right ) - \sin \left (x\right )}{2 \,{\left (2 \,{\left (\cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 2 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) - 1\right )}}\,{d x} - \frac{1}{24} \, \log \left (2 \,{\left (\cos \left (x\right ) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac{1}{24} \, \log \left (-2 \,{\left (\cos \left (x\right ) - 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + \cos \left (x\right )^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) - \frac{1}{12} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac{1}{12} \, \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.61589, size = 259, normalized size = 6.82 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (\frac{4 \, \cos \left (x\right )^{2} - 4 \, \sqrt{3} \cos \left (x\right ) + 3}{4 \, \cos \left (x\right )^{2} - 3}\right ) + \cos \left (x\right ) - \frac{1}{12} \, \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{12} \, \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) + \frac{1}{12} \, \log \left (-2 \, \cos \left (x\right ) + 1\right ) - \frac{1}{12} \, \log \left (-2 \, \cos \left (x\right ) - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.14683, size = 95, normalized size = 2.5 \begin{align*} \frac{1}{12} \, \sqrt{3} \log \left (\frac{{\left | -4 \, \sqrt{3} + 8 \, \cos \left (x\right ) \right |}}{{\left | 4 \, \sqrt{3} + 8 \, \cos \left (x\right ) \right |}}\right ) + \cos \left (x\right ) - \frac{1}{12} \, \log \left (\cos \left (x\right ) + 1\right ) + \frac{1}{12} \, \log \left (-\cos \left (x\right ) + 1\right ) - \frac{1}{12} \, \log \left ({\left | 2 \, \cos \left (x\right ) + 1 \right |}\right ) + \frac{1}{12} \, \log \left ({\left | 2 \, \cos \left (x\right ) - 1 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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