Optimal. Leaf size=45 \[ \cos (x)+\frac {1}{6} \log (1-2 \cos (x))+\frac {1}{6} \log (1-\cos (x))-\frac {1}{6} \log (\cos (x)+1)-\frac {1}{6} \log (2 \cos (x)+1) \]
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Rubi [A] time = 0.05, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {1279, 1161, 616, 31} \[ \cos (x)+\frac {1}{6} \log (1-2 \cos (x))+\frac {1}{6} \log (1-\cos (x))-\frac {1}{6} \log (\cos (x)+1)-\frac {1}{6} \log (2 \cos (x)+1) \]
Antiderivative was successfully verified.
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Rule 31
Rule 616
Rule 1161
Rule 1279
Rubi steps
\begin {align*} \int \cos (x) \cot (3 x) \, dx &=-\operatorname {Subst}\left (\int \frac {x^2 \left (3-4 x^2\right )}{1-5 x^2+4 x^4} \, dx,x,\cos (x)\right )\\ &=\cos (x)+\frac {1}{4} \operatorname {Subst}\left (\int \frac {-4+8 x^2}{1-5 x^2+4 x^4} \, dx,x,\cos (x)\right )\\ &=\cos (x)+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}-\frac {x}{2}+x^2} \, dx,x,\cos (x)\right )+\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}+\frac {x}{2}+x^2} \, dx,x,\cos (x)\right )\\ &=\cos (x)+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{-1+x} \, dx,x,\cos (x)\right )+\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2}+x} \, dx,x,\cos (x)\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2}+x} \, dx,x,\cos (x)\right )-\frac {1}{6} \operatorname {Subst}\left (\int \frac {1}{1+x} \, dx,x,\cos (x)\right )\\ &=\cos (x)+\frac {1}{6} \log (1-2 \cos (x))+\frac {1}{6} \log (1-\cos (x))-\frac {1}{6} \log (1+\cos (x))-\frac {1}{6} \log (1+2 \cos (x))\\ \end {align*}
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Mathematica [A] time = 0.02, size = 47, normalized size = 1.04 \[ \cos (x)+\frac {1}{3} \log \left (\sin \left (\frac {x}{2}\right )\right )-\frac {1}{3} \log \left (\cos \left (\frac {x}{2}\right )\right )+\frac {1}{6} \log (1-2 \cos (x))-\frac {1}{6} \log (2 \cos (x)+1) \]
Antiderivative was successfully verified.
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fricas [A] time = 3.04, size = 39, normalized size = 0.87 \[ \cos \relax (x) - \frac {1}{6} \, \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + \frac {1}{6} \, \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + \frac {1}{6} \, \log \left (-2 \, \cos \relax (x) + 1\right ) - \frac {1}{6} \, \log \left (-2 \, \cos \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.13, size = 39, normalized size = 0.87 \[ \cos \relax (x) - \frac {1}{6} \, \log \left (\cos \relax (x) + 1\right ) + \frac {1}{6} \, \log \left (-\cos \relax (x) + 1\right ) - \frac {1}{6} \, \log \left ({\left | 2 \, \cos \relax (x) + 1 \right |}\right ) + \frac {1}{6} \, \log \left ({\left | 2 \, \cos \relax (x) - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 36, normalized size = 0.80 \[ \frac {\ln \left (2 \cos \relax (x )-1\right )}{6}-\frac {\ln \left (1+2 \cos \relax (x )\right )}{6}+\frac {\ln \left (-1+\cos \relax (x )\right )}{6}-\frac {\ln \left (1+\cos \relax (x )\right )}{6}+\cos \relax (x ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.42, size = 131, normalized size = 2.91 \[ \cos \relax (x) - \frac {1}{12} \, \log \left (2 \, {\left (\cos \relax (x) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + \cos \relax (x)^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \left (2 \, x\right ) \sin \relax (x) + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + \frac {1}{12} \, \log \left (-2 \, {\left (\cos \relax (x) - 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + \cos \relax (x)^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \sin \left (2 \, x\right ) \sin \relax (x) + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) - \frac {1}{6} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + \frac {1}{6} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.37, size = 39, normalized size = 0.87 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{3}+\frac {\mathrm {atanh}\left (\frac {8}{183\,\left (\frac {488\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{243}-\frac {56}{81}\right )}+\frac {121}{122}\right )}{3}+\frac {2}{{\mathrm {tan}\left (\frac {x}{2}\right )}^2+1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cos {\relax (x )} \cot {\left (3 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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