Optimal. Leaf size=36 \[ -\frac {1}{6} \tanh ^{-1}(\cos (x))-\frac {1}{6} \tanh ^{-1}(2 \cos (x))+\frac {\tanh ^{-1}\left (\frac {2 \cos (x)}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
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Rubi [A] time = 0.04, antiderivative size = 36, normalized size of antiderivative = 1.00, 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, 2057, 207} \[ -\frac {1}{6} \tanh ^{-1}(\cos (x))-\frac {1}{6} \tanh ^{-1}(2 \cos (x))+\frac {\tanh ^{-1}\left (\frac {2 \cos (x)}{\sqrt {3}}\right )}{2 \sqrt {3}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 207
Rule 2057
Rubi steps
\begin {align*} \int \cos (x) \csc (6 x) \, dx &=-\operatorname {Subst}\left (\int \frac {1}{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}{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 (-\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 )\\ &=\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}}\\ \end {align*}
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Mathematica [B] time = 0.08, size = 83, normalized size = 2.31 \[ \frac {1}{12} \left (2 \log \left (\sin \left (\frac {x}{2}\right )\right )-2 \log \left (\cos \left (\frac {x}{2}\right )\right )+\log (1-2 \cos (x))-\log (2 \cos (x)+1)-2 \sqrt {3} \tanh ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )-2}{\sqrt {3}}\right )+2 \sqrt {3} \tanh ^{-1}\left (\frac {\tan \left (\frac {x}{2}\right )+2}{\sqrt {3}}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.71, size = 70, normalized size = 1.94 \[ \frac {1}{12} \, \sqrt {3} \log \left (-\frac {4 \, \cos \relax (x)^{2} + 4 \, \sqrt {3} \cos \relax (x) + 3}{4 \, \cos \relax (x)^{2} - 3}\right ) - \frac {1}{12} \, \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + \frac {1}{12} \, \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + \frac {1}{12} \, \log \left (-2 \, \cos \relax (x) + 1\right ) - \frac {1}{12} \, \log \left (-2 \, \cos \relax (x) - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 68, normalized size = 1.89 \[ -\frac {1}{12} \, \sqrt {3} \log \left (\frac {{\left | -4 \, \sqrt {3} + 8 \, \cos \relax (x) \right |}}{{\left | 4 \, \sqrt {3} + 8 \, \cos \relax (x) \right |}}\right ) - \frac {1}{12} \, \log \left (\cos \relax (x) + 1\right ) + \frac {1}{12} \, \log \left (-\cos \relax (x) + 1\right ) - \frac {1}{12} \, \log \left ({\left | 2 \, \cos \relax (x) + 1 \right |}\right ) + \frac {1}{12} \, \log \left ({\left | 2 \, \cos \relax (x) - 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 47, normalized size = 1.31 \[ \frac {\ln \left (2 \cos \relax (x )-1\right )}{12}-\frac {\ln \left (1+2 \cos \relax (x )\right )}{12}+\frac {\arctanh \left (\frac {2 \cos \relax (x ) \sqrt {3}}{3}\right ) \sqrt {3}}{6}+\frac {\ln \left (-1+\cos \relax (x )\right )}{12}-\frac {\ln \left (1+\cos \relax (x )\right )}{12} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {{\left (\sin \left (3 \, x\right ) - \sin \relax (x)\right )} \cos \left (4 \, x\right ) - {\left (\cos \left (3 \, x\right ) - \cos \relax (x)\right )} \sin \left (4 \, x\right ) - {\left (\cos \left (2 \, x\right ) - 1\right )} \sin \left (3 \, x\right ) + \cos \left (3 \, x\right ) \sin \left (2 \, x\right ) - \cos \relax (x) \sin \left (2 \, x\right ) + \cos \left (2 \, x\right ) \sin \relax (x) - \sin \relax (x)}{2 \, {\left (2 \, {\left (\cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 2 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - \sin \left (2 \, x\right )^{2} + 2 \, \cos \left (2 \, x\right ) - 1\right )}}\,{d x} - \frac {1}{24} \, \log \left (2 \, {\left (\cos \relax (x) + 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + \cos \relax (x)^{2} + \sin \left (2 \, x\right )^{2} + 2 \, \sin \left (2 \, x\right ) \sin \relax (x) + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + \frac {1}{24} \, \log \left (-2 \, {\left (\cos \relax (x) - 1\right )} \cos \left (2 \, x\right ) + \cos \left (2 \, x\right )^{2} + \cos \relax (x)^{2} + \sin \left (2 \, x\right )^{2} - 2 \, \sin \left (2 \, x\right ) \sin \relax (x) + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) - \frac {1}{12} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + \frac {1}{12} \, \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.37, size = 74, normalized size = 2.06 \[ \frac {\mathrm {atanh}\left (\frac {1073741824}{10761687\,\left (\frac {427973089951744\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{14348907}-\frac {47552804159488}{4782969}\right )}+\frac {797161}{797162}\right )}{6}+\frac {\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )}{6}+\frac {\sqrt {3}\,\mathrm {atanh}\left (\frac {4222769432625152\,\sqrt {3}}{4782969\,\left (\frac {101871591633190912\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{4782969}-\frac {7314051205955584}{4782969}\right )}-\frac {19605196950732800\,\sqrt {3}\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{1594323\,\left (\frac {101871591633190912\,{\mathrm {tan}\left (\frac {x}{2}\right )}^2}{4782969}-\frac {7314051205955584}{4782969}\right )}\right )}{6} \]
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 \[ \int \cos {\relax (x )} \csc {\left (6 x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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