Integrand size = 8, antiderivative size = 4 \[ \int \frac {\cot (x)}{\log (\sin (x))} \, dx=\log (\log (\sin (x))) \]
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Time = 0.02 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4423, 2339, 29} \[ \int \frac {\cot (x)}{\log (\sin (x))} \, dx=\log (\log (\sin (x))) \]
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Rule 29
Rule 2339
Rule 4423
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x \log (x)} \, dx,x,\sin (x)\right ) \\ & = \text {Subst}\left (\int \frac {1}{x} \, dx,x,\log (\sin (x))\right ) \\ & = \log (\log (\sin (x))) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x)}{\log (\sin (x))} \, dx=\log (\log (\sin (x))) \]
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Time = 0.20 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.25
method | result | size |
derivativedivides | \(\ln \left (\ln \left (\sin \left (x \right )\right )\right )\) | \(5\) |
default | \(\ln \left (\ln \left (\sin \left (x \right )\right )\right )\) | \(5\) |
risch | \(\ln \left (-\frac {i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) \operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{-i x}\right ) \operatorname {csgn}\left (\sin \left (x \right )\right )^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (i \sin \left (x \right )\right )}{2}-\frac {i \pi \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2}}{2}-\frac {i \pi \operatorname {csgn}\left (\sin \left (x \right )\right )^{3}}{2}+\frac {i \pi \,\operatorname {csgn}\left (\sin \left (x \right )\right ) \operatorname {csgn}\left (i \sin \left (x \right )\right )^{2}}{2}+\frac {i \pi \operatorname {csgn}\left (i \sin \left (x \right )\right )^{3}}{2}+\frac {i \pi }{2}+\ln \left (2\right )-\ln \left ({\mathrm e}^{2 i x}-1\right )+\ln \left ({\mathrm e}^{i x}\right )\right )\) | \(151\) |
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Time = 0.24 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x)}{\log (\sin (x))} \, dx=\log \left (\log \left (\sin \left (x\right )\right )\right ) \]
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\[ \int \frac {\cot (x)}{\log (\sin (x))} \, dx=\int \frac {\cot {\left (x \right )}}{\log {\left (\sin {\left (x \right )} \right )}}\, dx \]
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Time = 0.19 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x)}{\log (\sin (x))} \, dx=\log \left (\log \left (\sin \left (x\right )\right )\right ) \]
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Time = 0.27 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.25 \[ \int \frac {\cot (x)}{\log (\sin (x))} \, dx=\log \left ({\left | \log \left (\sin \left (x\right )\right ) \right |}\right ) \]
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Time = 0.43 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x)}{\log (\sin (x))} \, dx=\ln \left (\ln \left (\sin \left (x\right )\right )\right ) \]
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