Optimal. Leaf size=4 \[ \log (\log (\sin (x))) \]
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Rubi [A]
time = 0.01, antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {4423, 2339, 29}
\begin {gather*} \log (\log (\sin (x))) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2339
Rule 4423
Rubi steps
\begin {align*} \int \frac {\cot (x)}{\log (\sin (x))} \, dx &=\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*}
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Mathematica [A]
time = 0.01, size = 4, normalized size = 1.00 \begin {gather*} \log (\log (\sin (x))) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 5, normalized size = 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 | \(\ln \left (-\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) \mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \mathrm {csgn}\left (\sin \left (x \right )\right )}{2}-\frac {i \pi \,\mathrm {csgn}\left (i \left ({\mathrm e}^{2 i x}-1\right )\right ) \mathrm {csgn}\left (\sin \left (x \right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (i {\mathrm e}^{-i x}\right ) \mathrm {csgn}\left (\sin \left (x \right )\right )^{2}}{2}-\frac {i \pi \,\mathrm {csgn}\left (\sin \left (x \right )\right ) \mathrm {csgn}\left (i \sin \left (x \right )\right )}{2}-\frac {i \pi \mathrm {csgn}\left (i \sin \left (x \right )\right )^{2}}{2}-\frac {i \pi \mathrm {csgn}\left (\sin \left (x \right )\right )^{3}}{2}+\frac {i \pi \,\mathrm {csgn}\left (\sin \left (x \right )\right ) \mathrm {csgn}\left (i \sin \left (x \right )\right )^{2}}{2}+\frac {i \pi \mathrm {csgn}\left (i \sin \left (x \right )\right )^{3}}{2}+\frac {i \pi }{2}+\ln \left (2\right )-\ln \left ({\mathrm e}^{2 i x}-1\right )+\ln \left ({\mathrm e}^{i x}\right )\right )\) | \(151\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 3.39, size = 4, normalized size = 1.00 \begin {gather*} \log \left (\log \left (\sin \left (x\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.35, size = 4, normalized size = 1.00 \begin {gather*} \log \left (\log \left (\sin \left (x\right )\right )\right ) \end {gather*}
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 \begin {gather*} \int \frac {\cot {\left (x \right )}}{\log {\left (\sin {\left (x \right )} \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.64, size = 5, normalized size = 1.25 \begin {gather*} \log \left ({\left | \log \left (\sin \left (x\right )\right ) \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.40, size = 4, normalized size = 1.00 \begin {gather*} \ln \left (\ln \left (\sin \left (x\right )\right )\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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