Optimal. Leaf size=6 \[ \log (\log (e \sin (x))) \]
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Rubi [A] time = 0.0163892, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4338, 31} \[ \log (\log (\sin (x))+1) \]
Antiderivative was successfully verified.
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Rule 4338
Rule 31
Rubi steps
\begin{align*} \int \frac{\cot (x)}{\log (e \sin (x))} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x+x \log (x)} \, dx,x,\sin (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{1}{1+x} \, dx,x,\log (\sin (x))\right )\\ &=\log (1+\log (\sin (x)))\\ \end{align*}
Mathematica [A] time = 0.0135556, size = 6, normalized size = 1. \[ \log (\log (\sin (x))+1) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 7, normalized size = 1.2 \begin{align*} \ln \left ( \ln \left ( E\sin \left ( x \right ) \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00412, size = 8, normalized size = 1.33 \begin{align*} \log \left (\log \left (E \sin \left (x\right )\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.24403, size = 27, normalized size = 4.5 \begin{align*} \log \left (\log \left (E \sin \left (x\right )\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cot{\left (x \right )}}{\log{\left (\sin{\left (x \right )} \right )} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25981, size = 8, normalized size = 1.33 \begin{align*} \log \left (\log \left (E \sin \left (x\right )\right )\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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