Optimal. Leaf size=37 \[ \frac{\log \left (\log \left (e^{\sin (x)}\right )\right )}{\sin (x)-\log \left (e^{\sin (x)}\right )}-\frac{\log (\sin (x))}{\sin (x)-\log \left (e^{\sin (x)}\right )} \]
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Rubi [A] time = 0.0281907, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4338, 2160, 2157, 29} \[ \frac{\log \left (\log \left (e^{\sin (x)}\right )\right )}{\sin (x)-\log \left (e^{\sin (x)}\right )}-\frac{\log (\sin (x))}{\sin (x)-\log \left (e^{\sin (x)}\right )} \]
Antiderivative was successfully verified.
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Rule 4338
Rule 2160
Rule 2157
Rule 29
Rubi steps
\begin{align*} \int \frac{\cot (x)}{\log \left (e^{\sin (x)}\right )} \, dx &=\operatorname{Subst}\left (\int \frac{1}{x \log \left (e^x\right )} \, dx,x,\sin (x)\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\sin (x)\right )}{-\log \left (e^{\sin (x)}\right )+\sin (x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{\log \left (e^x\right )} \, dx,x,\sin (x)\right )}{-\log \left (e^{\sin (x)}\right )+\sin (x)}\\ &=\frac{\log (\sin (x))}{\log \left (e^{\sin (x)}\right )-\sin (x)}+\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\log \left (e^{\sin (x)}\right )\right )}{-\log \left (e^{\sin (x)}\right )+\sin (x)}\\ &=-\frac{\log \left (\log \left (e^{\sin (x)}\right )\right )}{\log \left (e^{\sin (x)}\right )-\sin (x)}+\frac{\log (\sin (x))}{\log \left (e^{\sin (x)}\right )-\sin (x)}\\ \end{align*}
Mathematica [A] time = 0.0347289, size = 25, normalized size = 0.68 \[ \frac{\log \left (\log \left (e^{\sin (x)}\right )\right )-\log (\sin (x))}{\sin (x)-\log \left (e^{\sin (x)}\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.02, size = 35, normalized size = 1. \begin{align*}{\frac{\ln \left ( \sin \left ( x \right ) \right ) }{\ln \left ({{\rm e}^{\sin \left ( x \right ) }} \right ) -\sin \left ( x \right ) }}-{\frac{\ln \left ( \ln \left ({{\rm e}^{\sin \left ( x \right ) }} \right ) \right ) }{\ln \left ({{\rm e}^{\sin \left ( x \right ) }} \right ) -\sin \left ( x \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00954, size = 8, normalized size = 0.22 \begin{align*} -\frac{1}{\sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21378, size = 15, normalized size = 0.41 \begin{align*} -\frac{1}{\sin \left (x\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 (e^{\sin{\left (x \right )}} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.29746, size = 8, normalized size = 0.22 \begin{align*} -\frac{1}{\sin \left (x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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