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 = 0.03, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, 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 29
Rule 2157
Rule 2160
Rule 4338
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*}
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Mathematica [A] time = 0.04, 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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fricas [A] time = 0.44, size = 6, normalized size = 0.16 \[ -\frac {1}{\sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 6, normalized size = 0.16 \[ -\frac {1}{\sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.45, size = 35, normalized size = 0.95 \[ -\frac {\ln \left (\ln \left ({\mathrm e}^{\sin \relax (x )}\right )\right )}{\ln \left ({\mathrm e}^{\sin \relax (x )}\right )-\sin \relax (x )}+\frac {\ln \left (\sin \relax (x )\right )}{\ln \left ({\mathrm e}^{\sin \relax (x )}\right )-\sin \relax (x )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 6, normalized size = 0.16 \[ -\frac {1}{\sin \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.37, size = 6, normalized size = 0.16 \[ -\frac {1}{\sin \relax (x)} \]
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 \[ \int \frac {\cot {\relax (x )}}{\log {\left (e^{\sin {\relax (x )}} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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