Optimal. Leaf size=37 \[ \frac {\log \left (\log \left (e^{\sin (x)}\right )\right )}{-\log \left (e^{\sin (x)}\right )+\sin (x)}-\frac {\log (\sin (x))}{-\log \left (e^{\sin (x)}\right )+\sin (x)} \]
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Rubi [A]
time = 0.02, 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 = {4423, 2191,
2188, 29} \begin {gather*} \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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 2188
Rule 2191
Rule 4423
Rubi steps
\begin {align*} \int \frac {\cot (x)}{\log \left (e^{\sin (x)}\right )} \, dx &=\text {Subst}\left (\int \frac {1}{x \log \left (e^x\right )} \, dx,x,\sin (x)\right )\\ &=-\frac {\text {Subst}\left (\int \frac {1}{x} \, dx,x,\sin (x)\right )}{-\log \left (e^{\sin (x)}\right )+\sin (x)}+\frac {\text {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 {\text {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 = 0.03, size = 25, normalized size = 0.68 \begin {gather*} \frac {\log \left (\log \left (e^{\sin (x)}\right )\right )-\log (\sin (x))}{-\log \left (e^{\sin (x)}\right )+\sin (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 35, normalized size = 0.95
method | result | size |
derivativedivides | \(-\frac {\ln \left (\ln \left ({\mathrm e}^{\sin \left (x \right )}\right )\right )}{\ln \left ({\mathrm e}^{\sin \left (x \right )}\right )-\sin \left (x \right )}+\frac {\ln \left (\sin \left (x \right )\right )}{\ln \left ({\mathrm e}^{\sin \left (x \right )}\right )-\sin \left (x \right )}\) | \(35\) |
default | \(-\frac {\ln \left (\ln \left ({\mathrm e}^{\sin \left (x \right )}\right )\right )}{\ln \left ({\mathrm e}^{\sin \left (x \right )}\right )-\sin \left (x \right )}+\frac {\ln \left (\sin \left (x \right )\right )}{\ln \left ({\mathrm e}^{\sin \left (x \right )}\right )-\sin \left (x \right )}\) | \(35\) |
risch | \(\frac {\ln \left ({\mathrm e}^{i x}+1\right )}{\ln \left ({\mathrm e}^{\sin \left (x \right )}\right )-\sin \left (x \right )}+\frac {\ln \left ({\mathrm e}^{i x}-1\right )}{\ln \left ({\mathrm e}^{\sin \left (x \right )}\right )-\sin \left (x \right )}-\frac {\ln \left (2 i {\mathrm e}^{i x} \ln \left ({\mathrm e}^{-\frac {i {\mathrm e}^{i x}}{2}} {\mathrm e}^{\frac {i {\mathrm e}^{-i x}}{2}}\right )\right )}{\ln \left ({\mathrm e}^{\sin \left (x \right )}\right )-\sin \left (x \right )}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 6, normalized size = 0.16 \begin {gather*} -\frac {1}{\sin \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 6, normalized size = 0.16 \begin {gather*} -\frac {1}{\sin \left (x\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 (e^{\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 = 5.21, size = 6, normalized size = 0.16 \begin {gather*} -\frac {1}{\sin \left (x\right )} \end {gather*}
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 \begin {gather*} -\frac {1}{\sin \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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