Optimal. Leaf size=9 \[ \log (\sin (x))-\tanh ^{-1}(\cos (x)) \]
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Rubi [A] time = 0.01, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3475, 3770} \[ \log (\sin (x))-\tanh ^{-1}(\cos (x)) \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3770
Rubi steps
\begin {align*} \int (\cot (x)+\csc (x)) \, dx &=\int \cot (x) \, dx+\int \csc (x) \, dx\\ &=-\tanh ^{-1}(\cos (x))+\log (\sin (x))\\ \end {align*}
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Mathematica [B] time = 0.00, size = 20, normalized size = 2.22 \[ \log \left (\sin \left (\frac {x}{2}\right )\right )+\log (\sin (x))-\log \left (\cos \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 7, normalized size = 0.78 \[ \log \left (-\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 11, normalized size = 1.22 \[ \log \left ({\left | \sin \relax (x) \right |}\right ) + \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 13, normalized size = 1.44 \[ \ln \left (\sin \relax (x )\right )-\ln \left (\cot \relax (x )+\csc \relax (x )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.30, size = 12, normalized size = 1.33 \[ -\log \left (\cot \relax (x) + \csc \relax (x)\right ) + \log \left (\sin \relax (x)\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.41, size = 19, normalized size = 2.11 \[ 2\,\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )-\ln \left ({\mathrm {tan}\left (\frac {x}{2}\right )}^2+1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 0.10, size = 20, normalized size = 2.22 \[ \frac {\log {\left (\cos {\relax (x )} - 1 \right )}}{2} - \frac {\log {\left (\cos {\relax (x )} + 1 \right )}}{2} + \log {\left (\sin {\relax (x )} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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