Optimal. Leaf size=51 \[ x \log (a \cot (x))+\frac {1}{2} i \text {Li}_2\left (-e^{2 i x}\right )-\frac {1}{2} i \text {Li}_2\left (e^{2 i x}\right )-2 x \tanh ^{-1}\left (e^{2 i x}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2548, 4419, 4183, 2279, 2391} \[ \frac {1}{2} i \text {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{2} i \text {PolyLog}\left (2,e^{2 i x}\right )+x \log (a \cot (x))-2 x \tanh ^{-1}\left (e^{2 i x}\right ) \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 2548
Rule 4183
Rule 4419
Rubi steps
\begin {align*} \int \log (a \cot (x)) \, dx &=x \log (a \cot (x))+\int x \csc (x) \sec (x) \, dx\\ &=x \log (a \cot (x))+2 \int x \csc (2 x) \, dx\\ &=-2 x \tanh ^{-1}\left (e^{2 i x}\right )+x \log (a \cot (x))-\int \log \left (1-e^{2 i x}\right ) \, dx+\int \log \left (1+e^{2 i x}\right ) \, dx\\ &=-2 x \tanh ^{-1}\left (e^{2 i x}\right )+x \log (a \cot (x))+\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i x}\right )-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i x}\right )\\ &=-2 x \tanh ^{-1}\left (e^{2 i x}\right )+x \log (a \cot (x))+\frac {1}{2} i \text {Li}_2\left (-e^{2 i x}\right )-\frac {1}{2} i \text {Li}_2\left (e^{2 i x}\right )\\ \end {align*}
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Mathematica [A] time = 0.01, size = 75, normalized size = 1.47 \[ \frac {1}{2} i \log (-i (-\cot (x)+i)) \log (a \cot (x))-\frac {1}{2} i \log (-i (\cot (x)+i)) \log (a \cot (x))+\frac {1}{2} i \text {Li}_2(-i \cot (x))-\frac {1}{2} i \text {Li}_2(i \cot (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 147, normalized size = 2.88 \[ x \log \left (\frac {a \cos \left (2 \, x\right ) + a}{\sin \left (2 \, x\right )}\right ) - \frac {1}{2} \, x \log \left (\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right ) + 1\right ) - \frac {1}{2} \, x \log \left (\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right ) + 1\right ) + \frac {1}{2} \, x \log \left (-\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right ) + 1\right ) + \frac {1}{2} \, x \log \left (-\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right ) + 1\right ) - \frac {1}{4} i \, {\rm Li}_2\left (\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right )\right ) + \frac {1}{4} i \, {\rm Li}_2\left (\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right )\right ) - \frac {1}{4} i \, {\rm Li}_2\left (-\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right )\right ) + \frac {1}{4} i \, {\rm Li}_2\left (-\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \log \left (a \cot \relax (x)\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.16, size = 82, normalized size = 1.61 \[ \frac {i \ln \left (a \cot \relax (x )\right ) \ln \left (\frac {i a \cot \relax (x )+a}{a}\right )}{2}-\frac {i \ln \left (a \cot \relax (x )\right ) \ln \left (-\frac {i a \cot \relax (x )-a}{a}\right )}{2}+\frac {i \dilog \left (\frac {i a \cot \relax (x )+a}{a}\right )}{2}-\frac {i \dilog \left (-\frac {i a \cot \relax (x )-a}{a}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.47, size = 43, normalized size = 0.84 \[ -\frac {1}{4} \, \pi \log \left (\tan \relax (x)^{2} + 1\right ) + x \log \left (\frac {a}{\tan \relax (x)}\right ) + x \log \left (\tan \relax (x)\right ) - \frac {1}{2} i \, {\rm Li}_2\left (i \, \tan \relax (x) + 1\right ) + \frac {1}{2} i \, {\rm Li}_2\left (-i \, \tan \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \ln \left (a\,\mathrm {cot}\relax (x)\right ) \,d x \]
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 \[ \int \log {\left (a \cot {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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