Optimal. Leaf size=46 \[ x \log (a \csc (x))-\frac {1}{2} i \text {Li}_2\left (e^{2 i x}\right )-\frac {i x^2}{2}+x \log \left (1-e^{2 i x}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2548, 3717, 2190, 2279, 2391} \[ -\frac {1}{2} i \text {PolyLog}\left (2,e^{2 i x}\right )+x \log (a \csc (x))-\frac {i x^2}{2}+x \log \left (1-e^{2 i x}\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 2548
Rule 3717
Rubi steps
\begin {align*} \int \log (a \csc (x)) \, dx &=x \log (a \csc (x))+\int x \cot (x) \, dx\\ &=-\frac {i x^2}{2}+x \log (a \csc (x))-2 i \int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx\\ &=-\frac {i x^2}{2}+x \log \left (1-e^{2 i x}\right )+x \log (a \csc (x))-\int \log \left (1-e^{2 i x}\right ) \, dx\\ &=-\frac {i x^2}{2}+x \log \left (1-e^{2 i x}\right )+x \log (a \csc (x))+\frac {1}{2} i \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i x}\right )\\ &=-\frac {i x^2}{2}+x \log \left (1-e^{2 i x}\right )+x \log (a \csc (x))-\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 = 41, normalized size = 0.89 \[ x \log (a \csc (x))-\frac {1}{2} i \left (x^2+\text {Li}_2\left (e^{2 i x}\right )\right )+x \log \left (1-e^{2 i x}\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 106, normalized size = 2.30 \[ x \log \left (\frac {a}{\sin \relax (x)}\right ) + \frac {1}{2} \, x \log \left (\cos \relax (x) + i \, \sin \relax (x) + 1\right ) + \frac {1}{2} \, x \log \left (\cos \relax (x) - i \, \sin \relax (x) + 1\right ) + \frac {1}{2} \, x \log \left (-\cos \relax (x) + i \, \sin \relax (x) + 1\right ) + \frac {1}{2} \, x \log \left (-\cos \relax (x) - i \, \sin \relax (x) + 1\right ) - \frac {1}{2} i \, {\rm Li}_2\left (\cos \relax (x) + i \, \sin \relax (x)\right ) + \frac {1}{2} i \, {\rm Li}_2\left (\cos \relax (x) - i \, \sin \relax (x)\right ) + \frac {1}{2} i \, {\rm Li}_2\left (-\cos \relax (x) + i \, \sin \relax (x)\right ) - \frac {1}{2} i \, {\rm Li}_2\left (-\cos \relax (x) - i \, \sin \relax (x)\right ) \]
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 \[ \int \log \left (a \csc \relax (x)\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.49, size = 89, normalized size = 1.93 \[ -i \ln \left (\frac {i a \,{\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\right ) \ln \left ({\mathrm e}^{i x}\right )-i \ln \left ({\mathrm e}^{i x}+1\right ) \ln \left ({\mathrm e}^{i x}\right )+\frac {i \ln \left ({\mathrm e}^{i x}\right )^{2}}{2}-i \dilog \left ({\mathrm e}^{i x}+1\right )+i \dilog \left ({\mathrm e}^{i x}\right )-i \ln \relax (2) \ln \left ({\mathrm e}^{i x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.26, size = 87, normalized size = 1.89 \[ -\frac {1}{2} i \, x^{2} + i \, x \arctan \left (\sin \relax (x), \cos \relax (x) + 1\right ) - i \, x \arctan \left (\sin \relax (x), -\cos \relax (x) + 1\right ) + \frac {1}{2} \, x \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} + 2 \, \cos \relax (x) + 1\right ) + \frac {1}{2} \, x \log \left (\cos \relax (x)^{2} + \sin \relax (x)^{2} - 2 \, \cos \relax (x) + 1\right ) + x \log \left (a \csc \relax (x)\right ) - i \, {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) - i \, {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) \]
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 \[ \int \ln \left (\frac {a}{\sin \relax (x)}\right ) \,d x \]
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 \[ \int \log {\left (a \csc {\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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