Integrand size = 5, antiderivative size = 46 \[ \int \log (a \csc (x)) \, dx=-\frac {i x^2}{2}+x \log \left (1-e^{2 i x}\right )+x \log (a \csc (x))-\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {2628, 3798, 2221, 2317, 2438} \[ \int \log (a \csc (x)) \, dx=x \log (a \csc (x))-\frac {1}{2} i \operatorname {PolyLog}\left (2,e^{2 i x}\right )-\frac {i x^2}{2}+x \log \left (1-e^{2 i x}\right ) \]
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Rule 2221
Rule 2317
Rule 2438
Rule 2628
Rule 3798
Rubi steps \begin{align*} \text {integral}& = 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 \text {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*}
Time = 0.01 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.89 \[ \int \log (a \csc (x)) \, dx=x \log \left (1-e^{2 i x}\right )+x \log (a \csc (x))-\frac {1}{2} i \left (x^2+\operatorname {PolyLog}\left (2,e^{2 i x}\right )\right ) \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 82 vs. \(2 (36 ) = 72\).
Time = 1.14 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.80
method | result | size |
default | \(-i \left (\ln \left (2\right ) \ln \left ({\mathrm e}^{i x}\right )+\ln \left ({\mathrm e}^{i x}\right ) \ln \left (\frac {i a \,{\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\right )-\frac {\ln \left ({\mathrm e}^{i x}\right )^{2}}{2}-\operatorname {dilog}\left ({\mathrm e}^{i x}\right )+\ln \left ({\mathrm e}^{i x}\right ) \ln \left ({\mathrm e}^{i x}+1\right )+\operatorname {dilog}\left ({\mathrm e}^{i x}+1\right )\right )\) | \(83\) |
risch | \(x \ln \left ({\mathrm e}^{i x}\right )-\frac {i x^{2}}{2}+\frac {i \pi x}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\right ) \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\right )^{2} x}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i {\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\right )^{3} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\right ) \operatorname {csgn}\left (\frac {a \,{\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\right )^{2} x}{2}+\frac {i \pi \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\right )^{2} \operatorname {csgn}\left (i a \right ) x}{2}-i \ln \left ({\mathrm e}^{i x}\right ) \ln \left ({\mathrm e}^{i x}+1\right )-\frac {i \pi \operatorname {csgn}\left (\frac {a \,{\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\right )^{2} x}{2}-i \operatorname {dilog}\left ({\mathrm e}^{i x}+1\right )+i \operatorname {dilog}\left ({\mathrm e}^{i x}\right )+\ln \left (a \right ) x +x \ln \left (2\right )+i \ln \left ({\mathrm e}^{i x}\right ) \ln \left ({\mathrm e}^{2 i x}-1\right )+\frac {i \pi \operatorname {csgn}\left (\frac {a \,{\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\right )^{3} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i x}-1}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\right )^{2} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i {\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\right ) \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\right ) \operatorname {csgn}\left (i a \right ) x}{2}-\frac {i \pi \,\operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\right ) \operatorname {csgn}\left (\frac {a \,{\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\right ) x}{2}-\frac {i \pi \operatorname {csgn}\left (\frac {i a \,{\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\right )^{3} x}{2}+\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{i x}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\right )^{2} x}{2}-\frac {i \pi \,\operatorname {csgn}\left (i {\mathrm e}^{i x}\right ) \operatorname {csgn}\left (\frac {i}{{\mathrm e}^{2 i x}-1}\right ) \operatorname {csgn}\left (\frac {i {\mathrm e}^{i x}}{{\mathrm e}^{2 i x}-1}\right ) x}{2}\) | \(505\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 106 vs. \(2 (31) = 62\).
Time = 0.37 (sec) , antiderivative size = 106, normalized size of antiderivative = 2.30 \[ \int \log (a \csc (x)) \, dx=x \log \left (\frac {a}{\sin \left (x\right )}\right ) + \frac {1}{2} \, x \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) + \frac {1}{2} \, x \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) + \frac {1}{2} \, x \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) + \frac {1}{2} \, x \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - \frac {1}{2} i \, {\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + \frac {1}{2} i \, {\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) + \frac {1}{2} i \, {\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac {1}{2} i \, {\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) \]
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\[ \int \log (a \csc (x)) \, dx=\int \log {\left (a \csc {\left (x \right )} \right )}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 87 vs. \(2 (31) = 62\).
Time = 0.43 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.89 \[ \int \log (a \csc (x)) \, dx=-\frac {1}{2} i \, x^{2} + i \, x \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - i \, x \arctan \left (\sin \left (x\right ), -\cos \left (x\right ) + 1\right ) + \frac {1}{2} \, x \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + \frac {1}{2} \, x \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + x \log \left (a \csc \left (x\right )\right ) - i \, {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) - i \, {\rm Li}_2\left (e^{\left (i \, x\right )}\right ) \]
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\[ \int \log (a \csc (x)) \, dx=\int { \log \left (a \csc \left (x\right )\right ) \,d x } \]
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Timed out. \[ \int \log (a \csc (x)) \, dx=\int \ln \left (\frac {a}{\sin \left (x\right )}\right ) \,d x \]
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