Integrand size = 7, antiderivative size = 51 \[ \int \log \left (a \csc ^n(x)\right ) \, dx=-\frac {1}{2} i n x^2+n x \log \left (1-e^{2 i x}\right )+x \log \left (a \csc ^n(x)\right )-\frac {1}{2} i n \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {2628, 12, 3798, 2221, 2317, 2438} \[ \int \log \left (a \csc ^n(x)\right ) \, dx=x \log \left (a \csc ^n(x)\right )-\frac {1}{2} i n \operatorname {PolyLog}\left (2,e^{2 i x}\right )-\frac {1}{2} i n x^2+n x \log \left (1-e^{2 i x}\right ) \]
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Rule 12
Rule 2221
Rule 2317
Rule 2438
Rule 2628
Rule 3798
Rubi steps \begin{align*} \text {integral}& = x \log \left (a \csc ^n(x)\right )+\int n x \cot (x) \, dx \\ & = x \log \left (a \csc ^n(x)\right )+n \int x \cot (x) \, dx \\ & = -\frac {1}{2} i n x^2+x \log \left (a \csc ^n(x)\right )-(2 i n) \int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx \\ & = -\frac {1}{2} i n x^2+n x \log \left (1-e^{2 i x}\right )+x \log \left (a \csc ^n(x)\right )-n \int \log \left (1-e^{2 i x}\right ) \, dx \\ & = -\frac {1}{2} i n x^2+n x \log \left (1-e^{2 i x}\right )+x \log \left (a \csc ^n(x)\right )+\frac {1}{2} (i n) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i x}\right ) \\ & = -\frac {1}{2} i n x^2+n x \log \left (1-e^{2 i x}\right )+x \log \left (a \csc ^n(x)\right )-\frac {1}{2} i n \text {Li}_2\left (e^{2 i x}\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.00 \[ \int \log \left (a \csc ^n(x)\right ) \, dx=-\frac {1}{2} i n x^2+n x \log \left (1-e^{2 i x}\right )+x \log \left (a \csc ^n(x)\right )-\frac {1}{2} i n \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \]
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\[\int \ln \left (a \left (\csc ^{n}\left (x \right )\right )\right )d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (36) = 72\).
Time = 0.36 (sec) , antiderivative size = 117, normalized size of antiderivative = 2.29 \[ \int \log \left (a \csc ^n(x)\right ) \, dx=n x \log \left (\frac {1}{\sin \left (x\right )}\right ) + \frac {1}{2} \, n x \log \left (\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) + \frac {1}{2} \, n x \log \left (\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) + \frac {1}{2} \, n x \log \left (-\cos \left (x\right ) + i \, \sin \left (x\right ) + 1\right ) + \frac {1}{2} \, n x \log \left (-\cos \left (x\right ) - i \, \sin \left (x\right ) + 1\right ) - \frac {1}{2} i \, n {\rm Li}_2\left (\cos \left (x\right ) + i \, \sin \left (x\right )\right ) + \frac {1}{2} i \, n {\rm Li}_2\left (\cos \left (x\right ) - i \, \sin \left (x\right )\right ) + \frac {1}{2} i \, n {\rm Li}_2\left (-\cos \left (x\right ) + i \, \sin \left (x\right )\right ) - \frac {1}{2} i \, n {\rm Li}_2\left (-\cos \left (x\right ) - i \, \sin \left (x\right )\right ) + x \log \left (a\right ) \]
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\[ \int \log \left (a \csc ^n(x)\right ) \, dx=\int \log {\left (a \csc ^{n}{\left (x \right )} \right )}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (36) = 72\).
Time = 0.46 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.78 \[ \int \log \left (a \csc ^n(x)\right ) \, dx=\frac {1}{2} \, {\left (-i \, x^{2} + 2 i \, x \arctan \left (\sin \left (x\right ), \cos \left (x\right ) + 1\right ) - 2 i \, x \arctan \left (\sin \left (x\right ), -\cos \left (x\right ) + 1\right ) + x \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + x \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) - 2 i \, {\rm Li}_2\left (-e^{\left (i \, x\right )}\right ) - 2 i \, {\rm Li}_2\left (e^{\left (i \, x\right )}\right )\right )} n + x \log \left (a \csc \left (x\right )^{n}\right ) \]
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\[ \int \log \left (a \csc ^n(x)\right ) \, dx=\int { \log \left (a \csc \left (x\right )^{n}\right ) \,d x } \]
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Timed out. \[ \int \log \left (a \csc ^n(x)\right ) \, dx=\int \ln \left (a\,{\left (\frac {1}{\sin \left (x\right )}\right )}^n\right ) \,d x \]
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