Optimal. Leaf size=56 \[ x \log \left (a \cot ^n(x)\right )+\frac {1}{2} i n \text {Li}_2\left (-e^{2 i x}\right )-\frac {1}{2} i n \text {Li}_2\left (e^{2 i x}\right )-2 n x \tanh ^{-1}\left (e^{2 i x}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {2548, 12, 4419, 4183, 2279, 2391} \[ \frac {1}{2} i n \text {PolyLog}\left (2,-e^{2 i x}\right )-\frac {1}{2} i n \text {PolyLog}\left (2,e^{2 i x}\right )+x \log \left (a \cot ^n(x)\right )-2 n x \tanh ^{-1}\left (e^{2 i x}\right ) \]
Antiderivative was successfully verified.
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Rule 12
Rule 2279
Rule 2391
Rule 2548
Rule 4183
Rule 4419
Rubi steps
\begin {align*} \int \log \left (a \cot ^n(x)\right ) \, dx &=x \log \left (a \cot ^n(x)\right )+\int n x \csc (x) \sec (x) \, dx\\ &=x \log \left (a \cot ^n(x)\right )+n \int x \csc (x) \sec (x) \, dx\\ &=x \log \left (a \cot ^n(x)\right )+(2 n) \int x \csc (2 x) \, dx\\ &=-2 n x \tanh ^{-1}\left (e^{2 i x}\right )+x \log \left (a \cot ^n(x)\right )-n \int \log \left (1-e^{2 i x}\right ) \, dx+n \int \log \left (1+e^{2 i x}\right ) \, dx\\ &=-2 n x \tanh ^{-1}\left (e^{2 i x}\right )+x \log \left (a \cot ^n(x)\right )+\frac {1}{2} (i n) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i x}\right )-\frac {1}{2} (i n) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i x}\right )\\ &=-2 n x \tanh ^{-1}\left (e^{2 i x}\right )+x \log \left (a \cot ^n(x)\right )+\frac {1}{2} i n \text {Li}_2\left (-e^{2 i x}\right )-\frac {1}{2} i n \text {Li}_2\left (e^{2 i x}\right )\\ \end {align*}
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Mathematica [A] time = 0.02, size = 81, normalized size = 1.45 \[ -\frac {1}{2} i \log (-i (-\tan (x)+i)) \log \left (a \cot ^n(x)\right )+\frac {1}{2} i \log (-i (\tan (x)+i)) \log \left (a \cot ^n(x)\right )+\frac {1}{2} i n \text {Li}_2(-i \tan (x))-\frac {1}{2} i n \text {Li}_2(i \tan (x)) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 158, normalized size = 2.82 \[ n x \log \left (\frac {\cos \left (2 \, x\right ) + 1}{\sin \left (2 \, x\right )}\right ) - \frac {1}{2} \, n x \log \left (\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right ) + 1\right ) - \frac {1}{2} \, n x \log \left (\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right ) + 1\right ) + \frac {1}{2} \, n x \log \left (-\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right ) + 1\right ) + \frac {1}{2} \, n x \log \left (-\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right ) + 1\right ) - \frac {1}{4} i \, n {\rm Li}_2\left (\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right )\right ) + \frac {1}{4} i \, n {\rm Li}_2\left (\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right )\right ) - \frac {1}{4} i \, n {\rm Li}_2\left (-\cos \left (2 \, x\right ) + i \, \sin \left (2 \, x\right )\right ) + \frac {1}{4} i \, n {\rm Li}_2\left (-\cos \left (2 \, x\right ) - i \, \sin \left (2 \, x\right )\right ) + x \log \relax (a) \]
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 \[ \int \log \left (a \cot \relax (x)^{n}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \ln \left (a \left (\cot ^{n}\relax (x )\right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.54, size = 49, normalized size = 0.88 \[ n x \log \left (\tan \relax (x)\right ) - \frac {1}{4} \, {\left (\pi \log \left (\tan \relax (x)^{2} + 1\right ) + 2 i \, {\rm Li}_2\left (i \, \tan \relax (x) + 1\right ) - 2 i \, {\rm Li}_2\left (-i \, \tan \relax (x) + 1\right )\right )} n + x \log \left (a \frac {1}{\tan \relax (x)}^{n}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 44, normalized size = 0.79 \[ x\,\ln \left (a\,{\mathrm {cot}\relax (x)}^n\right )-\frac {n\,\mathrm {polylog}\left (2,{\mathrm {e}}^{x\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}+\frac {n\,\mathrm {polylog}\left (2,-{\mathrm {e}}^{x\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}-2\,n\,x\,\mathrm {atanh}\left ({\mathrm {e}}^{x\,2{}\mathrm {i}}\right ) \]
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 \[ \int \log {\left (a \cot ^{n}{\relax (x )} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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