Optimal. Leaf size=56 \[ \frac{1}{2} i n \text{PolyLog}\left (2,-e^{2 i x}\right )-\frac{1}{2} i n \text{PolyLog}\left (2,e^{2 i x}\right )+x \log \left (a \cot ^n(x)\right )-2 n x \tanh ^{-1}\left (e^{2 i x}\right ) \]
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Rubi [A] time = 0.0486499, antiderivative size = 56, normalized size of antiderivative = 1., 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 2548
Rule 12
Rule 4419
Rule 4183
Rule 2279
Rule 2391
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*}
Mathematica [A] time = 0.0131143, size = 81, normalized size = 1.45 \[ \frac{1}{2} i n \text{PolyLog}(2,-i \tan (x))-\frac{1}{2} i n \text{PolyLog}(2,i \tan (x))-\frac{1}{2} i \log (-i (-\tan (x)+i)) \log \left (a \cot ^n(x)\right )+\frac{1}{2} i \log (-i (\tan (x)+i)) \log \left (a \cot ^n(x)\right ) \]
Antiderivative was successfully verified.
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Maple [C] time = 2.596, size = 6531, normalized size = 116.6 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.52, size = 66, normalized size = 1.18 \begin{align*} n x \log \left (\tan \left (x\right )\right ) - \frac{1}{4} \,{\left (\pi \log \left (\tan \left (x\right )^{2} + 1\right ) + 2 i \,{\rm Li}_2\left (i \, \tan \left (x\right ) + 1\right ) - 2 i \,{\rm Li}_2\left (-i \, \tan \left (x\right ) + 1\right )\right )} n + x \log \left (a \frac{1}{\tan \left (x\right )}^{n}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.14417, size = 498, normalized size = 8.89 \begin{align*} 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 \left (a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \log{\left (a \cot ^{n}{\left (x \right )} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \log \left (a \cot \left (x\right )^{n}\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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