Optimal. Leaf size=38 \[ \frac{\text{PolyLog}\left (2,-\frac{x^{-n}}{a}\right )}{2 n}-\frac{\text{PolyLog}\left (2,\frac{x^{-n}}{a}\right )}{2 n} \]
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Rubi [A] time = 0.023247, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6096, 5913} \[ \frac{\text{PolyLog}\left (2,-\frac{x^{-n}}{a}\right )}{2 n}-\frac{\text{PolyLog}\left (2,\frac{x^{-n}}{a}\right )}{2 n} \]
Antiderivative was successfully verified.
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Rule 6096
Rule 5913
Rubi steps
\begin{align*} \int \frac{\coth ^{-1}\left (a x^n\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\coth ^{-1}(a x)}{x} \, dx,x,x^n\right )}{n}\\ &=\frac{\text{Li}_2\left (-\frac{x^{-n}}{a}\right )}{2 n}-\frac{\text{Li}_2\left (\frac{x^{-n}}{a}\right )}{2 n}\\ \end{align*}
Mathematica [B] time = 0.0454459, size = 97, normalized size = 2.55 \[ \frac{-\text{PolyLog}\left (2,1-a x^n\right )+\text{PolyLog}\left (2,a x^n+1\right )+n \log (x) \log \left (a x^n-1\right )-n \log (x) \log \left (a x^n+1\right )-\log \left (a x^n\right ) \log \left (a x^n-1\right )+\log \left (-a x^n\right ) \log \left (a x^n+1\right )+2 n \log (x) \coth ^{-1}\left (a x^n\right )}{2 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.09, size = 61, normalized size = 1.6 \begin{align*}{\frac{\ln \left ( a{x}^{n} \right ){\rm arccoth} \left (a{x}^{n}\right )}{n}}-{\frac{{\it dilog} \left ( a{x}^{n} \right ) }{2\,n}}-{\frac{{\it dilog} \left ( a{x}^{n}+1 \right ) }{2\,n}}-{\frac{\ln \left ( a{x}^{n} \right ) \ln \left ( a{x}^{n}+1 \right ) }{2\,n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.26882, size = 198, normalized size = 5.21 \begin{align*} -\frac{1}{2} \, a n{\left (\frac{\log \left (\frac{a x^{n} + 1}{a}\right )}{a n} - \frac{\log \left (\frac{a x^{n} - 1}{a}\right )}{a n}\right )} \log \left (x\right ) + \frac{1}{2} \, a n{\left (\frac{\log \left (a x^{n} + 1\right ) \log \left (x\right ) - \log \left (a x^{n} - 1\right ) \log \left (x\right )}{a n} - \frac{n \log \left (a x^{n} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (-a x^{n}\right )}{a n^{2}} + \frac{n \log \left (-a x^{n} + 1\right ) \log \left (x\right ) +{\rm Li}_2\left (a x^{n}\right )}{a n^{2}}\right )} + \operatorname{arcoth}\left (a x^{n}\right ) \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.00394, size = 421, normalized size = 11.08 \begin{align*} -\frac{n \log \left (a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right ) + 1\right ) \log \left (x\right ) - n \log \left (-a \cosh \left (n \log \left (x\right )\right ) - a \sinh \left (n \log \left (x\right )\right ) + 1\right ) \log \left (x\right ) - n \log \left (x\right ) \log \left (\frac{a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right ) + 1}{a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right ) - 1}\right ) -{\rm Li}_2\left (a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right )\right ) +{\rm Li}_2\left (-a \cosh \left (n \log \left (x\right )\right ) - a \sinh \left (n \log \left (x\right )\right )\right )}{2 \, n} \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 \frac{\operatorname{acoth}{\left (a x^{n} \right )}}{x}\, 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 \frac{\operatorname{arcoth}\left (a x^{n}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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