Optimal. Leaf size=26 \[ -\frac{\text{PolyLog}\left (2,1-a x^{1-n}\right )}{a (1-n)} \]
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Rubi [A] time = 0.0903546, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {1593, 2336, 2315} \[ -\frac{\text{PolyLog}\left (2,1-a x^{1-n}\right )}{a (1-n)} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 2336
Rule 2315
Rubi steps
\begin{align*} \int \frac{\log \left (a x^{1-n}\right )}{a x-x^n} \, dx &=\int \frac{x^{-n} \log \left (a x^{1-n}\right )}{-1+a x^{1-n}} \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{\log (a x)}{-1+a x} \, dx,x,x^{1-n}\right )}{1-n}\\ &=-\frac{\text{Li}_2\left (1-a x^{1-n}\right )}{a (1-n)}\\ \end{align*}
Mathematica [A] time = 0.0095532, size = 23, normalized size = 0.88 \[ \frac{\text{PolyLog}\left (2,1-a x^{1-n}\right )}{a (n-1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.823, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( a{x}^{1-n} \right ) }{ax-{x}^{n}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (a x^{-n + 1}\right )}{a x - x^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.28845, size = 236, normalized size = 9.08 \begin{align*} \frac{2 \,{\left (n - 1\right )} \log \left (a\right ) \log \left (x\right ) -{\left (n^{2} - 2 \, n + 1\right )} \log \left (x\right )^{2} + 2 \,{\left (n - 1\right )} \log \left (x\right ) \log \left (\frac{a - x^{n - 1}}{a}\right ) - 2 \, \log \left (a\right ) \log \left (-a + x^{n - 1}\right ) + 2 \,{\rm Li}_2\left (-\frac{a - x^{n - 1}}{a} + 1\right )}{2 \,{\left (a n - 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 \frac{\log{\left (a x x^{- n} \right )}}{a x - x^{n}}\, 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{\log \left (a x^{-n + 1}\right )}{a x - x^{n}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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