Optimal. Leaf size=27 \[ \frac{\text{PolyLog}\left (2,\frac{(1-c) \left (a x^{-m}+b\right )}{b}\right )}{a m} \]
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Rubi [A] time = 0.130007, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2475, 2412, 2393, 2391} \[ \frac{\text{PolyLog}\left (2,\frac{(1-c) \left (a x^{-m}+b\right )}{b}\right )}{a m} \]
Antiderivative was successfully verified.
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Rule 2475
Rule 2412
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{\log \left (c-\frac{a (1-c) x^{-m}}{b}\right )}{x \left (a+b x^m\right )} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\log \left (c-\frac{a (1-c) x}{b}\right )}{\left (a+\frac{b}{x}\right ) x} \, dx,x,x^{-m}\right )}{m}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\log \left (c-\frac{a (1-c) x}{b}\right )}{b+a x} \, dx,x,x^{-m}\right )}{m}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\log \left (1-\frac{(1-c) x}{b}\right )}{x} \, dx,x,b+a x^{-m}\right )}{a m}\\ &=\frac{\text{Li}_2\left (\frac{(1-c) \left (b+a x^{-m}\right )}{b}\right )}{a m}\\ \end{align*}
Mathematica [A] time = 0.0204637, size = 29, normalized size = 1.07 \[ \frac{\text{PolyLog}\left (2,-\frac{(c-1) x^{-m} \left (a+b x^m\right )}{b}\right )}{a m} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.067, size = 24, normalized size = 0.9 \begin{align*}{\frac{1}{am}{\it dilog} \left ( c+{\frac{a \left ( -1+c \right ) }{b{x}^{m}}} \right ) } \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*}{\left (c m - m\right )} \int \frac{\log \left (x\right )}{b c x x^{m} + a{\left (c - 1\right )} x}\,{d x} + \frac{\log \left (b c x^{m} + a c - a\right ) \log \left (x\right ) - \log \left (b\right ) \log \left (x\right ) - \log \left (x\right ) \log \left (x^{m}\right )}{a} + \frac{\log \left (b\right ) \log \left (\frac{b x^{m} + a}{b}\right )}{a m} + \frac{\log \left (x^{m}\right ) \log \left (\frac{b x^{m}}{a} + 1\right ) +{\rm Li}_2\left (-\frac{b x^{m}}{a}\right )}{a m} - \frac{\log \left (b c x^{m} + a c - a\right ) \log \left (\frac{b c x^{m} + a{\left (c - 1\right )}}{a} + 1\right ) +{\rm Li}_2\left (-\frac{b c x^{m} + a{\left (c - 1\right )}}{a}\right )}{a m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.56231, size = 63, normalized size = 2.33 \begin{align*} \frac{{\rm Li}_2\left (-\frac{b c x^{m} + a c - a}{b x^{m}} + 1\right )}{a m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 (c + \frac{a{\left (c - 1\right )}}{b x^{m}}\right )}{{\left (b x^{m} + a\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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