Optimal. Leaf size=27 \[ \frac {\text {Li}_2\left (\frac {(1-c) \left (a x^{-m}+b\right )}{b}\right )}{a m} \]
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Rubi [A] time = 0.13, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 2391
Rule 2393
Rule 2412
Rule 2475
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*}
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Mathematica [A] time = 0.02, size = 29, normalized size = 1.07 \[ \frac {\text {Li}_2\left (-\frac {(c-1) x^{-m} \left (b x^m+a\right )}{b}\right )}{a m} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 33, normalized size = 1.22 \[ \frac {{\rm Li}_2\left (-\frac {b c x^{m} + a c - a}{b x^{m}} + 1\right )}{a m} \]
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 \[ \int \frac {\log \left (c + \frac {a {\left (c - 1\right )}}{b x^{m}}\right )}{{\left (b x^{m} + a\right )} x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 24, normalized size = 0.89 \[ \frac {\dilog \left (\frac {\left (c -1\right ) a \,x^{-m}}{b}+c \right )}{a m} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ {\left (c m - m\right )} \int \frac {\log \relax (x)}{b c x x^{m} + a {\left (c - 1\right )} x}\,{d x} + \frac {\log \left (b c x^{m} + a c - a\right ) \log \relax (x) - \log \relax (b) \log \relax (x) - \log \relax (x) \log \left (x^{m}\right )}{a} + \frac {\log \relax (b) \log \left (\frac {b x^{m} + a}{b}\right )}{a m} + \frac {\log \left (x^{m}\right ) \log \left (\frac {b x^{m}}{a} + 1\right ) + {\rm Li}_2\left (-\frac {b x^{m}}{a}\right )}{a m} - \frac {\log \left (b c x^{m} + a c - a\right ) \log \left (\frac {b c x^{m} + a {\left (c - 1\right )}}{a} + 1\right ) + {\rm Li}_2\left (-\frac {b c x^{m} + a {\left (c - 1\right )}}{a}\right )}{a m} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\ln \left (c+\frac {a\,\left (c-1\right )}{b\,x^m}\right )}{x\,\left (a+b\,x^m\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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