Optimal. Leaf size=28 \[ \frac{\text{PolyLog}\left (2,\frac{(1-a c) \left (d x^{-m}+e\right )}{e}\right )}{d m} \]
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Rubi [A] time = 0.182942, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.132, Rules used = {2480, 2475, 2412, 2393, 2391} \[ \frac{\text{PolyLog}\left (2,\frac{(1-a c) \left (d x^{-m}+e\right )}{e}\right )}{d m} \]
Antiderivative was successfully verified.
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Rule 2480
Rule 2475
Rule 2412
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{\log \left (\frac{x^{-m} \left (-d+a c d+a c e x^m\right )}{e}\right )}{x \left (d+e x^m\right )} \, dx &=\int \frac{\log \left (a c+\frac{(-d+a c d) x^{-m}}{e}\right )}{x \left (d+e x^m\right )} \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\log \left (a c+\frac{(-d+a c d) x}{e}\right )}{\left (d+\frac{e}{x}\right ) x} \, dx,x,x^{-m}\right )}{m}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\log \left (a c+\frac{(-d+a c d) x}{e}\right )}{e+d x} \, dx,x,x^{-m}\right )}{m}\\ &=-\frac{\operatorname{Subst}\left (\int \frac{\log \left (1+\frac{(-d+a c d) x}{d e}\right )}{x} \, dx,x,e+d x^{-m}\right )}{d m}\\ &=\frac{\text{Li}_2\left (\frac{(1-a c) \left (e+d x^{-m}\right )}{e}\right )}{d m}\\ \end{align*}
Mathematica [A] time = 0.006654, size = 31, normalized size = 1.11 \[ \frac{\text{PolyLog}\left (2,-\frac{(a c-1) x^{-m} \left (d+e x^m\right )}{e}\right )}{d m} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.064, size = 28, normalized size = 1. \begin{align*}{\frac{1}{md}{\it dilog} \left ( ac+{\frac{d \left ( ac-1 \right ) }{e{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 (a c m - m\right )} \int \frac{\log \left (x\right )}{a c e x x^{m} +{\left (a c d - d\right )} x}\,{d x} + \frac{\log \left (a c e x^{m} +{\left (a c - 1\right )} d\right ) \log \left (x\right ) - \log \left (e\right ) \log \left (x\right ) - \log \left (x\right ) \log \left (x^{m}\right )}{d} + \frac{\log \left (e\right ) \log \left (\frac{e x^{m} + d}{e}\right )}{d m} + \frac{\log \left (x^{m}\right ) \log \left (\frac{e x^{m}}{d} + 1\right ) +{\rm Li}_2\left (-\frac{e x^{m}}{d}\right )}{d m} - \frac{\log \left (a c e x^{m} +{\left (a c - 1\right )} d\right ) \log \left (\frac{a c e x^{m} + a c d - d}{d} + 1\right ) +{\rm Li}_2\left (-\frac{a c e x^{m} + a c d - d}{d}\right )}{d m} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.66443, size = 72, normalized size = 2.57 \begin{align*} \frac{{\rm Li}_2\left (-\frac{a c e x^{m} +{\left (a c - 1\right )} d}{e x^{m}} + 1\right )}{d 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 (\frac{a c e x^{m} + a c d - d}{e x^{m}}\right )}{{\left (e x^{m} + d\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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