Optimal. Leaf size=33 \[ \frac{p \text{PolyLog}\left (3,-e x^m\right )}{m^2}-\frac{\log \left (f x^p\right ) \text{PolyLog}\left (2,-e x^m\right )}{m} \]
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Rubi [A] time = 0.0260934, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2374, 6589} \[ \frac{p \text{PolyLog}\left (3,-e x^m\right )}{m^2}-\frac{\log \left (f x^p\right ) \text{PolyLog}\left (2,-e x^m\right )}{m} \]
Antiderivative was successfully verified.
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Rule 2374
Rule 6589
Rubi steps
\begin{align*} \int \frac{\log \left (f x^p\right ) \log \left (1+e x^m\right )}{x} \, dx &=-\frac{\log \left (f x^p\right ) \text{Li}_2\left (-e x^m\right )}{m}+\frac{p \int \frac{\text{Li}_2\left (-e x^m\right )}{x} \, dx}{m}\\ &=-\frac{\log \left (f x^p\right ) \text{Li}_2\left (-e x^m\right )}{m}+\frac{p \text{Li}_3\left (-e x^m\right )}{m^2}\\ \end{align*}
Mathematica [A] time = 0.0115781, size = 33, normalized size = 1. \[ \frac{p \text{PolyLog}\left (3,-e x^m\right )}{m^2}-\frac{\log \left (f x^p\right ) \text{PolyLog}\left (2,-e x^m\right )}{m} \]
Antiderivative was successfully verified.
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Maple [C] time = 2.384, size = 191, normalized size = 5.8 \begin{align*} -{\frac{p\ln \left ( x \right ){\it polylog} \left ( 2,-e{x}^{m} \right ) }{m}}+{\frac{p{\it polylog} \left ( 3,-e{x}^{m} \right ) }{{m}^{2}}}-{\frac{ \left ( \ln \left ({x}^{p} \right ) -p\ln \left ( x \right ) \right ){\it dilog} \left ( 1+e{x}^{m} \right ) }{m}}+{\frac{{\frac{i}{2}}{\it dilog} \left ( 1+e{x}^{m} \right ) \pi \,{\it csgn} \left ( if \right ){\it csgn} \left ( i{x}^{p} \right ){\it csgn} \left ( if{x}^{p} \right ) }{m}}-{\frac{{\frac{i}{2}}{\it dilog} \left ( 1+e{x}^{m} \right ) \pi \,{\it csgn} \left ( if \right ) \left ({\it csgn} \left ( if{x}^{p} \right ) \right ) ^{2}}{m}}-{\frac{{\frac{i}{2}}{\it dilog} \left ( 1+e{x}^{m} \right ) \pi \,{\it csgn} \left ( i{x}^{p} \right ) \left ({\it csgn} \left ( if{x}^{p} \right ) \right ) ^{2}}{m}}+{\frac{{\frac{i}{2}}{\it dilog} \left ( 1+e{x}^{m} \right ) \pi \, \left ({\it csgn} \left ( if{x}^{p} \right ) \right ) ^{3}}{m}}-{\frac{{\it dilog} \left ( 1+e{x}^{m} \right ) \ln \left ( f \right ) }{m}} \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*} -\frac{1}{2} \,{\left (p \log \left (x\right )^{2} - 2 \, \log \left (f\right ) \log \left (x\right ) - 2 \, \log \left (x\right ) \log \left (x^{p}\right )\right )} \log \left (e x^{m} + 1\right ) - \int \frac{2 \, e m x^{m} \log \left (x\right ) \log \left (x^{p}\right ) -{\left (e m p \log \left (x\right )^{2} - 2 \, e m \log \left (f\right ) \log \left (x\right )\right )} x^{m}}{2 \,{\left (e x x^{m} + x\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.56477, size = 93, normalized size = 2.82 \begin{align*} -\frac{{\left (m p \log \left (x\right ) + m \log \left (f\right )\right )}{\rm Li}_2\left (-e x^{m}\right ) - p{\rm polylog}\left (3, -e x^{m}\right )}{m^{2}} \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 (e x^{m} + 1\right ) \log \left (f x^{p}\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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