Optimal. Leaf size=33 \[ \frac {p \text {Li}_3\left (-e x^m\right )}{m^2}-\frac {\text {Li}_2\left (-e x^m\right ) \log \left (f x^p\right )}{m} \]
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Rubi [A] time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, size = 33, normalized size = 1.00 \[ \frac {p \text {Li}_3\left (-e x^m\right )}{m^2}-\frac {\text {Li}_2\left (-e x^m\right ) \log \left (f x^p\right )}{m} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.70, size = 35, normalized size = 1.06 \[ -\frac {{\left (m p \log \relax (x) + m \log \relax (f)\right )} {\rm Li}_2\left (-e x^{m}\right ) - p {\rm polylog}\left (3, -e x^{m}\right )}{m^{2}} \]
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 \[ \int \frac {\log \left (e x^{m} + 1\right ) \log \left (f x^{p}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.06, size = 191, normalized size = 5.79 \[ \frac {i \pi \dilog \left (e \,x^{m}+1\right ) \mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i x^{p}\right ) \mathrm {csgn}\left (i f \,x^{p}\right )}{2 m}-\frac {i \pi \dilog \left (e \,x^{m}+1\right ) \mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i f \,x^{p}\right )^{2}}{2 m}-\frac {i \pi \dilog \left (e \,x^{m}+1\right ) \mathrm {csgn}\left (i x^{p}\right ) \mathrm {csgn}\left (i f \,x^{p}\right )^{2}}{2 m}+\frac {i \pi \dilog \left (e \,x^{m}+1\right ) \mathrm {csgn}\left (i f \,x^{p}\right )^{3}}{2 m}-\frac {p \polylog \left (2, -e \,x^{m}\right ) \ln \relax (x )}{m}-\frac {\dilog \left (e \,x^{m}+1\right ) \ln \relax (f )}{m}-\frac {\left (-p \ln \relax (x )+\ln \left (x^{p}\right )\right ) \dilog \left (e \,x^{m}+1\right )}{m}+\frac {p \polylog \left (3, -e \,x^{m}\right )}{m^{2}} \]
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 \[ -\frac {1}{2} \, {\left (p \log \relax (x)^{2} - 2 \, \log \relax (f) \log \relax (x) - 2 \, \log \relax (x) \log \left (x^{p}\right )\right )} \log \left (e x^{m} + 1\right ) - \int \frac {2 \, e m x^{m} \log \relax (x) \log \left (x^{p}\right ) - {\left (e m p \log \relax (x)^{2} - 2 \, e m \log \relax (f) \log \relax (x)\right )} x^{m}}{2 \, {\left (e x x^{m} + x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\ln \left (f\,x^p\right )\,\ln \left (e\,x^m+1\right )}{x} \,d x \]
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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