Optimal. Leaf size=33 \[ -\frac {\log \left (f x^p\right ) \text {Li}_2\left (-e x^m\right )}{m}+\frac {p \text {Li}_3\left (-e x^m\right )}{m^2} \]
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Rubi [A]
time = 0.02, 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 = {2421, 6724}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2421
Rule 6724
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 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.45, size = 191, normalized size = 5.79
method | result | size |
risch | \(-\frac {p \ln \left (x \right ) \polylog \left (2, -e \,x^{m}\right )}{m}+\frac {p \polylog \left (3, -e \,x^{m}\right )}{m^{2}}-\frac {\left (\ln \left (x^{p}\right )-\ln \left (x \right ) p \right ) \dilog \left (1+e \,x^{m}\right )}{m}+\frac {i \dilog \left (1+e \,x^{m}\right ) \pi \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i x^{p}\right ) \mathrm {csgn}\left (i f \,x^{p}\right )}{2 m}-\frac {i \dilog \left (1+e \,x^{m}\right ) \pi \,\mathrm {csgn}\left (i f \right ) \mathrm {csgn}\left (i f \,x^{p}\right )^{2}}{2 m}-\frac {i \dilog \left (1+e \,x^{m}\right ) \pi \,\mathrm {csgn}\left (i x^{p}\right ) \mathrm {csgn}\left (i f \,x^{p}\right )^{2}}{2 m}+\frac {i \dilog \left (1+e \,x^{m}\right ) \pi \mathrm {csgn}\left (i f \,x^{p}\right )^{3}}{2 m}-\frac {\dilog \left (1+e \,x^{m}\right ) \ln \left (f \right )}{m}\) | \(191\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 37, normalized size = 1.12 \begin {gather*} -\frac {{\left (m p \log \left (x\right ) + m \log \left (f\right )\right )} {\rm Li}_2\left (-x^{m} e\right ) - p {\rm polylog}\left (3, -x^{m} e\right )}{m^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\ln \left (f\,x^p\right )\,\ln \left (e\,x^m+1\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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