Optimal. Leaf size=69 \[ \frac {\text {Li}_3\left (-\frac {e x^m}{d}\right )}{m^2}-\frac {\log (x) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac {1}{2} \log ^2(x) \log \left (\frac {e x^m}{d}+1\right ) \]
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Rubi [A] time = 0.12, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2375, 2337, 2374, 6589} \[ \frac {\text {PolyLog}\left (3,-\frac {e x^m}{d}\right )}{m^2}-\frac {\log (x) \text {PolyLog}\left (2,-\frac {e x^m}{d}\right )}{m}+\frac {1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac {1}{2} \log ^2(x) \log \left (\frac {e x^m}{d}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2337
Rule 2374
Rule 2375
Rule 6589
Rubi steps
\begin {align*} \int \frac {\log (x) \log \left (d+e x^m\right )}{x} \, dx &=\frac {1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac {1}{2} (e m) \int \frac {x^{-1+m} \log ^2(x)}{d+e x^m} \, dx\\ &=\frac {1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac {1}{2} \log ^2(x) \log \left (1+\frac {e x^m}{d}\right )+\int \frac {\log (x) \log \left (1+\frac {e x^m}{d}\right )}{x} \, dx\\ &=\frac {1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac {1}{2} \log ^2(x) \log \left (1+\frac {e x^m}{d}\right )-\frac {\log (x) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {\int \frac {\text {Li}_2\left (-\frac {e x^m}{d}\right )}{x} \, dx}{m}\\ &=\frac {1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac {1}{2} \log ^2(x) \log \left (1+\frac {e x^m}{d}\right )-\frac {\log (x) \text {Li}_2\left (-\frac {e x^m}{d}\right )}{m}+\frac {\text {Li}_3\left (-\frac {e x^m}{d}\right )}{m^2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 75, normalized size = 1.09 \[ \frac {\text {Li}_3\left (-\frac {d x^{-m}}{e}\right )}{m^2}+\frac {\log (x) \text {Li}_2\left (-\frac {d x^{-m}}{e}\right )}{m}-\frac {1}{6} \log ^2(x) \left (3 \log \left (\frac {d x^{-m}}{e}+1\right )-3 \log \left (d+e x^m\right )+m \log (x)\right ) \]
Antiderivative was successfully verified.
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fricas [C] time = 0.43, size = 76, normalized size = 1.10 \[ \frac {m^{2} \log \left (e x^{m} + d\right ) \log \relax (x)^{2} - m^{2} \log \relax (x)^{2} \log \left (\frac {e x^{m} + d}{d}\right ) - 2 \, m {\rm Li}_2\left (-\frac {e x^{m} + d}{d} + 1\right ) \log \relax (x) + 2 \, {\rm polylog}\left (3, -\frac {e x^{m}}{d}\right )}{2 \, m^{2}} \]
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 \[ \int \frac {\log \left (e x^{m} + d\right ) \log \relax (x)}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.78, size = 66, normalized size = 0.96 \[ -\frac {\ln \relax (x )^{2} \ln \left (\frac {e \,x^{m}}{d}+1\right )}{2}+\frac {\ln \relax (x )^{2} \ln \left (e \,x^{m}+d \right )}{2}-\frac {\polylog \left (2, -\frac {e \,x^{m}}{d}\right ) \ln \relax (x )}{m}+\frac {\polylog \left (3, -\frac {e \,x^{m}}{d}\right )}{m^{2}} \]
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 \[ -\frac {1}{6} \, m \log \relax (x)^{3} + d m \int \frac {\log \relax (x)^{2}}{2 \, {\left (e x x^{m} + d x\right )}}\,{d x} + \frac {1}{2} \, \log \left (e x^{m} + d\right ) \log \relax (x)^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (d+e\,x^m\right )\,\ln \relax (x)}{x} \,d x \]
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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