Optimal. Leaf size=69 \[ \frac{\text{PolyLog}\left (3,-\frac{e x^m}{d}\right )}{m^2}-\frac{\log (x) \text{PolyLog}\left (2,-\frac{e x^m}{d}\right )}{m}+\frac{1}{2} \log ^2(x) \log \left (d+e x^m\right )-\frac{1}{2} \log ^2(x) \log \left (\frac{e x^m}{d}+1\right ) \]
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Rubi [A] time = 0.124492, antiderivative size = 69, normalized size of antiderivative = 1., 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 2375
Rule 2337
Rule 2374
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*}
Mathematica [A] time = 0.0579992, size = 75, normalized size = 1.09 \[ \frac{\text{PolyLog}\left (3,-\frac{d x^{-m}}{e}\right )}{m^2}+\frac{\log (x) \text{PolyLog}\left (2,-\frac{d x^{-m}}{e}\right )}{m}-\frac{1}{6} \log ^2(x) \left (3 \log \left (\frac{d x^{-m}}{e}+1\right )-3 \log \left (d+e x^m\right )+m \log (x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 2.256, size = 66, normalized size = 1. \begin{align*}{\frac{ \left ( \ln \left ( x \right ) \right ) ^{2}\ln \left ( d+e{x}^{m} \right ) }{2}}-{\frac{ \left ( \ln \left ( x \right ) \right ) ^{2}}{2}\ln \left ( 1+{\frac{e{x}^{m}}{d}} \right ) }-{\frac{\ln \left ( x \right ) }{m}{\it polylog} \left ( 2,-{\frac{e{x}^{m}}{d}} \right ) }+{\frac{1}{{m}^{2}}{\it polylog} \left ( 3,-{\frac{e{x}^{m}}{d}} \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*} -\frac{1}{6} \, m \log \left (x\right )^{3} + d m \int \frac{\log \left (x\right )^{2}}{2 \,{\left (e x x^{m} + d x\right )}}\,{d x} + \frac{1}{2} \, \log \left (e x^{m} + d\right ) \log \left (x\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.72426, size = 185, normalized size = 2.68 \begin{align*} \frac{m^{2} \log \left (e x^{m} + d\right ) \log \left (x\right )^{2} - m^{2} \log \left (x\right )^{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 \left (x\right ) + 2 \,{\rm polylog}\left (3, -\frac{e x^{m}}{d}\right )}{2 \, 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} + d\right ) \log \left (x\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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