Optimal. Leaf size=70 \[ \frac{p \text{PolyLog}\left (2,\frac{f \left (d+e x^n\right )}{d f-e g}\right )}{f n}+\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac{e \left (f x^n+g\right )}{d f-e g}\right )}{f n} \]
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Rubi [A] time = 0.163625, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2475, 2412, 2394, 2393, 2391} \[ \frac{p \text{PolyLog}\left (2,\frac{f \left (d+e x^n\right )}{d f-e g}\right )}{f n}+\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac{e \left (f x^n+g\right )}{d f-e g}\right )}{f n} \]
Antiderivative was successfully verified.
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Rule 2475
Rule 2412
Rule 2394
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-n}\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{\left (f+\frac{g}{x}\right ) x} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{g+f x} \, dx,x,x^n\right )}{n}\\ &=\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac{e \left (g+f x^n\right )}{d f-e g}\right )}{f n}-\frac{(e p) \operatorname{Subst}\left (\int \frac{\log \left (\frac{e (g+f x)}{-d f+e g}\right )}{d+e x} \, dx,x,x^n\right )}{f n}\\ &=\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac{e \left (g+f x^n\right )}{d f-e g}\right )}{f n}-\frac{p \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{f x}{-d f+e g}\right )}{x} \, dx,x,d+e x^n\right )}{f n}\\ &=\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac{e \left (g+f x^n\right )}{d f-e g}\right )}{f n}+\frac{p \text{Li}_2\left (\frac{f \left (d+e x^n\right )}{d f-e g}\right )}{f n}\\ \end{align*}
Mathematica [A] time = 0.0227354, size = 64, normalized size = 0.91 \[ \frac{p \text{PolyLog}\left (2,\frac{f \left (d+e x^n\right )}{d f-e g}\right )+\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (f x^n+g\right )}{e g-d f}\right )}{f n} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.198, size = 298, normalized size = 4.3 \begin{align*}{\frac{\ln \left ( g+f{x}^{n} \right ) \ln \left ( \left ( d+e{x}^{n} \right ) ^{p} \right ) }{nf}}-{\frac{p}{nf}{\it dilog} \left ({\frac{ \left ( g+f{x}^{n} \right ) e+df-eg}{df-eg}} \right ) }-{\frac{p\ln \left ( g+f{x}^{n} \right ) }{nf}\ln \left ({\frac{ \left ( g+f{x}^{n} \right ) e+df-eg}{df-eg}} \right ) }+{\frac{{\frac{i}{2}}\ln \left ( g+f{x}^{n} \right ) \pi \,{\it csgn} \left ( i \left ( d+e{x}^{n} \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{2}}{nf}}-{\frac{{\frac{i}{2}}\ln \left ( g+f{x}^{n} \right ) \pi \,{\it csgn} \left ( i \left ( d+e{x}^{n} \right ) ^{p} \right ){\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ){\it csgn} \left ( ic \right ) }{nf}}-{\frac{{\frac{i}{2}}\ln \left ( g+f{x}^{n} \right ) \pi \, \left ({\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{3}}{nf}}+{\frac{{\frac{i}{2}}\ln \left ( g+f{x}^{n} \right ) \pi \, \left ({\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) }{nf}}+{\frac{\ln \left ( g+f{x}^{n} \right ) \ln \left ( c \right ) }{nf}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.40701, size = 151, normalized size = 2.16 \begin{align*}{\left (\frac{\log \left (f + \frac{g}{x^{n}}\right )}{f n} - \frac{\log \left (\frac{1}{x^{n}}\right )}{f n}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) - \frac{{\left (\log \left (f x^{n} + g\right ) \log \left (\frac{e f x^{n} + e g}{d f - e g} + 1\right ) +{\rm Li}_2\left (-\frac{e f x^{n} + e g}{d f - e g}\right )\right )} p}{f n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{n} \log \left ({\left (e x^{n} + d\right )}^{p} c\right )}{f x x^{n} + g x}, x\right ) \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 ({\left (e x^{n} + d\right )}^{p} c\right )}{{\left (f + \frac{g}{x^{n}}\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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