Optimal. Leaf size=41 \[ \frac{\text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{n}+\frac{\log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )\right )}{n} \]
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Rubi [A] time = 0.0392548, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {2454, 2394, 2315} \[ \frac{\text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{n}+\frac{\log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )\right )}{n} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2394
Rule 2315
Rubi steps
\begin{align*} \int \frac{\log \left (c \left (d+e x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\log (c (d+e x))}{x} \, dx,x,x^n\right )}{n}\\ &=\frac{\log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )\right )}{n}-\frac{e \operatorname{Subst}\left (\int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx,x,x^n\right )}{n}\\ &=\frac{\log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )\right )}{n}+\frac{\text{Li}_2\left (1+\frac{e x^n}{d}\right )}{n}\\ \end{align*}
Mathematica [A] time = 0.0075831, size = 39, normalized size = 0.95 \[ \frac{\text{PolyLog}\left (2,\frac{d+e x^n}{d}\right )+\log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )\right )}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.114, size = 41, normalized size = 1. \begin{align*}{\frac{\ln \left ( ce{x}^{n}+cd \right ) }{n}\ln \left ( -{\frac{e{x}^{n}}{d}} \right ) }+{\frac{1}{n}{\it dilog} \left ( -{\frac{e{x}^{n}}{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*} d n \int \frac{\log \left (x\right )}{e x x^{n} + d x}\,{d x} - \frac{1}{2} \, n \log \left (x\right )^{2} + \log \left (e x^{n} + d\right ) \log \left (x\right ) + \log \left (c\right ) \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.13791, size = 123, normalized size = 3. \begin{align*} \frac{n \log \left (c e x^{n} + c d\right ) \log \left (x\right ) - n \log \left (x\right ) \log \left (\frac{e x^{n} + d}{d}\right ) -{\rm Li}_2\left (-\frac{e x^{n} + d}{d} + 1\right )}{n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (c d + c e x^{n} \right )}}{x}\, dx \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 )} c\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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