Optimal. Leaf size=44 \[ \frac{p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{n}+\frac{\log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n} \]
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Rubi [A] time = 0.0398748, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {2454, 2394, 2315} \[ \frac{p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{n}+\frac{\log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n} \]
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 )^p\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{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 )^p\right )}{n}-\frac{(e p) \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 )^p\right )}{n}+\frac{p \text{Li}_2\left (1+\frac{e x^n}{d}\right )}{n}\\ \end{align*}
Mathematica [A] time = 0.0029561, size = 43, normalized size = 0.98 \[ \frac{p \text{PolyLog}\left (2,\frac{d+e x^n}{d}\right )+\log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n} \]
Antiderivative was successfully verified.
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Maple [C] time = 2.52, size = 177, normalized size = 4. \begin{align*} \ln \left ( x \right ) \ln \left ( \left ( d+e{x}^{n} \right ) ^{p} \right ) +{\frac{i}{2}}\ln \left ( x \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}-{\frac{i}{2}}\ln \left ( x \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 ) -{\frac{i}{2}}\ln \left ( x \right ) \pi \, \left ({\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{3}+{\frac{i}{2}}\ln \left ( x \right ) \pi \, \left ({\it csgn} \left ( ic \left ( d+e{x}^{n} \right ) ^{p} \right ) \right ) ^{2}{\it csgn} \left ( ic \right ) +\ln \left ( c \right ) \ln \left ( x \right ) -{\frac{p}{n}{\it dilog} \left ({\frac{d+e{x}^{n}}{d}} \right ) }-p\ln \left ( x \right ) \ln \left ({\frac{d+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 p \int \frac{\log \left (x\right )}{e x x^{n} + d x}\,{d x} - \frac{1}{2} \, n p \log \left (x\right )^{2} + \log \left ({\left (e x^{n} + d\right )}^{p}\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 = 2.04783, size = 150, normalized size = 3.41 \begin{align*} \frac{n p \log \left (e x^{n} + d\right ) \log \left (x\right ) - n p \log \left (x\right ) \log \left (\frac{e x^{n} + d}{d}\right ) + n \log \left (c\right ) \log \left (x\right ) - p{\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 \left (d + e x^{n}\right )^{p} \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 )}^{p} c\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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