Optimal. Leaf size=113 \[ -\frac{6 p^2 \text{PolyLog}\left (3,\frac{e x^n}{d}+1\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac{3 p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{n}+\frac{6 p^3 \text{PolyLog}\left (4,\frac{e x^n}{d}+1\right )}{n}+\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n} \]
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Rubi [A] time = 0.148454, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2454, 2396, 2433, 2374, 2383, 6589} \[ -\frac{6 p^2 \text{PolyLog}\left (3,\frac{e x^n}{d}+1\right ) \log \left (c \left (d+e x^n\right )^p\right )}{n}+\frac{3 p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{n}+\frac{6 p^3 \text{PolyLog}\left (4,\frac{e x^n}{d}+1\right )}{n}+\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n} \]
Antiderivative was successfully verified.
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Rule 2454
Rule 2396
Rule 2433
Rule 2374
Rule 2383
Rule 6589
Rubi steps
\begin{align*} \int \frac{\log ^3\left (c \left (d+e x^n\right )^p\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\log ^3\left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{n}\\ &=\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n}-\frac{(3 e p) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{e x}{d}\right ) \log ^2\left (c (d+e x)^p\right )}{d+e x} \, dx,x,x^n\right )}{n}\\ &=\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n}-\frac{(3 p) \operatorname{Subst}\left (\int \frac{\log ^2\left (c x^p\right ) \log \left (-\frac{e \left (-\frac{d}{e}+\frac{x}{e}\right )}{d}\right )}{x} \, dx,x,d+e x^n\right )}{n}\\ &=\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n}+\frac{3 p \log ^2\left (c \left (d+e x^n\right )^p\right ) \text{Li}_2\left (1+\frac{e x^n}{d}\right )}{n}-\frac{\left (6 p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^p\right ) \text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+e x^n\right )}{n}\\ &=\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n}+\frac{3 p \log ^2\left (c \left (d+e x^n\right )^p\right ) \text{Li}_2\left (1+\frac{e x^n}{d}\right )}{n}-\frac{6 p^2 \log \left (c \left (d+e x^n\right )^p\right ) \text{Li}_3\left (1+\frac{e x^n}{d}\right )}{n}+\frac{\left (6 p^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{x}{d}\right )}{x} \, dx,x,d+e x^n\right )}{n}\\ &=\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^3\left (c \left (d+e x^n\right )^p\right )}{n}+\frac{3 p \log ^2\left (c \left (d+e x^n\right )^p\right ) \text{Li}_2\left (1+\frac{e x^n}{d}\right )}{n}-\frac{6 p^2 \log \left (c \left (d+e x^n\right )^p\right ) \text{Li}_3\left (1+\frac{e x^n}{d}\right )}{n}+\frac{6 p^3 \text{Li}_4\left (1+\frac{e x^n}{d}\right )}{n}\\ \end{align*}
Mathematica [B] time = 0.0996806, size = 270, normalized size = 2.39 \[ \frac{-6 p^2 \text{PolyLog}\left (3,\frac{e x^n}{d}+1\right ) \log \left (c \left (d+e x^n\right )^p\right )+3 p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )+6 p^3 \text{PolyLog}\left (4,\frac{e x^n}{d}+1\right )+3 n p^2 \log (x) \log ^2\left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )-3 p^2 \log \left (-\frac{e x^n}{d}\right ) \log ^2\left (d+e x^n\right ) \log \left (c \left (d+e x^n\right )^p\right )-3 n p \log (x) \log \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )+3 p \log \left (-\frac{e x^n}{d}\right ) \log \left (d+e x^n\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )+n \log (x) \log ^3\left (c \left (d+e x^n\right )^p\right )-n p^3 \log (x) \log ^3\left (d+e x^n\right )+p^3 \log \left (-\frac{e x^n}{d}\right ) \log ^3\left (d+e x^n\right )}{n} \]
Antiderivative was successfully verified.
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Maple [C] time = 4.84, size = 6131, normalized size = 54.3 \begin{align*} \text{output too large to display} \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*} \log \left ({\left (e x^{n} + d\right )}^{p}\right )^{3} \log \left (x\right ) - \int -\frac{e x^{n} \log \left (c\right )^{3} + d \log \left (c\right )^{3} - 3 \,{\left ({\left (e n p \log \left (x\right ) - e \log \left (c\right )\right )} x^{n} - d \log \left (c\right )\right )} \log \left ({\left (e x^{n} + d\right )}^{p}\right )^{2} + 3 \,{\left (e x^{n} \log \left (c\right )^{2} + d \log \left (c\right )^{2}\right )} \log \left ({\left (e x^{n} + d\right )}^{p}\right )}{e x x^{n} + d x}\,{d x} \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{\log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{3}}{x}, x\right ) \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 )}^{3}}{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 )^{3}}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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