Optimal. Leaf size=121 \[ -\frac{p \text{PolyLog}\left (2,-\frac{g \left (d+e x^n\right )}{e f-d g}\right )}{f n}+\frac{p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{f n}-\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (f+g x^n\right )}{e f-d g}\right )}{f n}+\frac{\log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n} \]
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Rubi [A] time = 0.195171, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {2475, 36, 29, 31, 2416, 2394, 2315, 2393, 2391} \[ -\frac{p \text{PolyLog}\left (2,-\frac{g \left (d+e x^n\right )}{e f-d g}\right )}{f n}+\frac{p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )}{f n}-\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (f+g x^n\right )}{e f-d g}\right )}{f n}+\frac{\log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n} \]
Antiderivative was successfully verified.
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Rule 2475
Rule 36
Rule 29
Rule 31
Rule 2416
Rule 2394
Rule 2315
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 )}{x (f+g x)} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{\log \left (c (d+e x)^p\right )}{f x}-\frac{g \log \left (c (d+e x)^p\right )}{f (f+g x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{f n}-\frac{g \operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{f+g x} \, dx,x,x^n\right )}{f n}\\ &=\frac{\log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (f+g x^n\right )}{e f-d g}\right )}{f n}-\frac{(e p) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{e x}{d}\right )}{d+e x} \, dx,x,x^n\right )}{f n}+\frac{(e p) \operatorname{Subst}\left (\int \frac{\log \left (\frac{e (f+g x)}{e f-d g}\right )}{d+e x} \, dx,x,x^n\right )}{f n}\\ &=\frac{\log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (f+g x^n\right )}{e f-d g}\right )}{f n}+\frac{p \text{Li}_2\left (1+\frac{e x^n}{d}\right )}{f n}+\frac{p \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{g x}{e f-d g}\right )}{x} \, dx,x,d+e x^n\right )}{f n}\\ &=\frac{\log \left (-\frac{e x^n}{d}\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (f+g x^n\right )}{e f-d g}\right )}{f n}-\frac{p \text{Li}_2\left (-\frac{g \left (d+e x^n\right )}{e f-d g}\right )}{f n}+\frac{p \text{Li}_2\left (1+\frac{e x^n}{d}\right )}{f n}\\ \end{align*}
Mathematica [A] time = 0.0749918, size = 92, normalized size = 0.76 \[ \frac{-p \text{PolyLog}\left (2,\frac{g \left (d+e x^n\right )}{d g-e f}\right )+p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )+\log \left (c \left (d+e x^n\right )^p\right ) \left (\log \left (-\frac{e x^n}{d}\right )-\log \left (\frac{e \left (f+g x^n\right )}{e f-d g}\right )\right )}{f n} \]
Antiderivative was successfully verified.
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Maple [C] time = 1.197, size = 532, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.43952, size = 208, normalized size = 1.72 \begin{align*} -e n p{\left (\frac{\log \left (x^{n}\right ) \log \left (\frac{e x^{n}}{d} + 1\right ) +{\rm Li}_2\left (-\frac{e x^{n}}{d}\right )}{e f n^{2}} - \frac{\log \left (g x^{n} + f\right ) \log \left (-\frac{e g x^{n} + e f}{e f - d g} + 1\right ) +{\rm Li}_2\left (\frac{e g x^{n} + e f}{e f - d g}\right )}{e f n^{2}}\right )} -{\left (\frac{\log \left (g x^{n} + f\right )}{f n} - \frac{\log \left (x^{n}\right )}{f n}\right )} \log \left ({\left (e x^{n} + d\right )}^{p} c\right ) \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 )}{g x x^{n} + f 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 (g x^{n} + f\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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