Optimal. Leaf size=377 \[ \frac{p \text{PolyLog}\left (2,\frac{\sqrt{-f} \left (d+e x^n\right )}{d \sqrt{-f}-e \sqrt{g}}\right )}{2 f^2 n}+\frac{p \text{PolyLog}\left (2,\frac{\sqrt{-f} \left (d+e x^n\right )}{d \sqrt{-f}+e \sqrt{g}}\right )}{2 f^2 n}+\frac{g \log \left (c \left (d+e x^n\right )^p\right )}{2 f^2 n \left (f x^{2 n}+g\right )}+\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (\sqrt{g}-\sqrt{-f} x^n\right )}{d \sqrt{-f}+e \sqrt{g}}\right )}{2 f^2 n}+\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac{e \left (\sqrt{-f} x^n+\sqrt{g}\right )}{d \sqrt{-f}-e \sqrt{g}}\right )}{2 f^2 n}-\frac{e^2 g p \log \left (d+e x^n\right )}{2 f^2 n \left (d^2 f+e^2 g\right )}+\frac{e^2 g p \log \left (f x^{2 n}+g\right )}{4 f^2 n \left (d^2 f+e^2 g\right )}-\frac{d e \sqrt{g} p \tan ^{-1}\left (\frac{\sqrt{f} x^n}{\sqrt{g}}\right )}{2 f^{3/2} n \left (d^2 f+e^2 g\right )} \]
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Rubi [A] time = 0.608658, antiderivative size = 377, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 14, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.518, Rules used = {2475, 263, 266, 43, 2416, 2413, 706, 31, 635, 205, 260, 2394, 2393, 2391} \[ \frac{p \text{PolyLog}\left (2,\frac{\sqrt{-f} \left (d+e x^n\right )}{d \sqrt{-f}-e \sqrt{g}}\right )}{2 f^2 n}+\frac{p \text{PolyLog}\left (2,\frac{\sqrt{-f} \left (d+e x^n\right )}{d \sqrt{-f}+e \sqrt{g}}\right )}{2 f^2 n}+\frac{g \log \left (c \left (d+e x^n\right )^p\right )}{2 f^2 n \left (f x^{2 n}+g\right )}+\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (\sqrt{g}-\sqrt{-f} x^n\right )}{d \sqrt{-f}+e \sqrt{g}}\right )}{2 f^2 n}+\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac{e \left (\sqrt{-f} x^n+\sqrt{g}\right )}{d \sqrt{-f}-e \sqrt{g}}\right )}{2 f^2 n}-\frac{e^2 g p \log \left (d+e x^n\right )}{2 f^2 n \left (d^2 f+e^2 g\right )}+\frac{e^2 g p \log \left (f x^{2 n}+g\right )}{4 f^2 n \left (d^2 f+e^2 g\right )}-\frac{d e \sqrt{g} p \tan ^{-1}\left (\frac{\sqrt{f} x^n}{\sqrt{g}}\right )}{2 f^{3/2} n \left (d^2 f+e^2 g\right )} \]
Antiderivative was successfully verified.
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Rule 2475
Rule 263
Rule 266
Rule 43
Rule 2416
Rule 2413
Rule 706
Rule 31
Rule 635
Rule 205
Rule 260
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^{-2 n}\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{\left (f+\frac{g}{x^2}\right )^2 x} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{g x \log \left (c (d+e x)^p\right )}{f \left (g+f x^2\right )^2}+\frac{x \log \left (c (d+e x)^p\right )}{f \left (g+f x^2\right )}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \frac{x \log \left (c (d+e x)^p\right )}{g+f x^2} \, dx,x,x^n\right )}{f n}-\frac{g \operatorname{Subst}\left (\int \frac{x \log \left (c (d+e x)^p\right )}{\left (g+f x^2\right )^2} \, dx,x,x^n\right )}{f n}\\ &=\frac{g \log \left (c \left (d+e x^n\right )^p\right )}{2 f^2 n \left (g+f x^{2 n}\right )}+\frac{\operatorname{Subst}\left (\int \left (-\frac{\sqrt{-f} \log \left (c (d+e x)^p\right )}{2 f \left (\sqrt{g}-\sqrt{-f} x\right )}+\frac{\sqrt{-f} \log \left (c (d+e x)^p\right )}{2 f \left (\sqrt{g}+\sqrt{-f} x\right )}\right ) \, dx,x,x^n\right )}{f n}-\frac{(e g p) \operatorname{Subst}\left (\int \frac{1}{(d+e x) \left (g+f x^2\right )} \, dx,x,x^n\right )}{2 f^2 n}\\ &=\frac{g \log \left (c \left (d+e x^n\right )^p\right )}{2 f^2 n \left (g+f x^{2 n}\right )}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{\sqrt{g}-\sqrt{-f} x} \, dx,x,x^n\right )}{2 (-f)^{3/2} n}+\frac{\operatorname{Subst}\left (\int \frac{\log \left (c (d+e x)^p\right )}{\sqrt{g}+\sqrt{-f} x} \, dx,x,x^n\right )}{2 (-f)^{3/2} n}-\frac{(e g p) \operatorname{Subst}\left (\int \frac{d f-e f x}{g+f x^2} \, dx,x,x^n\right )}{2 f^2 \left (d^2 f+e^2 g\right ) n}-\frac{\left (e^3 g p\right ) \operatorname{Subst}\left (\int \frac{1}{d+e x} \, dx,x,x^n\right )}{2 f^2 \left (d^2 f+e^2 g\right ) n}\\ &=-\frac{e^2 g p \log \left (d+e x^n\right )}{2 f^2 \left (d^2 f+e^2 g\right ) n}+\frac{g \log \left (c \left (d+e x^n\right )^p\right )}{2 f^2 n \left (g+f x^{2 n}\right )}+\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (\sqrt{g}-\sqrt{-f} x^n\right )}{d \sqrt{-f}+e \sqrt{g}}\right )}{2 f^2 n}+\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac{e \left (\sqrt{g}+\sqrt{-f} x^n\right )}{d \sqrt{-f}-e \sqrt{g}}\right )}{2 f^2 n}-\frac{(e p) \operatorname{Subst}\left (\int \frac{\log \left (\frac{e \left (\sqrt{g}-\sqrt{-f} x\right )}{d \sqrt{-f}+e \sqrt{g}}\right )}{d+e x} \, dx,x,x^n\right )}{2 f^2 n}-\frac{(e p) \operatorname{Subst}\left (\int \frac{\log \left (\frac{e \left (\sqrt{g}+\sqrt{-f} x\right )}{-d \sqrt{-f}+e \sqrt{g}}\right )}{d+e x} \, dx,x,x^n\right )}{2 f^2 n}-\frac{(d e g p) \operatorname{Subst}\left (\int \frac{1}{g+f x^2} \, dx,x,x^n\right )}{2 f \left (d^2 f+e^2 g\right ) n}+\frac{\left (e^2 g p\right ) \operatorname{Subst}\left (\int \frac{x}{g+f x^2} \, dx,x,x^n\right )}{2 f \left (d^2 f+e^2 g\right ) n}\\ &=-\frac{d e \sqrt{g} p \tan ^{-1}\left (\frac{\sqrt{f} x^n}{\sqrt{g}}\right )}{2 f^{3/2} \left (d^2 f+e^2 g\right ) n}-\frac{e^2 g p \log \left (d+e x^n\right )}{2 f^2 \left (d^2 f+e^2 g\right ) n}+\frac{g \log \left (c \left (d+e x^n\right )^p\right )}{2 f^2 n \left (g+f x^{2 n}\right )}+\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (\sqrt{g}-\sqrt{-f} x^n\right )}{d \sqrt{-f}+e \sqrt{g}}\right )}{2 f^2 n}+\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac{e \left (\sqrt{g}+\sqrt{-f} x^n\right )}{d \sqrt{-f}-e \sqrt{g}}\right )}{2 f^2 n}+\frac{e^2 g p \log \left (g+f x^{2 n}\right )}{4 f^2 \left (d^2 f+e^2 g\right ) n}-\frac{p \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{-f} x}{-d \sqrt{-f}+e \sqrt{g}}\right )}{x} \, dx,x,d+e x^n\right )}{2 f^2 n}-\frac{p \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{-f} x}{d \sqrt{-f}+e \sqrt{g}}\right )}{x} \, dx,x,d+e x^n\right )}{2 f^2 n}\\ &=-\frac{d e \sqrt{g} p \tan ^{-1}\left (\frac{\sqrt{f} x^n}{\sqrt{g}}\right )}{2 f^{3/2} \left (d^2 f+e^2 g\right ) n}-\frac{e^2 g p \log \left (d+e x^n\right )}{2 f^2 \left (d^2 f+e^2 g\right ) n}+\frac{g \log \left (c \left (d+e x^n\right )^p\right )}{2 f^2 n \left (g+f x^{2 n}\right )}+\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (\frac{e \left (\sqrt{g}-\sqrt{-f} x^n\right )}{d \sqrt{-f}+e \sqrt{g}}\right )}{2 f^2 n}+\frac{\log \left (c \left (d+e x^n\right )^p\right ) \log \left (-\frac{e \left (\sqrt{g}+\sqrt{-f} x^n\right )}{d \sqrt{-f}-e \sqrt{g}}\right )}{2 f^2 n}+\frac{e^2 g p \log \left (g+f x^{2 n}\right )}{4 f^2 \left (d^2 f+e^2 g\right ) n}+\frac{p \text{Li}_2\left (\frac{\sqrt{-f} \left (d+e x^n\right )}{d \sqrt{-f}-e \sqrt{g}}\right )}{2 f^2 n}+\frac{p \text{Li}_2\left (\frac{\sqrt{-f} \left (d+e x^n\right )}{d \sqrt{-f}+e \sqrt{g}}\right )}{2 f^2 n}\\ \end{align*}
Mathematica [F] time = 1.2449, size = 0, normalized size = 0. \[ \int \frac{\log \left (c \left (d+e x^n\right )^p\right )}{x \left (f+g x^{-2 n}\right )^2} \, dx \]
Verification is Not applicable to the result.
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Maple [C] time = 1.082, size = 810, normalized size = 2.2 \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 [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^{2 \, n}}\right )}^{2} 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 )}{f^{2} x + \frac{2 \, f g x x^{2 \, n}}{x^{4 \, n}} + \frac{g^{2} x}{x^{4 \, n}}}, 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^{2 \, n}}\right )}^{2} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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