Optimal. Leaf size=88 \[ \frac{2 p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac{2 p^2 \text{PolyLog}\left (3,\frac{e x^n}{d}+1\right )}{f n}+\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{f n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.113938, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {12, 2454, 2396, 2433, 2374, 6589} \[ \frac{2 p \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right ) \log \left (c \left (d+e x^n\right )^p\right )}{f n}-\frac{2 p^2 \text{PolyLog}\left (3,\frac{e x^n}{d}+1\right )}{f n}+\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{f n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2454
Rule 2396
Rule 2433
Rule 2374
Rule 6589
Rubi steps
\begin{align*} \int \frac{\log ^2\left (c \left (d+e x^n\right )^p\right )}{f x} \, dx &=\frac{\int \frac{\log ^2\left (c \left (d+e x^n\right )^p\right )}{x} \, dx}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\log ^2\left (c (d+e x)^p\right )}{x} \, dx,x,x^n\right )}{f n}\\ &=\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{f n}-\frac{(2 e p) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{e x}{d}\right ) \log \left (c (d+e x)^p\right )}{d+e x} \, dx,x,x^n\right )}{f n}\\ &=\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{f n}-\frac{(2 p) \operatorname{Subst}\left (\int \frac{\log \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 )}{f n}\\ &=\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{f n}+\frac{2 p \log \left (c \left (d+e x^n\right )^p\right ) \text{Li}_2\left (1+\frac{e x^n}{d}\right )}{f n}-\frac{\left (2 p^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{d}\right )}{x} \, dx,x,d+e x^n\right )}{f n}\\ &=\frac{\log \left (-\frac{e x^n}{d}\right ) \log ^2\left (c \left (d+e x^n\right )^p\right )}{f n}+\frac{2 p \log \left (c \left (d+e x^n\right )^p\right ) \text{Li}_2\left (1+\frac{e x^n}{d}\right )}{f n}-\frac{2 p^2 \text{Li}_3\left (1+\frac{e x^n}{d}\right )}{f n}\\ \end{align*}
Mathematica [A] time = 0.0993098, size = 168, normalized size = 1.91 \[ \frac{2 p \left (\log (x) \left (\log \left (d+e x^n\right )-\log \left (\frac{e x^n}{d}+1\right )\right )-\frac{\text{PolyLog}\left (2,-\frac{e x^n}{d}\right )}{n}\right ) \left (\log \left (c \left (d+e x^n\right )^p\right )-p \log \left (d+e x^n\right )\right )+\frac{p^2 \left (-2 \text{PolyLog}\left (3,\frac{e x^n}{d}+1\right )+2 \log \left (d+e x^n\right ) \text{PolyLog}\left (2,\frac{e x^n}{d}+1\right )+\log \left (-\frac{e x^n}{d}\right ) \log ^2\left (d+e x^n\right )\right )}{n}+\log (x) \left (\log \left (c \left (d+e x^n\right )^p\right )-p \log \left (d+e x^n\right )\right )^2}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 4.615, size = 1473, normalized size = 16.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{d{\left (\frac{\log \left (x\right )}{d} - \frac{\log \left (\frac{e x^{n} + d}{e}\right )}{d n}\right )} \log \left (c\right )^{2} + \log \left ({\left (e x^{n} + d\right )}^{p}\right )^{2} \log \left (x\right ) + \frac{\log \left (c\right )^{2} \log \left (\frac{e x^{n} + d}{e}\right )}{n} - \int \frac{2 \,{\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 )}{e x x^{n} + d x}\,{d x}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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 )^{2}}{f x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\log{\left (c \left (d + e x^{n}\right )^{p} \right )}^{2}}{x}\, dx}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (e x^{n} + d\right )}^{p} c\right )^{2}}{f x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]