Optimal. Leaf size=25 \[ -\frac{\text{PolyLog}\left (2,1-c \left (d+e x^n\right )\right )}{c e n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.157073, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.121, Rules used = {2475, 2412, 2393, 2391} \[ -\frac{\text{PolyLog}\left (2,1-c \left (d+e x^n\right )\right )}{c e n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2475
Rule 2412
Rule 2393
Rule 2391
Rubi steps
\begin{align*} \int \frac{\log \left (c \left (d+e x^n\right )\right )}{x \left (c e-(1-c d) x^{-n}\right )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\log (c (d+e x))}{\left (c e+\frac{-1+c d}{x}\right ) x} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\log (c (d+e x))}{-1+c d+c e x} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,-1+c d+c e x^n\right )}{c e n}\\ &=-\frac{\text{Li}_2\left (1-c \left (d+e x^n\right )\right )}{c e n}\\ \end{align*}
Mathematica [A] time = 0.0686126, size = 26, normalized size = 1.04 \[ -\frac{\text{PolyLog}\left (2,-c d-c e x^n+1\right )}{c e n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.083, size = 23, normalized size = 0.9 \begin{align*} -{\frac{{\it dilog} \left ( ce{x}^{n}+cd \right ) }{nce}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.40197, size = 143, normalized size = 5.72 \begin{align*}{\left (\frac{\log \left (c e + \frac{c d - 1}{x^{n}}\right )}{c e n} - \frac{\log \left (\frac{1}{x^{n}}\right )}{c e n}\right )} \log \left ({\left (e x^{n} + d\right )} c\right ) - \frac{\log \left (c e x^{n} + c d\right ) \log \left (c e x^{n} + c d - 1\right ) +{\rm Li}_2\left (-c e x^{n} - c d + 1\right )}{c e n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.69567, size = 49, normalized size = 1.96 \begin{align*} -\frac{{\rm Li}_2\left (-c e x^{n} - c d + 1\right )}{c e n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (e x^{n} + d\right )} c\right )}{{\left (c e + \frac{c d - 1}{x^{n}}\right )} x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]