Optimal. Leaf size=24 \[ \frac{\text{PolyLog}\left (2,1-\frac{2 e}{e+f x}\right )}{2 e f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0314029, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {2402, 2315} \[ \frac{\text{PolyLog}\left (2,1-\frac{2 e}{e+f x}\right )}{2 e f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{\log \left (\frac{2 e}{e+f x}\right )}{e^2-f^2 x^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\log (2 e x)}{1-2 e x} \, dx,x,\frac{1}{e+f x}\right )}{f}\\ &=\frac{\text{Li}_2\left (1-\frac{2 e}{e+f x}\right )}{2 e f}\\ \end{align*}
Mathematica [A] time = 0.0056395, size = 27, normalized size = 1.12 \[ \frac{\text{PolyLog}\left (2,\frac{f x-e}{e+f x}\right )}{2 e f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.06, size = 20, normalized size = 0.8 \begin{align*}{\frac{1}{2\,fe}{\it dilog} \left ( 2\,{\frac{e}{fx+e}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.19639, size = 162, normalized size = 6.75 \begin{align*} \frac{1}{4} \, f{\left (\frac{\log \left (f x + e\right )^{2} - 2 \, \log \left (f x + e\right ) \log \left (f x - e\right )}{e f^{2}} + \frac{2 \,{\left (\log \left (f x + e\right ) \log \left (-\frac{f x + e}{2 \, e} + 1\right ) +{\rm Li}_2\left (\frac{f x + e}{2 \, e}\right )\right )}}{e f^{2}}\right )} + \frac{1}{2} \,{\left (\frac{\log \left (f x + e\right )}{e f} - \frac{\log \left (f x - e\right )}{e f}\right )} \log \left (\frac{2 \, e}{f x + e}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.60103, size = 50, normalized size = 2.08 \begin{align*} \frac{{\rm Li}_2\left (-\frac{2 \, e}{f x + e} + 1\right )}{2 \, e f} \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*} - \int \frac{\log{\left (2 \right )}}{- e^{2} + f^{2} x^{2}}\, dx - \int \frac{\log{\left (\frac{e}{e + f x} \right )}}{- e^{2} + f^{2} x^{2}}\, dx \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 (\frac{2 \, e}{f x + e}\right )}{f^{2} x^{2} - e^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]