Optimal. Leaf size=41 \[ \frac{b \text{PolyLog}\left (2,1-\frac{2 e}{e+f x}\right )}{2 e f}+\frac{a \tanh ^{-1}\left (\frac{f x}{e}\right )}{e f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0605801, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2403, 208, 2402, 2315} \[ \frac{b \text{PolyLog}\left (2,1-\frac{2 e}{e+f x}\right )}{2 e f}+\frac{a \tanh ^{-1}\left (\frac{f x}{e}\right )}{e f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2403
Rule 208
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{a+b \log \left (\frac{2 e}{e+f x}\right )}{e^2-f^2 x^2} \, dx &=a \int \frac{1}{e^2-f^2 x^2} \, dx+b \int \frac{\log \left (\frac{2 e}{e+f x}\right )}{e^2-f^2 x^2} \, dx\\ &=\frac{a \tanh ^{-1}\left (\frac{f x}{e}\right )}{e f}+\frac{b \operatorname{Subst}\left (\int \frac{\log (2 e x)}{1-2 e x} \, dx,x,\frac{1}{e+f x}\right )}{f}\\ &=\frac{a \tanh ^{-1}\left (\frac{f x}{e}\right )}{e f}+\frac{b \text{Li}_2\left (1-\frac{2 e}{e+f x}\right )}{2 e f}\\ \end{align*}
Mathematica [A] time = 0.032232, size = 82, normalized size = 2. \[ \frac{2 b^2 \text{PolyLog}\left (2,\frac{e+f x}{2 e}\right )-\left (a+b \log \left (\frac{2 e}{e+f x}\right )\right ) \left (a+2 b \log \left (\frac{e-f x}{2 e}\right )+b \log \left (\frac{2 e}{e+f x}\right )\right )}{4 b e f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.063, size = 44, normalized size = 1.1 \begin{align*} -{\frac{a}{2\,fe}\ln \left ( 2\,{\frac{e}{fx+e}}-1 \right ) }+{\frac{b}{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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a{\left (\frac{\log \left (f x + e\right )}{e f} - \frac{\log \left (f x - e\right )}{e f}\right )} + b \int -\frac{\log \left (2\right ) - \log \left (f x + e\right ) + \log \left (e\right )}{f^{2} x^{2} - e^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.75179, size = 101, normalized size = 2.46 \begin{align*} \frac{b{\rm Li}_2\left (-\frac{2 \, e}{f x + e} + 1\right ) + a \log \left (f x + e\right ) - a \log \left (f x - e\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{a}{- e^{2} + f^{2} x^{2}}\, dx - \int \frac{b \log{\left (2 \right )}}{- e^{2} + f^{2} x^{2}}\, dx - \int \frac{b \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{b \log \left (\frac{2 \, e}{f x + e}\right ) + a}{f^{2} x^{2} - e^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]