Optimal. Leaf size=24 \[ \frac {\text {Li}_2\left (1-\frac {2 e}{e+f x}\right )}{2 e f} \]
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Rubi [A] time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, 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.
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Rule 2315
Rule 2402
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*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 1.12 \[ \frac {\text {Li}_2\left (\frac {f x-e}{e+f x}\right )}{2 e f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 21, normalized size = 0.88 \[ \frac {{\rm Li}_2\left (-\frac {2 \, e}{f x + e} + 1\right )}{2 \, e f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {\log \left (\frac {2 \, e}{f x + e}\right )}{f^{2} x^{2} - e^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 20, normalized size = 0.83 \[ \frac {\dilog \left (\frac {2 e}{f x +e}\right )}{2 e f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.64, size = 120, normalized size = 5.00 \[ \frac {1}{4} \, f {\left (\frac {\log \left (f x + e\right )^{2} - 2 \, \log \left (f x + e\right ) \log \left (f x - e\right )}{e f^{2}} + \frac {2 \, {\left (\log \left (f x + e\right ) \log \left (-\frac {f x + e}{2 \, e} + 1\right ) + {\rm Li}_2\left (\frac {f x + e}{2 \, e}\right )\right )}}{e f^{2}}\right )} + \frac {1}{2} \, {\left (\frac {\log \left (f x + e\right )}{e f} - \frac {\log \left (f x - e\right )}{e f}\right )} \log \left (\frac {2 \, e}{f x + e}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.29, size = 19, normalized size = 0.79 \[ \frac {{\mathrm {Li}}_{\mathrm {2}}\left (\frac {2\,e}{e+f\,x}\right )}{2\,e\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\log {\relax (2 )}}{- e^{2} + f^{2} x^{2}}\, dx - \int \frac {\log {\left (\frac {e}{e + f x} \right )}}{- e^{2} + f^{2} x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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