Optimal. Leaf size=41 \[ \frac {a \tanh ^{-1}\left (\frac {f x}{e}\right )}{e f}+\frac {b \text {Li}_2\left (1-\frac {2 e}{e+f x}\right )}{2 e f} \]
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Rubi [A] time = 0.06, antiderivative size = 41, normalized size of antiderivative = 1.00, 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.
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Rule 208
Rule 2315
Rule 2402
Rule 2403
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*}
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Mathematica [A] time = 0.03, size = 82, normalized size = 2.00 \[ \frac {2 b^2 \text {Li}_2\left (\frac {e+f x}{2 e}\right )-\left (a+b \log \left (\frac {2 e}{e+f x}\right )\right ) \left (a+2 b \log \left (\frac {e-f x}{2 e}\right )+b \log \left (\frac {2 e}{e+f x}\right )\right )}{4 b e f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 43, normalized size = 1.05 \[ \frac {b {\rm Li}_2\left (-\frac {2 \, e}{f x + e} + 1\right ) + a \log \left (f x + e\right ) - a \log \left (f x - e\right )}{2 \, e f} \]
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 \[ \int -\frac {b \log \left (\frac {2 \, e}{f x + e}\right ) + a}{f^{2} x^{2} - e^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 44, normalized size = 1.07 \[ -\frac {a \ln \left (\frac {2 e}{f x +e}-1\right )}{2 e f}+\frac {b \dilog \left (\frac {2 e}{f x +e}\right )}{2 e f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a {\left (\frac {\log \left (f x + e\right )}{e f} - \frac {\log \left (f x - e\right )}{e f}\right )} + b \int -\frac {\log \relax (2) - \log \left (f x + e\right ) + \log \relax (e)}{f^{2} x^{2} - e^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 43, normalized size = 1.05 \[ -\frac {a\,\ln \left (f\,x-e\right )-b\,{\mathrm {Li}}_{\mathrm {2}}\left (\frac {2\,e}{e+f\,x}\right )+a\,\ln \left (\frac {1}{e+f\,x}\right )}{2\,e\,f} \]
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 \[ - \int \frac {a}{- e^{2} + f^{2} x^{2}}\, dx - \int \frac {b \log {\relax (2 )}}{- e^{2} + f^{2} x^{2}}\, dx - \int \frac {b \log {\left (\frac {e}{e + f x} \right )}}{- e^{2} + f^{2} x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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