Optimal. Leaf size=47 \[ \frac {(a-b \log (2)) \tanh ^{-1}\left (\frac {f x}{e}\right )}{e f}+\frac {b \text {Li}_2\left (1-\frac {2 e}{e+f x}\right )}{2 e f} \]
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Rubi [A] time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2403, 208, 2402, 2315} \[ \frac {b \text {PolyLog}\left (2,1-\frac {2 e}{e+f x}\right )}{2 e f}+\frac {(a-b \log (2)) \tanh ^{-1}\left (\frac {f x}{e}\right )}{e f} \]
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 {e}{e+f x}\right )}{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+(a-b \log (2)) \int \frac {1}{e^2-f^2 x^2} \, dx\\ &=\frac {\tanh ^{-1}\left (\frac {f x}{e}\right ) (a-b \log (2))}{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 {\tanh ^{-1}\left (\frac {f x}{e}\right ) (a-b \log (2))}{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 = 80, normalized size = 1.70 \[ \frac {2 b^2 \text {Li}_2\left (\frac {e+f x}{2 e}\right )-\left (a+b \log \left (\frac {e}{e+f x}\right )\right ) \left (a+2 b \log \left (\frac {e-f x}{2 e}\right )+b \log \left (\frac {e}{e+f x}\right )\right )}{4 b e f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b \log \left (\frac {e}{f x + e}\right ) + a}{f^{2} x^{2} - e^{2}}, x\right ) \]
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 \[ \int -\frac {b \log \left (\frac {e}{f x + e}\right ) + a}{f^{2} x^{2} - e^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 109, normalized size = 2.32 \[ -\frac {b \ln \left (\frac {e}{f x +e}\right ) \ln \left (-\frac {2 e}{f x +e}+1\right )}{2 e f}+\frac {b \ln \left (\frac {2 e}{f x +e}\right ) \ln \left (-\frac {2 e}{f x +e}+1\right )}{2 e f}-\frac {a \ln \left (\frac {2 e}{f x +e}-1\right )}{2 e f}+\frac {b \dilog \left (\frac {2 e}{f x +e}\right )}{2 e f} \]
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 \[ \frac {1}{2} \, a {\left (\frac {\log \left (f x + e\right )}{e f} - \frac {\log \left (f x - e\right )}{e f}\right )} + b \int \frac {\log \left (f x + e\right ) - \log \relax (e)}{f^{2} x^{2} - e^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {a+b\,\ln \left (\frac {e}{e+f\,x}\right )}{e^2-f^2\,x^2} \,d x \]
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 \[ - \int \frac {a}{- e^{2} + f^{2} x^{2}}\, dx - \int \frac {b \log {\left (\frac {e}{e + f x} \right )}}{- e^{2} + f^{2} x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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