Optimal. Leaf size=42 \[ -\frac {\tanh ^{-1}\left (\frac {f x}{e}\right ) \log (2)}{e f}+\frac {\text {Li}_2\left (1-\frac {2 e}{e+f x}\right )}{2 e f} \]
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Rubi [A]
time = 0.04, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2450, 214,
2449, 2352} \begin {gather*} \frac {\text {PolyLog}\left (2,1-\frac {2 e}{e+f x}\right )}{2 e f}-\frac {\log (2) \tanh ^{-1}\left (\frac {f x}{e}\right )}{e f} \end {gather*}
Antiderivative was successfully verified.
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Rule 214
Rule 2352
Rule 2449
Rule 2450
Rubi steps
\begin {align*} \int \frac {\log \left (\frac {e}{e+f x}\right )}{e^2-f^2 x^2} \, dx &=-\left (\log (2) \int \frac {1}{e^2-f^2 x^2} \, dx\right )+\int \frac {\log \left (\frac {2 e}{e+f x}\right )}{e^2-f^2 x^2} \, dx\\ &=-\frac {\tanh ^{-1}\left (\frac {f x}{e}\right ) \log (2)}{e f}+\frac {\text {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 ) \log (2)}{e 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 = 81, normalized size = 1.93 \begin {gather*} -\frac {\log \left (\frac {e-f x}{2 e}\right ) \log \left (\frac {e}{e+f x}\right )}{2 e f}-\frac {\log ^2\left (\frac {e}{e+f x}\right )}{4 e f}+\frac {\text {Li}_2\left (\frac {e+f x}{2 e}\right )}{2 e f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.66, size = 62, normalized size = 1.48
method | result | size |
derivativedivides | \(-\frac {\frac {\left (\ln \left (\frac {e}{f x +e}\right )-\ln \left (\frac {2 e}{f x +e}\right )\right ) \ln \left (1-\frac {2 e}{f x +e}\right )}{2}-\frac {\dilog \left (\frac {2 e}{f x +e}\right )}{2}}{f e}\) | \(62\) |
default | \(-\frac {\frac {\left (\ln \left (\frac {e}{f x +e}\right )-\ln \left (\frac {2 e}{f x +e}\right )\right ) \ln \left (1-\frac {2 e}{f x +e}\right )}{2}-\frac {\dilog \left (\frac {2 e}{f x +e}\right )}{2}}{f e}\) | \(62\) |
risch | \(-\frac {\ln \left (1-\frac {2 e}{f x +e}\right ) \ln \left (\frac {e}{f x +e}\right )}{2 e f}+\frac {\ln \left (1-\frac {2 e}{f x +e}\right ) \ln \left (\frac {2 e}{f x +e}\right )}{2 e f}+\frac {\dilog \left (\frac {2 e}{f x +e}\right )}{2 f e}\) | \(84\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 123 vs.
\(2 (38) = 76\).
time = 0.28, size = 123, normalized size = 2.93 \begin {gather*} \frac {1}{4} \, f {\left (\frac {{\left (\log \left (f x + e\right )^{2} - 2 \, \log \left (f x + e\right ) \log \left (f x - e\right )\right )} e^{\left (-1\right )}}{f^{2}} + \frac {2 \, {\left (\log \left (f x + e\right ) \log \left (-\frac {1}{2} \, {\left (f x + e\right )} e^{\left (-1\right )} + 1\right ) + {\rm Li}_2\left (\frac {1}{2} \, {\left (f x + e\right )} e^{\left (-1\right )}\right )\right )} e^{\left (-1\right )}}{f^{2}}\right )} + \frac {1}{2} \, {\left (\frac {e^{\left (-1\right )} \log \left (f x + e\right )}{f} - \frac {e^{\left (-1\right )} \log \left (f x - e\right )}{f}\right )} \log \left (\frac {e}{f x + e}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} - \int \frac {\log {\left (\frac {e}{e + f x} \right )}}{- e^{2} + f^{2} x^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\ln \left (\frac {e}{e+f\,x}\right )}{e^2-f^2\,x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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