Optimal. Leaf size=37 \[ \frac{\text{PolyLog}\left (2,1-\frac{a (1-c)+b (c+1) x}{a+b x}\right )}{2 a b} \]
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Rubi [A] time = 0.0250285, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 38, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.026, Rules used = {2447} \[ \frac{\text{PolyLog}\left (2,1-\frac{a (1-c)+b (c+1) x}{a+b x}\right )}{2 a b} \]
Antiderivative was successfully verified.
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Rule 2447
Rubi steps
\begin{align*} \int \frac{\log \left (\frac{a (1-c)+b (1+c) x}{a+b x}\right )}{a^2-b^2 x^2} \, dx &=\frac{\text{Li}_2\left (1-\frac{a (1-c)+b (1+c) x}{a+b x}\right )}{2 a b}\\ \end{align*}
Mathematica [B] time = 0.187185, size = 252, normalized size = 6.81 \[ \frac{2 \text{PolyLog}\left (2,\frac{(c+1) (a-b x)}{2 a}\right )-2 \text{PolyLog}\left (2,\frac{(c+1) (a+b x)}{2 a c}\right )-2 \text{PolyLog}\left (2,\frac{a-b x}{2 a}\right )+\log ^2\left (\frac{2 a c}{(c+1) (a+b x)}\right )+2 \log \left (-\frac{a (-c)+a+b (c+1) x}{2 a c}\right ) \log \left (\frac{2 a c}{(c+1) (a+b x)}\right )-2 \log \left (\frac{a (-c)+a+b (c+1) x}{a+b x}\right ) \log \left (\frac{2 a c}{(c+1) (a+b x)}\right )+2 \log (a-b x) \log \left (\frac{a (-c)+a+b (c+1) x}{2 a}\right )-2 \log (a-b x) \log \left (\frac{a (-c)+a+b (c+1) x}{a+b x}\right )-2 \log (a-b x) \log \left (\frac{a+b x}{2 a}\right )}{4 a b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.064, size = 24, normalized size = 0.7 \begin{align*}{\frac{1}{2\,ab}{\it dilog} \left ( 1+c-2\,{\frac{ac}{bx+a}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.26546, size = 332, normalized size = 8.97 \begin{align*} \frac{1}{2} \,{\left (\frac{\log \left (b x + a\right )}{a b} - \frac{\log \left (b x - a\right )}{a b}\right )} \log \left (\frac{b{\left (c + 1\right )} x - a{\left (c - 1\right )}}{b x + a}\right ) + \frac{\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (b x - a\right )}{4 \, a b} + \frac{\log \left (b x - a\right ) \log \left (\frac{b{\left (c + 1\right )} x - a{\left (c + 1\right )}}{2 \, a} + 1\right ) +{\rm Li}_2\left (-\frac{b{\left (c + 1\right )} x - a{\left (c + 1\right )}}{2 \, a}\right )}{2 \, a b} + \frac{\log \left (b x + a\right ) \log \left (-\frac{b x + a}{2 \, a} + 1\right ) +{\rm Li}_2\left (\frac{b x + a}{2 \, a}\right )}{2 \, a b} - \frac{\log \left (b x + a\right ) \log \left (-\frac{b{\left (c + 1\right )} x + a{\left (c + 1\right )}}{2 \, a c} + 1\right ) +{\rm Li}_2\left (\frac{b{\left (c + 1\right )} x + a{\left (c + 1\right )}}{2 \, a c}\right )}{2 \, a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.61004, size = 76, normalized size = 2.05 \begin{align*} \frac{{\rm Li}_2\left (\frac{a c -{\left (b c + b\right )} x - a}{b x + a} + 1\right )}{2 \, a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{\log \left (\frac{b{\left (c + 1\right )} x - a{\left (c - 1\right )}}{b x + a}\right )}{b^{2} x^{2} - a^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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