Optimal. Leaf size=13 \[ \frac{\text{PolyLog}\left (2,\frac{1}{a+b x}\right )}{b} \]
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Rubi [A] time = 0.013087, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {2447} \[ \frac{\text{PolyLog}\left (2,\frac{1}{a+b x}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2447
Rubi steps
\begin{align*} \int \frac{\log \left (1-\frac{1}{a+b x}\right )}{a+b x} \, dx &=\frac{\text{Li}_2\left (\frac{1}{a+b x}\right )}{b}\\ \end{align*}
Mathematica [B] time = 0.0144664, size = 133, normalized size = 10.23 \[ -\frac{\text{PolyLog}(2,a+b x)}{b}+\frac{\log ^2\left (\frac{a b-(a-1) b}{b (a+b x)}\right )}{2 b}+\frac{\log \left (\frac{b (a+b x-1)}{(a-1) b-a b}\right ) \log \left (\frac{a b-(a-1) b}{b (a+b x)}\right )}{b}-\frac{\log \left (\frac{a+b x-1}{a+b x}\right ) \log \left (\frac{a b-(a-1) b}{b (a+b x)}\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 17, normalized size = 1.3 \begin{align*}{\frac{1}{b}{\it dilog} \left ( 1- \left ( bx+a \right ) ^{-1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.08632, size = 80, normalized size = 6.15 \begin{align*} -\frac{\log \left (b x + a\right )^{2} - 2 \, \log \left (b x + a\right ) \log \left (b x + a - 1\right )}{2 \, b} - \frac{\log \left (b x + a\right ) \log \left (-b x - a + 1\right ) +{\rm Li}_2\left (b x + a\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.980251, size = 53, normalized size = 4.08 \begin{align*} \frac{{\rm Li}_2\left (-\frac{b x + a - 1}{b x + a} + 1\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (1 - \frac{1}{a + b x} \right )}}{a + b x}\, dx \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{1}{b x + a} + 1\right )}{b x + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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