Optimal. Leaf size=30 \[ \frac{1}{2} b \text{PolyLog}\left (2,-\frac{c}{x}\right )-\frac{1}{2} b \text{PolyLog}\left (2,\frac{c}{x}\right )+a \log (x) \]
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Rubi [A] time = 0.031678, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6095, 5912} \[ \frac{1}{2} b \text{PolyLog}\left (2,-\frac{c}{x}\right )-\frac{1}{2} b \text{PolyLog}\left (2,\frac{c}{x}\right )+a \log (x) \]
Antiderivative was successfully verified.
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Rule 6095
Rule 5912
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}\left (\frac{c}{x}\right )}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx,x,\frac{1}{x}\right )\\ &=a \log (x)+\frac{1}{2} b \text{Li}_2\left (-\frac{c}{x}\right )-\frac{1}{2} b \text{Li}_2\left (\frac{c}{x}\right )\\ \end{align*}
Mathematica [A] time = 0.012366, size = 28, normalized size = 0.93 \[ \frac{1}{2} b \left (\text{PolyLog}\left (2,-\frac{c}{x}\right )-\text{PolyLog}\left (2,\frac{c}{x}\right )\right )+a \log (x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 63, normalized size = 2.1 \begin{align*} -a\ln \left ({\frac{c}{x}} \right ) -b\ln \left ({\frac{c}{x}} \right ){\it Artanh} \left ({\frac{c}{x}} \right ) +{\frac{b}{2}{\it dilog} \left ({\frac{c}{x}} \right ) }+{\frac{b}{2}{\it dilog} \left ( 1+{\frac{c}{x}} \right ) }+{\frac{b}{2}\ln \left ({\frac{c}{x}} \right ) \ln \left ( 1+{\frac{c}{x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, b \int \frac{\log \left (\frac{c}{x} + 1\right ) - \log \left (-\frac{c}{x} + 1\right )}{x}\,{d x} + a \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{artanh}\left (\frac{c}{x}\right ) + a}{x}, x\right ) \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{a + b \operatorname{atanh}{\left (\frac{c}{x} \right )}}{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{b \operatorname{artanh}\left (\frac{c}{x}\right ) + a}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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