Optimal. Leaf size=29 \[ -b \text{PolyLog}\left (2,-c \sqrt{x}\right )+b \text{PolyLog}\left (2,c \sqrt{x}\right )+a \log (x) \]
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Rubi [A] time = 0.0315208, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6095, 5912} \[ -b \text{PolyLog}\left (2,-c \sqrt{x}\right )+b \text{PolyLog}\left (2,c \sqrt{x}\right )+a \log (x) \]
Antiderivative was successfully verified.
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Rule 6095
Rule 5912
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}\left (c \sqrt{x}\right )}{x} \, dx &=2 \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx,x,\sqrt{x}\right )\\ &=a \log (x)-b \text{Li}_2\left (-c \sqrt{x}\right )+b \text{Li}_2\left (c \sqrt{x}\right )\\ \end{align*}
Mathematica [A] time = 0.0114324, size = 29, normalized size = 1. \[ -b \text{PolyLog}\left (2,-c \sqrt{x}\right )+b \text{PolyLog}\left (2,c \sqrt{x}\right )+a \log (x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.036, size = 63, normalized size = 2.2 \begin{align*} 2\,a\ln \left ( c\sqrt{x} \right ) +2\,b\ln \left ( c\sqrt{x} \right ){\it Artanh} \left ( c\sqrt{x} \right ) -b{\it dilog} \left ( c\sqrt{x} \right ) -b{\it dilog} \left ( 1+c\sqrt{x} \right ) -b\ln \left ( c\sqrt{x} \right ) \ln \left ( 1+c\sqrt{x} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.49312, size = 82, normalized size = 2.83 \begin{align*} -{\left (\log \left (c \sqrt{x}\right ) \log \left (-c \sqrt{x} + 1\right ) +{\rm Li}_2\left (-c \sqrt{x} + 1\right )\right )} b +{\left (\log \left (c \sqrt{x} + 1\right ) \log \left (-c \sqrt{x}\right ) +{\rm Li}_2\left (c \sqrt{x} + 1\right )\right )} b + 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 (c \sqrt{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 (c \sqrt{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 (c \sqrt{x}\right ) + a}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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