Optimal. Leaf size=30 \[ -\frac{1}{4} b \text{PolyLog}\left (2,-c x^2\right )+\frac{1}{4} b \text{PolyLog}\left (2,c x^2\right )+a \log (x) \]
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Rubi [A] time = 0.0331072, 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} \[ -\frac{1}{4} b \text{PolyLog}\left (2,-c x^2\right )+\frac{1}{4} b \text{PolyLog}\left (2,c x^2\right )+a \log (x) \]
Antiderivative was successfully verified.
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Rule 6095
Rule 5912
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}\left (c x^2\right )}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx,x,x^2\right )\\ &=a \log (x)-\frac{1}{4} b \text{Li}_2\left (-c x^2\right )+\frac{1}{4} b \text{Li}_2\left (c x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0133833, size = 28, normalized size = 0.93 \[ \frac{1}{4} b \left (\text{PolyLog}\left (2,c x^2\right )-\text{PolyLog}\left (2,-c x^2\right )\right )+a \log (x) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.029, size = 124, normalized size = 4.1 \begin{align*} a\ln \left ( x \right ) +b\ln \left ( x \right ){\it Artanh} \left ( c{x}^{2} \right ) -{\frac{b\ln \left ( x \right ) }{2}\ln \left ( 1+x\sqrt{-c} \right ) }-{\frac{b\ln \left ( x \right ) }{2}\ln \left ( 1-x\sqrt{-c} \right ) }-{\frac{b}{2}{\it dilog} \left ( 1+x\sqrt{-c} \right ) }-{\frac{b}{2}{\it dilog} \left ( 1-x\sqrt{-c} \right ) }+{\frac{b\ln \left ( x \right ) }{2}\ln \left ( 1-x\sqrt{c} \right ) }+{\frac{b\ln \left ( x \right ) }{2}\ln \left ( 1+x\sqrt{c} \right ) }+{\frac{b}{2}{\it dilog} \left ( 1-x\sqrt{c} \right ) }+{\frac{b}{2}{\it dilog} \left ( 1+x\sqrt{c} \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 (c x^{2} + 1\right ) - \log \left (-c x^{2} + 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 (c x^{2}\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 x^{2} \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 x^{2}\right ) + a}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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