Optimal. Leaf size=17 \[ -\frac{\log \left (1-2 \tanh ^{-1}(x)\right )}{2 a b} \]
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Rubi [A] time = 0.0444294, antiderivative size = 17, 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 = {5946} \[ -\frac{\log \left (1-2 \tanh ^{-1}(x)\right )}{2 a b} \]
Antiderivative was successfully verified.
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Rule 5946
Rubi steps
\begin{align*} \int \frac{1}{\left (a-a x^2\right ) \left (b-2 b \tanh ^{-1}(x)\right )} \, dx &=-\frac{\log \left (1-2 \tanh ^{-1}(x)\right )}{2 a b}\\ \end{align*}
Mathematica [A] time = 0.0546647, size = 17, normalized size = 1. \[ -\frac{\log \left (2 \tanh ^{-1}(x)-1\right )}{2 a b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 19, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( 2\,b{\it Artanh} \left ( x \right ) -b \right ) }{2\,ab}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.970007, size = 31, normalized size = 1.82 \begin{align*} -\frac{\log \left (-\log \left (x + 1\right ) + \log \left (-x + 1\right ) + 1\right )}{2 \, a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91186, size = 58, normalized size = 3.41 \begin{align*} -\frac{\log \left (\log \left (-\frac{x + 1}{x - 1}\right ) - 1\right )}{2 \, a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.57957, size = 14, normalized size = 0.82 \begin{align*} - \frac{\log{\left (\operatorname{atanh}{\left (x \right )} - \frac{1}{2} \right )}}{2 a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16613, size = 107, normalized size = 6.29 \begin{align*} -\frac{\log \left (\frac{1}{4} \,{\left (\pi{\left (\mathrm{sgn}\left (x + 1\right ) - 1\right )} - \pi{\left (\mathrm{sgn}\left (x - 1\right ) + 1\right )} + 4 \, \pi \left \lfloor -\frac{\pi{\left (\mathrm{sgn}\left (x + 1\right ) - 1\right )} - \pi{\left (\mathrm{sgn}\left (x - 1\right ) + 1\right )}}{4 \, \pi } + \frac{1}{2} \right \rfloor \right )}^{2} +{\left (\log \left (\frac{{\left | x + 1 \right |}}{{\left | -x + 1 \right |}}\right ) - 1\right )}^{2}\right )}{4 \, a b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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