Optimal. Leaf size=17 \[ -\frac {\log \left (1-2 \tanh ^{-1}(x)\right )}{2 a b} \]
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Rubi [A] time = 0.04, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, 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*}
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Mathematica [A] time = 0.06, size = 17, normalized size = 1.00 \[ -\frac {\log \left (2 \tanh ^{-1}(x)-1\right )}{2 a b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 22, normalized size = 1.29 \[ -\frac {\log \left (\log \left (-\frac {x + 1}{x - 1}\right ) - 1\right )}{2 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 48, normalized size = 2.82 \[ -\frac {\log \left (\frac {1}{4} \, \pi ^{2} {\left (\mathrm {sgn}\left (x - 1\right ) \mathrm {sgn}\left (-x - 1\right ) - 1\right )}^{2} + {\left (\log \left (\frac {{\left | -x - 1 \right |}}{{\left | x - 1 \right |}}\right ) - 1\right )}^{2}\right )}{4 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 19, normalized size = 1.12 \[ -\frac {\ln \left (2 b \arctanh \relax (x )-b \right )}{2 a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 23, normalized size = 1.35 \[ -\frac {\log \left (-\log \left (x + 1\right ) + \log \left (-x + 1\right ) + 1\right )}{2 \, a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.98, size = 15, normalized size = 0.88 \[ -\frac {\ln \left (2\,\mathrm {atanh}\relax (x)-1\right )}{2\,a\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.56, size = 14, normalized size = 0.82 \[ - \frac {\log {\left (\operatorname {atanh}{\relax (x )} - \frac {1}{2} \right )}}{2 a b} \]
Verification of antiderivative is not currently implemented for this CAS.
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