Optimal. Leaf size=36 \[ \frac {1}{2 (1-a x)}-\frac {3}{4} \log (1-a x)-\frac {1}{4} \log (a x+1)+\log (x) \]
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Rubi [A] time = 0.11, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {6150, 72} \[ \frac {1}{2 (1-a x)}-\frac {3}{4} \log (1-a x)-\frac {1}{4} \log (a x+1)+\log (x) \]
Antiderivative was successfully verified.
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Rule 72
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x \left (1-a^2 x^2\right )^{3/2}} \, dx &=\int \frac {1}{x (1-a x)^2 (1+a x)} \, dx\\ &=\int \left (\frac {1}{x}+\frac {a}{2 (-1+a x)^2}-\frac {3 a}{4 (-1+a x)}-\frac {a}{4 (1+a x)}\right ) \, dx\\ &=\frac {1}{2 (1-a x)}+\log (x)-\frac {3}{4} \log (1-a x)-\frac {1}{4} \log (1+a x)\\ \end {align*}
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Mathematica [A] time = 0.03, size = 32, normalized size = 0.89 \[ \frac {1}{2-2 a x}-\frac {3}{4} \log (1-a x)-\frac {1}{4} \log (a x+1)+\log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.67, size = 45, normalized size = 1.25 \[ -\frac {{\left (a x - 1\right )} \log \left (a x + 1\right ) + 3 \, {\left (a x - 1\right )} \log \left (a x - 1\right ) - 4 \, {\left (a x - 1\right )} \log \relax (x) + 2}{4 \, {\left (a x - 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 31, normalized size = 0.86 \[ -\frac {1}{2 \, {\left (a x - 1\right )}} - \frac {1}{4} \, \log \left ({\left | a x + 1 \right |}\right ) - \frac {3}{4} \, \log \left ({\left | a x - 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 29, normalized size = 0.81 \[ \ln \relax (x )-\frac {1}{2 \left (a x -1\right )}-\frac {3 \ln \left (a x -1\right )}{4}-\frac {\ln \left (a x +1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 28, normalized size = 0.78 \[ -\frac {1}{2 \, {\left (a x - 1\right )}} - \frac {1}{4} \, \log \left (a x + 1\right ) - \frac {3}{4} \, \log \left (a x - 1\right ) + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 30, normalized size = 0.83 \[ \ln \relax (x)-\frac {3\,\ln \left (1-a\,x\right )}{4}-\frac {\ln \left (a\,x+1\right )}{4}-\frac {1}{2\,\left (a\,x-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 29, normalized size = 0.81 \[ \log {\relax (x )} - \frac {3 \log {\left (x - \frac {1}{a} \right )}}{4} - \frac {\log {\left (x + \frac {1}{a} \right )}}{4} - \frac {1}{2 a x - 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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