Optimal. Leaf size=32 \[ a^2 \log (x)-a^2 \log (1-a x)-\frac {a}{x}-\frac {1}{2 x^2} \]
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Rubi [A] time = 0.09, antiderivative size = 32, 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, 44} \[ a^2 \log (x)-a^2 \log (1-a x)-\frac {a}{x}-\frac {1}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^3 \sqrt {1-a^2 x^2}} \, dx &=\int \frac {1}{x^3 (1-a x)} \, dx\\ &=\int \left (\frac {1}{x^3}+\frac {a}{x^2}+\frac {a^2}{x}-\frac {a^3}{-1+a x}\right ) \, dx\\ &=-\frac {1}{2 x^2}-\frac {a}{x}+a^2 \log (x)-a^2 \log (1-a x)\\ \end {align*}
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Mathematica [A] time = 0.01, size = 32, normalized size = 1.00 \[ a^2 \log (x)-a^2 \log (1-a x)-\frac {a}{x}-\frac {1}{2 x^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 35, normalized size = 1.09 \[ -\frac {2 \, a^{2} x^{2} \log \left (a x - 1\right ) - 2 \, a^{2} x^{2} \log \relax (x) + 2 \, a x + 1}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 31, normalized size = 0.97 \[ -a^{2} \log \left ({\left | a x - 1 \right |}\right ) + a^{2} \log \left ({\left | x \right |}\right ) - \frac {2 \, a x + 1}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 30, normalized size = 0.94 \[ -\frac {1}{2 x^{2}}-\frac {a}{x}+a^{2} \ln \relax (x )-a^{2} \ln \left (a x -1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 29, normalized size = 0.91 \[ -a^{2} \log \left (a x - 1\right ) + a^{2} \log \relax (x) - \frac {2 \, a x + 1}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 23, normalized size = 0.72 \[ 2\,a^2\,\mathrm {atanh}\left (2\,a\,x-1\right )-\frac {a\,x+\frac {1}{2}}{x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.17, size = 26, normalized size = 0.81 \[ - a^{2} \left (- \log {\relax (x )} + \log {\left (x - \frac {1}{a} \right )}\right ) - \frac {2 a x + 1}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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