Optimal. Leaf size=46 \[ \frac {a}{2 (1-a x)}+a \log (x)-\frac {5}{4} a \log (1-a x)+\frac {1}{4} a \log (a x+1)-\frac {1}{x} \]
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Rubi [A] time = 0.11, antiderivative size = 46, 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, 88} \[ \frac {a}{2 (1-a x)}+a \log (x)-\frac {5}{4} a \log (1-a x)+\frac {1}{4} a \log (a x+1)-\frac {1}{x} \]
Antiderivative was successfully verified.
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Rule 88
Rule 6150
Rubi steps
\begin {align*} \int \frac {e^{\tanh ^{-1}(a x)}}{x^2 \left (1-a^2 x^2\right )^{3/2}} \, dx &=\int \frac {1}{x^2 (1-a x)^2 (1+a x)} \, dx\\ &=\int \left (\frac {1}{x^2}+\frac {a}{x}+\frac {a^2}{2 (-1+a x)^2}-\frac {5 a^2}{4 (-1+a x)}+\frac {a^2}{4 (1+a x)}\right ) \, dx\\ &=-\frac {1}{x}+\frac {a}{2 (1-a x)}+a \log (x)-\frac {5}{4} a \log (1-a x)+\frac {1}{4} a \log (1+a x)\\ \end {align*}
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Mathematica [A] time = 0.04, size = 46, normalized size = 1.00 \[ \frac {a}{2 (1-a x)}+a \log (x)-\frac {5}{4} a \log (1-a x)+\frac {1}{4} a \log (a x+1)-\frac {1}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 75, normalized size = 1.63 \[ -\frac {6 \, a x - {\left (a^{2} x^{2} - a x\right )} \log \left (a x + 1\right ) + 5 \, {\left (a^{2} x^{2} - a x\right )} \log \left (a x - 1\right ) - 4 \, {\left (a^{2} x^{2} - a x\right )} \log \relax (x) - 4}{4 \, {\left (a x^{2} - x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 44, normalized size = 0.96 \[ \frac {1}{4} \, a \log \left ({\left | a x + 1 \right |}\right ) - \frac {5}{4} \, a \log \left ({\left | a x - 1 \right |}\right ) + a \log \left ({\left | x \right |}\right ) - \frac {3 \, a x - 2}{2 \, {\left (a x - 1\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 39, normalized size = 0.85 \[ -\frac {1}{x}+a \ln \relax (x )-\frac {a}{2 \left (a x -1\right )}-\frac {5 a \ln \left (a x -1\right )}{4}+\frac {a \ln \left (a x +1\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 42, normalized size = 0.91 \[ \frac {1}{4} \, a \log \left (a x + 1\right ) - \frac {5}{4} \, a \log \left (a x - 1\right ) + a \log \relax (x) - \frac {3 \, a x - 2}{2 \, {\left (a x^{2} - x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.89, size = 40, normalized size = 0.87 \[ a\,\ln \relax (x)-\frac {5\,a\,\ln \left (a\,x-1\right )}{4}+\frac {a\,\ln \left (a\,x+1\right )}{4}+\frac {\frac {3\,a\,x}{2}-1}{x-a\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 42, normalized size = 0.91 \[ a \log {\relax (x )} - \frac {5 a \log {\left (x - \frac {1}{a} \right )}}{4} + \frac {a \log {\left (x + \frac {1}{a} \right )}}{4} + \frac {- 3 a x + 2}{2 a x^{2} - 2 x} \]
Verification of antiderivative is not currently implemented for this CAS.
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