Optimal. Leaf size=27 \[ \frac {1}{2 a^2 (1-a x)}-\frac {\tanh ^{-1}(a x)}{2 a^2} \]
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Rubi [A] time = 0.08, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6150, 77, 207} \[ \frac {1}{2 a^2 (1-a x)}-\frac {\tanh ^{-1}(a x)}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 77
Rule 207
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 {x}{(1-a x)^2 (1+a x)} \, dx\\ &=\int \left (\frac {1}{2 a (-1+a x)^2}+\frac {1}{2 a \left (-1+a^2 x^2\right )}\right ) \, dx\\ &=\frac {1}{2 a^2 (1-a x)}+\frac {\int \frac {1}{-1+a^2 x^2} \, dx}{2 a}\\ &=\frac {1}{2 a^2 (1-a x)}-\frac {\tanh ^{-1}(a x)}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 22, normalized size = 0.81 \[ \frac {\frac {1}{1-a x}-\tanh ^{-1}(a x)}{2 a^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 42, normalized size = 1.56 \[ -\frac {{\left (a x - 1\right )} \log \left (a x + 1\right ) - {\left (a x - 1\right )} \log \left (a x - 1\right ) + 2}{4 \, {\left (a^{3} x - a^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 37, normalized size = 1.37 \[ -\frac {\log \left ({\left | a x + 1 \right |}\right )}{4 \, a^{2}} + \frac {\log \left ({\left | a x - 1 \right |}\right )}{4 \, a^{2}} - \frac {1}{2 \, {\left (a x - 1\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 36, normalized size = 1.33 \[ -\frac {1}{2 a^{2} \left (a x -1\right )}+\frac {\ln \left (a x -1\right )}{4 a^{2}}-\frac {\ln \left (a x +1\right )}{4 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 38, normalized size = 1.41 \[ -\frac {1}{2 \, {\left (a^{3} x - a^{2}\right )}} - \frac {\log \left (a x + 1\right )}{4 \, a^{2}} + \frac {\log \left (a x - 1\right )}{4 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 22, normalized size = 0.81 \[ -\frac {1}{2\,a^2\,\left (a\,x-1\right )}-\frac {\mathrm {atanh}\left (a\,x\right )}{2\,a^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 32, normalized size = 1.19 \[ - \frac {1}{2 a^{3} x - 2 a^{2}} + \frac {\frac {\log {\left (x - \frac {1}{a} \right )}}{4} - \frac {\log {\left (x + \frac {1}{a} \right )}}{4}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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