Optimal. Leaf size=41 \[ \frac{1}{8 a^2 (a x+1)}+\frac{1}{8 a^2 (1-a x)^2}-\frac{\tanh ^{-1}(a x)}{8 a^2} \]
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Rubi [A] time = 0.0841084, antiderivative size = 41, normalized size of antiderivative = 1., 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} \[ \frac{1}{8 a^2 (a x+1)}+\frac{1}{8 a^2 (1-a x)^2}-\frac{\tanh ^{-1}(a x)}{8 a^2} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 77
Rule 207
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x}{\left (1-a^2 x^2\right )^{5/2}} \, dx &=\int \frac{x}{(1-a x)^3 (1+a x)^2} \, dx\\ &=\int \left (-\frac{1}{4 a (-1+a x)^3}-\frac{1}{8 a (1+a x)^2}+\frac{1}{8 a \left (-1+a^2 x^2\right )}\right ) \, dx\\ &=\frac{1}{8 a^2 (1-a x)^2}+\frac{1}{8 a^2 (1+a x)}+\frac{\int \frac{1}{-1+a^2 x^2} \, dx}{8 a}\\ &=\frac{1}{8 a^2 (1-a x)^2}+\frac{1}{8 a^2 (1+a x)}-\frac{\tanh ^{-1}(a x)}{8 a^2}\\ \end{align*}
Mathematica [A] time = 0.0276091, size = 28, normalized size = 0.68 \[ \frac{\frac{1}{a x+1}+\frac{1}{(a x-1)^2}-\tanh ^{-1}(a x)}{8 a^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 48, normalized size = 1.2 \begin{align*}{\frac{1}{8\,{a}^{2} \left ( ax+1 \right ) }}-{\frac{\ln \left ( ax+1 \right ) }{16\,{a}^{2}}}+{\frac{1}{8\,{a}^{2} \left ( ax-1 \right ) ^{2}}}+{\frac{\ln \left ( ax-1 \right ) }{16\,{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.967148, size = 88, normalized size = 2.15 \begin{align*} \frac{a^{2} x^{2} - a x + 2}{8 \,{\left (a^{5} x^{3} - a^{4} x^{2} - a^{3} x + a^{2}\right )}} - \frac{\log \left (a x + 1\right )}{16 \, a^{2}} + \frac{\log \left (a x - 1\right )}{16 \, a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.13842, size = 208, normalized size = 5.07 \begin{align*} \frac{2 \, a^{2} x^{2} - 2 \, a x -{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right ) +{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x - 1\right ) + 4}{16 \,{\left (a^{5} x^{3} - a^{4} x^{2} - a^{3} x + a^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.479255, size = 60, normalized size = 1.46 \begin{align*} \frac{a^{2} x^{2} - a x + 2}{8 a^{5} x^{3} - 8 a^{4} x^{2} - 8 a^{3} x + 8 a^{2}} - \frac{- \frac{\log{\left (x - \frac{1}{a} \right )}}{16} + \frac{\log{\left (x + \frac{1}{a} \right )}}{16}}{a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20794, size = 77, normalized size = 1.88 \begin{align*} -\frac{\log \left ({\left | a x + 1 \right |}\right )}{16 \, a^{2}} + \frac{\log \left ({\left | a x - 1 \right |}\right )}{16 \, a^{2}} + \frac{a^{2} x^{2} - a x + 2}{8 \,{\left (a x + 1\right )}{\left (a x - 1\right )}^{2} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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