Optimal. Leaf size=89 \[ \frac{a^2}{1-a x}+\frac{a^2}{8 (a x+1)}+\frac{a^2}{8 (1-a x)^2}+3 a^2 \log (x)-\frac{39}{16} a^2 \log (1-a x)-\frac{9}{16} a^2 \log (a x+1)-\frac{a}{x}-\frac{1}{2 x^2} \]
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Rubi [A] time = 0.13547, antiderivative size = 89, normalized size of antiderivative = 1., 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} \[ \frac{a^2}{1-a x}+\frac{a^2}{8 (a x+1)}+\frac{a^2}{8 (1-a x)^2}+3 a^2 \log (x)-\frac{39}{16} a^2 \log (1-a x)-\frac{9}{16} a^2 \log (a x+1)-\frac{a}{x}-\frac{1}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x^3 \left (1-a^2 x^2\right )^{5/2}} \, dx &=\int \frac{1}{x^3 (1-a x)^3 (1+a x)^2} \, dx\\ &=\int \left (\frac{1}{x^3}+\frac{a}{x^2}+\frac{3 a^2}{x}-\frac{a^3}{4 (-1+a x)^3}+\frac{a^3}{(-1+a x)^2}-\frac{39 a^3}{16 (-1+a x)}-\frac{a^3}{8 (1+a x)^2}-\frac{9 a^3}{16 (1+a x)}\right ) \, dx\\ &=-\frac{1}{2 x^2}-\frac{a}{x}+\frac{a^2}{8 (1-a x)^2}+\frac{a^2}{1-a x}+\frac{a^2}{8 (1+a x)}+3 a^2 \log (x)-\frac{39}{16} a^2 \log (1-a x)-\frac{9}{16} a^2 \log (1+a x)\\ \end{align*}
Mathematica [A] time = 0.071539, size = 83, normalized size = 0.93 \[ \frac{1}{16} \left (\frac{16 a^2}{1-a x}+\frac{2 a^2}{a x+1}+\frac{2 a^2}{(a x-1)^2}+48 a^2 \log (x)-39 a^2 \log (1-a x)-9 a^2 \log (a x+1)-\frac{16 a}{x}-\frac{8}{x^2}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 78, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,{x}^{2}}}-{\frac{a}{x}}+3\,{a}^{2}\ln \left ( x \right ) +{\frac{{a}^{2}}{8\,ax+8}}-{\frac{9\,{a}^{2}\ln \left ( ax+1 \right ) }{16}}-{\frac{{a}^{2}}{ax-1}}+{\frac{{a}^{2}}{8\, \left ( ax-1 \right ) ^{2}}}-{\frac{39\,{a}^{2}\ln \left ( ax-1 \right ) }{16}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.961134, size = 120, normalized size = 1.35 \begin{align*} -\frac{9}{16} \, a^{2} \log \left (a x + 1\right ) - \frac{39}{16} \, a^{2} \log \left (a x - 1\right ) + 3 \, a^{2} \log \left (x\right ) - \frac{15 \, a^{4} x^{4} - 3 \, a^{3} x^{3} - 22 \, a^{2} x^{2} + 4 \, a x + 4}{8 \,{\left (a^{3} x^{5} - a^{2} x^{4} - a x^{3} + x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.14004, size = 348, normalized size = 3.91 \begin{align*} -\frac{30 \, a^{4} x^{4} - 6 \, a^{3} x^{3} - 44 \, a^{2} x^{2} + 8 \, a x + 9 \,{\left (a^{5} x^{5} - a^{4} x^{4} - a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (a x + 1\right ) + 39 \,{\left (a^{5} x^{5} - a^{4} x^{4} - a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (a x - 1\right ) - 48 \,{\left (a^{5} x^{5} - a^{4} x^{4} - a^{3} x^{3} + a^{2} x^{2}\right )} \log \left (x\right ) + 8}{16 \,{\left (a^{3} x^{5} - a^{2} x^{4} - a x^{3} + x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.917268, size = 95, normalized size = 1.07 \begin{align*} 3 a^{2} \log{\left (x \right )} - \frac{39 a^{2} \log{\left (x - \frac{1}{a} \right )}}{16} - \frac{9 a^{2} \log{\left (x + \frac{1}{a} \right )}}{16} - \frac{15 a^{4} x^{4} - 3 a^{3} x^{3} - 22 a^{2} x^{2} + 4 a x + 4}{8 a^{3} x^{5} - 8 a^{2} x^{4} - 8 a x^{3} + 8 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23689, size = 111, normalized size = 1.25 \begin{align*} -\frac{9}{16} \, a^{2} \log \left ({\left | a x + 1 \right |}\right ) - \frac{39}{16} \, a^{2} \log \left ({\left | a x - 1 \right |}\right ) + 3 \, a^{2} \log \left ({\left | x \right |}\right ) - \frac{15 \, a^{4} x^{4} - 3 \, a^{3} x^{3} - 22 \, a^{2} x^{2} + 4 \, a x + 4}{8 \,{\left (a x + 1\right )}{\left (a x - 1\right )}^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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