Optimal. Leaf size=59 \[ \frac{1}{2 (1-a x)}+\frac{1}{8 (a x+1)}+\frac{1}{8 (1-a x)^2}-\frac{11}{16} \log (1-a x)-\frac{5}{16} \log (a x+1)+\log (x) \]
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Rubi [A] time = 0.117907, antiderivative size = 59, 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} \[ \frac{1}{2 (1-a x)}+\frac{1}{8 (a x+1)}+\frac{1}{8 (1-a x)^2}-\frac{11}{16} \log (1-a x)-\frac{5}{16} \log (a x+1)+\log (x) \]
Antiderivative was successfully verified.
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Rule 6150
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)}}{x \left (1-a^2 x^2\right )^{5/2}} \, dx &=\int \frac{1}{x (1-a x)^3 (1+a x)^2} \, dx\\ &=\int \left (\frac{1}{x}-\frac{a}{4 (-1+a x)^3}+\frac{a}{2 (-1+a x)^2}-\frac{11 a}{16 (-1+a x)}-\frac{a}{8 (1+a x)^2}-\frac{5 a}{16 (1+a x)}\right ) \, dx\\ &=\frac{1}{8 (1-a x)^2}+\frac{1}{2 (1-a x)}+\frac{1}{8 (1+a x)}+\log (x)-\frac{11}{16} \log (1-a x)-\frac{5}{16} \log (1+a x)\\ \end{align*}
Mathematica [A] time = 0.0452678, size = 54, normalized size = 0.92 \[ \frac{1}{16} \left (\frac{8}{1-a x}+\frac{2}{a x+1}+\frac{2}{(a x-1)^2}-11 \log (1-a x)-5 \log (a x+1)+16 \log (x)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.038, size = 47, normalized size = 0.8 \begin{align*} \ln \left ( x \right ) +{\frac{1}{8\,ax+8}}-{\frac{5\,\ln \left ( ax+1 \right ) }{16}}+{\frac{1}{8\, \left ( ax-1 \right ) ^{2}}}-{\frac{1}{2\,ax-2}}-{\frac{11\,\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 = 0.958549, size = 77, normalized size = 1.31 \begin{align*} -\frac{3 \, a^{2} x^{2} + a x - 6}{8 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )}} - \frac{5}{16} \, \log \left (a x + 1\right ) - \frac{11}{16} \, \log \left (a x - 1\right ) + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.13587, size = 269, normalized size = 4.56 \begin{align*} -\frac{6 \, a^{2} x^{2} + 2 \, a x + 5 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x + 1\right ) + 11 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (a x - 1\right ) - 16 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )} \log \left (x\right ) - 12}{16 \,{\left (a^{3} x^{3} - a^{2} x^{2} - a x + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.590731, size = 60, normalized size = 1.02 \begin{align*} - \frac{3 a^{2} x^{2} + a x - 6}{8 a^{3} x^{3} - 8 a^{2} x^{2} - 8 a x + 8} + \log{\left (x \right )} - \frac{11 \log{\left (x - \frac{1}{a} \right )}}{16} - \frac{5 \log{\left (x + \frac{1}{a} \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16475, size = 69, normalized size = 1.17 \begin{align*} -\frac{3 \, a^{2} x^{2} + a x - 6}{8 \,{\left (a x + 1\right )}{\left (a x - 1\right )}^{2}} - \frac{5}{16} \, \log \left ({\left | a x + 1 \right |}\right ) - \frac{11}{16} \, \log \left ({\left | a x - 1 \right |}\right ) + \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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