Optimal. Leaf size=48 \[ \frac{x}{a^3}+\frac{1}{2 a^4 (1-a x)}+\frac{5 \log (1-a x)}{4 a^4}-\frac{\log (a x+1)}{4 a^4} \]
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Rubi [A] time = 0.113325, antiderivative size = 48, 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} \[ \frac{x}{a^3}+\frac{1}{2 a^4 (1-a x)}+\frac{5 \log (1-a x)}{4 a^4}-\frac{\log (a x+1)}{4 a^4} \]
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 )^{3/2}} \, dx &=\int \frac{x^3}{(1-a x)^2 (1+a x)} \, dx\\ &=\int \left (\frac{1}{a^3}+\frac{1}{2 a^3 (-1+a x)^2}+\frac{5}{4 a^3 (-1+a x)}-\frac{1}{4 a^3 (1+a x)}\right ) \, dx\\ &=\frac{x}{a^3}+\frac{1}{2 a^4 (1-a x)}+\frac{5 \log (1-a x)}{4 a^4}-\frac{\log (1+a x)}{4 a^4}\\ \end{align*}
Mathematica [A] time = 0.0306449, size = 39, normalized size = 0.81 \[ \frac{4 a x+\frac{2}{1-a x}+5 \log (1-a x)-\log (a x+1)}{4 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 41, normalized size = 0.9 \begin{align*}{\frac{x}{{a}^{3}}}-{\frac{\ln \left ( ax+1 \right ) }{4\,{a}^{4}}}-{\frac{1}{2\,{a}^{4} \left ( ax-1 \right ) }}+{\frac{5\,\ln \left ( ax-1 \right ) }{4\,{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.948588, size = 58, normalized size = 1.21 \begin{align*} -\frac{1}{2 \,{\left (a^{5} x - a^{4}\right )}} + \frac{x}{a^{3}} - \frac{\log \left (a x + 1\right )}{4 \, a^{4}} + \frac{5 \, \log \left (a x - 1\right )}{4 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.77264, size = 128, normalized size = 2.67 \begin{align*} \frac{4 \, a^{2} x^{2} - 4 \, a x -{\left (a x - 1\right )} \log \left (a x + 1\right ) + 5 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 2}{4 \,{\left (a^{5} x - a^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.412554, size = 39, normalized size = 0.81 \begin{align*} - \frac{1}{2 a^{5} x - 2 a^{4}} + \frac{x}{a^{3}} + \frac{\frac{5 \log{\left (x - \frac{1}{a} \right )}}{4} - \frac{\log{\left (x + \frac{1}{a} \right )}}{4}}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15815, size = 57, normalized size = 1.19 \begin{align*} \frac{x}{a^{3}} - \frac{\log \left ({\left | a x + 1 \right |}\right )}{4 \, a^{4}} + \frac{5 \, \log \left ({\left | a x - 1 \right |}\right )}{4 \, a^{4}} - \frac{1}{2 \,{\left (a x - 1\right )} a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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