Optimal. Leaf size=39 \[ -\frac{x^2}{2 a^2}-\frac{x}{a^3}-\frac{\log (1-a x)}{a^4}-\frac{x^3}{3 a} \]
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Rubi [A] time = 0.0960698, antiderivative size = 39, 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, 43} \[ -\frac{x^2}{2 a^2}-\frac{x}{a^3}-\frac{\log (1-a x)}{a^4}-\frac{x^3}{3 a} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^3}{\sqrt{1-a^2 x^2}} \, dx &=\int \frac{x^3}{1-a x} \, dx\\ &=\int \left (-\frac{1}{a^3}-\frac{x}{a^2}-\frac{x^2}{a}-\frac{1}{a^3 (-1+a x)}\right ) \, dx\\ &=-\frac{x}{a^3}-\frac{x^2}{2 a^2}-\frac{x^3}{3 a}-\frac{\log (1-a x)}{a^4}\\ \end{align*}
Mathematica [A] time = 0.019688, size = 39, normalized size = 1. \[ -\frac{x^2}{2 a^2}-\frac{x}{a^3}-\frac{\log (1-a x)}{a^4}-\frac{x^3}{3 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 35, normalized size = 0.9 \begin{align*} -{\frac{{x}^{3}}{3\,a}}-{\frac{{x}^{2}}{2\,{a}^{2}}}-{\frac{x}{{a}^{3}}}-{\frac{\ln \left ( ax-1 \right ) }{{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.951527, size = 47, normalized size = 1.21 \begin{align*} -\frac{2 \, a^{2} x^{3} + 3 \, a x^{2} + 6 \, x}{6 \, a^{3}} - \frac{\log \left (a x - 1\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.48202, size = 80, normalized size = 2.05 \begin{align*} -\frac{2 \, a^{3} x^{3} + 3 \, a^{2} x^{2} + 6 \, a x + 6 \, \log \left (a x - 1\right )}{6 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.25716, size = 31, normalized size = 0.79 \begin{align*} - \frac{x^{3}}{3 a} - \frac{x^{2}}{2 a^{2}} - \frac{x}{a^{3}} - \frac{\log{\left (a x - 1 \right )}}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13088, size = 49, normalized size = 1.26 \begin{align*} -\frac{2 \, a^{2} x^{3} + 3 \, a x^{2} + 6 \, x}{6 \, a^{3}} - \frac{\log \left ({\left | a x - 1 \right |}\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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