Optimal. Leaf size=49 \[ -\frac{x^3}{3 a^2}-\frac{x^2}{2 a^3}-\frac{x}{a^4}-\frac{\log (1-a x)}{a^5}-\frac{x^4}{4 a} \]
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Rubi [A] time = 0.0999109, antiderivative size = 49, 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} \[ -\frac{x^3}{3 a^2}-\frac{x^2}{2 a^3}-\frac{x}{a^4}-\frac{\log (1-a x)}{a^5}-\frac{x^4}{4 a} \]
Antiderivative was successfully verified.
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Rule 6150
Rule 43
Rubi steps
\begin{align*} \int \frac{e^{\tanh ^{-1}(a x)} x^4}{\sqrt{1-a^2 x^2}} \, dx &=\int \frac{x^4}{1-a x} \, dx\\ &=\int \left (-\frac{1}{a^4}-\frac{x}{a^3}-\frac{x^2}{a^2}-\frac{x^3}{a}-\frac{1}{a^4 (-1+a x)}\right ) \, dx\\ &=-\frac{x}{a^4}-\frac{x^2}{2 a^3}-\frac{x^3}{3 a^2}-\frac{x^4}{4 a}-\frac{\log (1-a x)}{a^5}\\ \end{align*}
Mathematica [A] time = 0.024949, size = 49, normalized size = 1. \[ -\frac{x^3}{3 a^2}-\frac{x^2}{2 a^3}-\frac{x}{a^4}-\frac{\log (1-a x)}{a^5}-\frac{x^4}{4 a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 43, normalized size = 0.9 \begin{align*} -{\frac{{x}^{4}}{4\,a}}-{\frac{{x}^{3}}{3\,{a}^{2}}}-{\frac{{x}^{2}}{2\,{a}^{3}}}-{\frac{x}{{a}^{4}}}-{\frac{\ln \left ( ax-1 \right ) }{{a}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.9511, size = 58, normalized size = 1.18 \begin{align*} -\frac{3 \, a^{3} x^{4} + 4 \, a^{2} x^{3} + 6 \, a x^{2} + 12 \, x}{12 \, a^{4}} - \frac{\log \left (a x - 1\right )}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.45332, size = 100, normalized size = 2.04 \begin{align*} -\frac{3 \, a^{4} x^{4} + 4 \, a^{3} x^{3} + 6 \, a^{2} x^{2} + 12 \, a x + 12 \, \log \left (a x - 1\right )}{12 \, a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.263434, size = 39, normalized size = 0.8 \begin{align*} - \frac{x^{4}}{4 a} - \frac{x^{3}}{3 a^{2}} - \frac{x^{2}}{2 a^{3}} - \frac{x}{a^{4}} - \frac{\log{\left (a x - 1 \right )}}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.21649, size = 59, normalized size = 1.2 \begin{align*} -\frac{3 \, a^{3} x^{4} + 4 \, a^{2} x^{3} + 6 \, a x^{2} + 12 \, x}{12 \, a^{4}} - \frac{\log \left ({\left | a x - 1 \right |}\right )}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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