Optimal. Leaf size=44 \[ -\frac{x^2}{a^2}-\frac{2 x}{a^3}-\frac{2 \log (1-a x)}{a^4}-\frac{2 x^3}{3 a}-\frac{x^4}{4} \]
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Rubi [A] time = 0.0423027, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6126, 77} \[ -\frac{x^2}{a^2}-\frac{2 x}{a^3}-\frac{2 \log (1-a x)}{a^4}-\frac{2 x^3}{3 a}-\frac{x^4}{4} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 77
Rubi steps
\begin{align*} \int e^{2 \tanh ^{-1}(a x)} x^3 \, dx &=\int \frac{x^3 (1+a x)}{1-a x} \, dx\\ &=\int \left (-\frac{2}{a^3}-\frac{2 x}{a^2}-\frac{2 x^2}{a}-x^3-\frac{2}{a^3 (-1+a x)}\right ) \, dx\\ &=-\frac{2 x}{a^3}-\frac{x^2}{a^2}-\frac{2 x^3}{3 a}-\frac{x^4}{4}-\frac{2 \log (1-a x)}{a^4}\\ \end{align*}
Mathematica [A] time = 0.0183638, size = 44, normalized size = 1. \[ -\frac{x^2}{a^2}-\frac{2 x}{a^3}-\frac{2 \log (1-a x)}{a^4}-\frac{2 x^3}{3 a}-\frac{x^4}{4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 40, normalized size = 0.9 \begin{align*} -{\frac{{x}^{4}}{4}}-{\frac{2\,{x}^{3}}{3\,a}}-{\frac{{x}^{2}}{{a}^{2}}}-2\,{\frac{x}{{a}^{3}}}-2\,{\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 = 0.939016, size = 58, normalized size = 1.32 \begin{align*} -\frac{3 \, a^{3} x^{4} + 8 \, a^{2} x^{3} + 12 \, a x^{2} + 24 \, x}{12 \, a^{3}} - \frac{2 \, \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 = 1.6383, size = 101, normalized size = 2.3 \begin{align*} -\frac{3 \, a^{4} x^{4} + 8 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 24 \, a x + 24 \, \log \left (a x - 1\right )}{12 \, a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.429389, size = 39, normalized size = 0.89 \begin{align*} - \frac{x^{4}}{4} - \frac{2 x^{3}}{3 a} - \frac{x^{2}}{a^{2}} - \frac{2 x}{a^{3}} - \frac{2 \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 = 1.15617, size = 63, normalized size = 1.43 \begin{align*} -\frac{3 \, a^{4} x^{4} + 8 \, a^{3} x^{3} + 12 \, a^{2} x^{2} + 24 \, a x}{12 \, a^{4}} - \frac{2 \, \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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