Optimal. Leaf size=43 \[ -\frac{x^2}{a^2}+\frac{2 x}{a^3}-\frac{2 \log (a x+1)}{a^4}+\frac{2 x^3}{3 a}-\frac{x^4}{4} \]
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Rubi [A] time = 0.0392676, antiderivative size = 43, 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} \[ -\frac{x^2}{a^2}+\frac{2 x}{a^3}-\frac{2 \log (a x+1)}{a^4}+\frac{2 x^3}{3 a}-\frac{x^4}{4} \]
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 = 0.0187701, size = 43, normalized size = 1. \[ -\frac{x^2}{a^2}+\frac{2 x}{a^3}-\frac{2 \log (a x+1)}{a^4}+\frac{2 x^3}{3 a}-\frac{x^4}{4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 40, normalized size = 0.9 \begin{align*} 2\,{\frac{x}{{a}^{3}}}-{\frac{{x}^{2}}{{a}^{2}}}+{\frac{2\,{x}^{3}}{3\,a}}-{\frac{{x}^{4}}{4}}-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 = 0.945287, size = 58, normalized size = 1.35 \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 = 1.76151, size = 101, normalized size = 2.35 \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 = 0.338888, size = 37, normalized size = 0.86 \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 = 1.23158, size = 89, normalized size = 2.07 \begin{align*} \frac{{\left (a x + 1\right )}^{4}{\left (\frac{20}{a x + 1} - \frac{54}{{\left (a x + 1\right )}^{2}} + \frac{84}{{\left (a x + 1\right )}^{3}} - 3\right )}}{12 \, a^{4}} + \frac{2 \, \log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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