Optimal. Leaf size=32 \[ -\frac{2 x}{a^2}+\frac{2 \log (a x+1)}{a^3}+\frac{x^2}{a}-\frac{x^3}{3} \]
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Rubi [A] time = 0.0331909, antiderivative size = 32, 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} \[ -\frac{2 x}{a^2}+\frac{2 \log (a x+1)}{a^3}+\frac{x^2}{a}-\frac{x^3}{3} \]
Antiderivative was successfully verified.
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Rule 6126
Rule 77
Rubi steps
\begin{align*} \int e^{-2 \tanh ^{-1}(a x)} x^2 \, dx &=\int \frac{x^2 (1-a x)}{1+a x} \, dx\\ &=\int \left (-\frac{2}{a^2}+\frac{2 x}{a}-x^2+\frac{2}{a^2 (1+a x)}\right ) \, dx\\ &=-\frac{2 x}{a^2}+\frac{x^2}{a}-\frac{x^3}{3}+\frac{2 \log (1+a x)}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0138062, size = 32, normalized size = 1. \[ -\frac{2 x}{a^2}+\frac{2 \log (a x+1)}{a^3}+\frac{x^2}{a}-\frac{x^3}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 31, normalized size = 1. \begin{align*} -2\,{\frac{x}{{a}^{2}}}+{\frac{{x}^{2}}{a}}-{\frac{{x}^{3}}{3}}+2\,{\frac{\ln \left ( ax+1 \right ) }{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.974067, size = 46, normalized size = 1.44 \begin{align*} -\frac{a^{2} x^{3} - 3 \, a x^{2} + 6 \, x}{3 \, a^{2}} + \frac{2 \, \log \left (a x + 1\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.91128, size = 77, normalized size = 2.41 \begin{align*} -\frac{a^{3} x^{3} - 3 \, a^{2} x^{2} + 6 \, a x - 6 \, \log \left (a x + 1\right )}{3 \, a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.356007, size = 27, normalized size = 0.84 \begin{align*} - \frac{x^{3}}{3} + \frac{x^{2}}{a} - \frac{2 x}{a^{2}} + \frac{2 \log{\left (a x + 1 \right )}}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17283, size = 77, normalized size = 2.41 \begin{align*} \frac{{\left (a x + 1\right )}^{3}{\left (\frac{6}{a x + 1} - \frac{15}{{\left (a x + 1\right )}^{2}} - 1\right )}}{3 \, a^{3}} - \frac{2 \, \log \left (\frac{{\left | a x + 1 \right |}}{{\left (a x + 1\right )}^{2}{\left | a \right |}}\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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