Optimal. Leaf size=33 \[ \frac{2 x}{a^2}-\frac{2 \log (a x+1)}{a^3}-\frac{x^2}{a}+\frac{x^3}{3} \]
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Rubi [A] time = 0.050608, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6167, 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 6167
Rule 6126
Rule 77
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} x^2 \, dx &=-\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 = 0.0137995, size = 33, 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 = 0.039, size = 32, 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 = 1.02015, size = 46, normalized size = 1.39 \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 = 1.58506, size = 76, normalized size = 2.3 \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 = 0.270656, size = 27, normalized size = 0.82 \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 = 1.14682, size = 51, normalized size = 1.55 \begin{align*} \frac{a^{3} x^{3} - 3 \, a^{2} x^{2} + 6 \, a x}{3 \, a^{3}} - \frac{2 \, \log \left ({\left | a x + 1 \right |}\right )}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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