Optimal. Leaf size=47 \[ \frac{8 x}{a^2}+\frac{4}{a^3 (1-a x)}+\frac{12 \log (1-a x)}{a^3}+\frac{2 x^2}{a}+\frac{x^3}{3} \]
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Rubi [A] time = 0.0595973, antiderivative size = 47, 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, 88} \[ \frac{8 x}{a^2}+\frac{4}{a^3 (1-a x)}+\frac{12 \log (1-a x)}{a^3}+\frac{2 x^2}{a}+\frac{x^3}{3} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6126
Rule 88
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} x^2 \, dx &=\int e^{4 \tanh ^{-1}(a x)} x^2 \, dx\\ &=\int \frac{x^2 (1+a x)^2}{(1-a x)^2} \, dx\\ &=\int \left (\frac{8}{a^2}+\frac{4 x}{a}+x^2+\frac{4}{a^2 (-1+a x)^2}+\frac{12}{a^2 (-1+a x)}\right ) \, dx\\ &=\frac{8 x}{a^2}+\frac{2 x^2}{a}+\frac{x^3}{3}+\frac{4}{a^3 (1-a x)}+\frac{12 \log (1-a x)}{a^3}\\ \end{align*}
Mathematica [A] time = 0.0361972, size = 47, normalized size = 1. \[ \frac{8 x}{a^2}+\frac{4}{a^3 (1-a x)}+\frac{12 \log (1-a x)}{a^3}+\frac{2 x^2}{a}+\frac{x^3}{3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 44, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3}}+2\,{\frac{{x}^{2}}{a}}+8\,{\frac{x}{{a}^{2}}}-4\,{\frac{1}{{a}^{3} \left ( ax-1 \right ) }}+12\,{\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 = 1.07234, size = 66, normalized size = 1.4 \begin{align*} -\frac{4}{a^{4} x - a^{3}} + \frac{a^{2} x^{3} + 6 \, a x^{2} + 24 \, x}{3 \, a^{2}} + \frac{12 \, \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 = 1.71211, size = 130, normalized size = 2.77 \begin{align*} \frac{a^{4} x^{4} + 5 \, a^{3} x^{3} + 18 \, a^{2} x^{2} - 24 \, a x + 36 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 12}{3 \,{\left (a^{4} x - a^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.320149, size = 39, normalized size = 0.83 \begin{align*} \frac{x^{3}}{3} - \frac{4}{a^{4} x - a^{3}} + \frac{2 x^{2}}{a} + \frac{8 x}{a^{2}} + \frac{12 \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 = 1.1178, size = 93, normalized size = 1.98 \begin{align*} \frac{{\left (a x - 1\right )}^{3}{\left (\frac{9}{a x - 1} + \frac{39}{{\left (a x - 1\right )}^{2}} + 1\right )}}{3 \, a^{3}} - \frac{12 \, \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a^{3}} - \frac{4}{{\left (a x - 1\right )} a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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