Optimal. Leaf size=42 \[ \frac{x^2}{a^2}-\frac{2 x}{a^3}+\frac{2 \log (a x+1)}{a^4}-\frac{2 x^3}{3 a}+\frac{x^4}{4} \]
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Rubi [A] time = 0.0570863, antiderivative size = 42, 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} \[ \frac{x^2}{a^2}-\frac{2 x}{a^3}+\frac{2 \log (a x+1)}{a^4}-\frac{2 x^3}{3 a}+\frac{x^4}{4} \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6126
Rule 77
Rubi steps
\begin{align*} \int e^{-2 \coth ^{-1}(a x)} x^3 \, dx &=-\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 = 0.0193023, size = 42, 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 = 0.039, size = 39, 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 = 0.985613, size = 58, normalized size = 1.38 \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 = 1.7463, size = 100, normalized size = 2.38 \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 = 0.276682, size = 37, normalized size = 0.88 \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 = 1.13709, size = 63, normalized size = 1.5 \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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