Optimal. Leaf size=57 \[ \frac{4 x^2}{a^2}+\frac{12 x}{a^3}+\frac{4}{a^4 (1-a x)}+\frac{16 \log (1-a x)}{a^4}+\frac{4 x^3}{3 a}+\frac{x^4}{4} \]
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Rubi [A] time = 0.0658147, antiderivative size = 57, 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} \[ \frac{4 x^2}{a^2}+\frac{12 x}{a^3}+\frac{4}{a^4 (1-a x)}+\frac{16 \log (1-a x)}{a^4}+\frac{4 x^3}{3 a}+\frac{x^4}{4} \]
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^3 \, dx &=\int e^{4 \tanh ^{-1}(a x)} x^3 \, dx\\ &=\int \frac{x^3 (1+a x)^2}{(1-a x)^2} \, dx\\ &=\int \left (\frac{12}{a^3}+\frac{8 x}{a^2}+\frac{4 x^2}{a}+x^3+\frac{4}{a^3 (-1+a x)^2}+\frac{16}{a^3 (-1+a x)}\right ) \, dx\\ &=\frac{12 x}{a^3}+\frac{4 x^2}{a^2}+\frac{4 x^3}{3 a}+\frac{x^4}{4}+\frac{4}{a^4 (1-a x)}+\frac{16 \log (1-a x)}{a^4}\\ \end{align*}
Mathematica [A] time = 0.045886, size = 57, normalized size = 1. \[ \frac{4 x^2}{a^2}+\frac{12 x}{a^3}+\frac{4}{a^4 (1-a x)}+\frac{16 \log (1-a x)}{a^4}+\frac{4 x^3}{3 a}+\frac{x^4}{4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 52, normalized size = 0.9 \begin{align*}{\frac{{x}^{4}}{4}}+{\frac{4\,{x}^{3}}{3\,a}}+4\,{\frac{{x}^{2}}{{a}^{2}}}+12\,{\frac{x}{{a}^{3}}}-4\,{\frac{1}{{a}^{4} \left ( ax-1 \right ) }}+16\,{\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 = 0.984836, size = 78, normalized size = 1.37 \begin{align*} -\frac{4}{a^{5} x - a^{4}} + \frac{3 \, a^{3} x^{4} + 16 \, a^{2} x^{3} + 48 \, a x^{2} + 144 \, x}{12 \, a^{3}} + \frac{16 \, \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 = 1.70644, size = 155, normalized size = 2.72 \begin{align*} \frac{3 \, a^{5} x^{5} + 13 \, a^{4} x^{4} + 32 \, a^{3} x^{3} + 96 \, a^{2} x^{2} - 144 \, a x + 192 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 48}{12 \,{\left (a^{5} x - a^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.353346, size = 49, normalized size = 0.86 \begin{align*} \frac{x^{4}}{4} - \frac{4}{a^{5} x - a^{4}} + \frac{4 x^{3}}{3 a} + \frac{4 x^{2}}{a^{2}} + \frac{12 x}{a^{3}} + \frac{16 \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 = 1.21016, size = 105, normalized size = 1.84 \begin{align*} \frac{{\left (a x - 1\right )}^{4}{\left (\frac{28}{a x - 1} + \frac{114}{{\left (a x - 1\right )}^{2}} + \frac{300}{{\left (a x - 1\right )}^{3}} + 3\right )}}{12 \, a^{4}} - \frac{16 \, \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a^{4}} - \frac{4}{{\left (a x - 1\right )} a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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