Optimal. Leaf size=27 \[ \frac{4}{a (1-a x)}+\frac{4 \log (1-a x)}{a}+x \]
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Rubi [A] time = 0.0191366, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {6167, 6125, 43} \[ \frac{4}{a (1-a x)}+\frac{4 \log (1-a x)}{a}+x \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6125
Rule 43
Rubi steps
\begin{align*} \int e^{4 \coth ^{-1}(a x)} \, dx &=\int e^{4 \tanh ^{-1}(a x)} \, dx\\ &=\int \frac{(1+a x)^2}{(1-a x)^2} \, dx\\ &=\int \left (1+\frac{4}{(-1+a x)^2}+\frac{4}{-1+a x}\right ) \, dx\\ &=x+\frac{4}{a (1-a x)}+\frac{4 \log (1-a x)}{a}\\ \end{align*}
Mathematica [A] time = 0.0185358, size = 26, normalized size = 0.96 \[ -\frac{4}{a (a x-1)}+\frac{4 \log (1-a x)}{a}+x \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 26, normalized size = 1. \begin{align*} x-4\,{\frac{1}{a \left ( ax-1 \right ) }}+4\,{\frac{\ln \left ( ax-1 \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.998486, size = 35, normalized size = 1.3 \begin{align*} x + \frac{4 \, \log \left (a x - 1\right )}{a} - \frac{4}{a^{2} x - a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71151, size = 81, normalized size = 3. \begin{align*} \frac{a^{2} x^{2} - a x + 4 \,{\left (a x - 1\right )} \log \left (a x - 1\right ) - 4}{a^{2} x - a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.319388, size = 19, normalized size = 0.7 \begin{align*} x - \frac{4}{a^{2} x - a} + \frac{4 \log{\left (a x - 1 \right )}}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12444, size = 62, normalized size = 2.3 \begin{align*} \frac{a x - 1}{a} - \frac{4 \, \log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a} - \frac{4}{{\left (a x - 1\right )} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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