Optimal. Leaf size=13 \[ \frac{4}{1-a x}+\log (x) \]
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Rubi [A] time = 0.0392385, antiderivative size = 13, 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} \[ \frac{4}{1-a x}+\log (x) \]
Antiderivative was successfully verified.
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Rule 6167
Rule 6126
Rule 88
Rubi steps
\begin{align*} \int \frac{e^{4 \coth ^{-1}(a x)}}{x} \, dx &=\int \frac{e^{4 \tanh ^{-1}(a x)}}{x} \, dx\\ &=\int \frac{(1+a x)^2}{x (1-a x)^2} \, dx\\ &=\int \left (\frac{1}{x}+\frac{4 a}{(-1+a x)^2}\right ) \, dx\\ &=\frac{4}{1-a x}+\log (x)\\ \end{align*}
Mathematica [A] time = 0.0097751, size = 13, normalized size = 1. \[ \frac{4}{1-a x}+\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 13, normalized size = 1. \begin{align*} -4\, \left ( ax-1 \right ) ^{-1}+\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01998, size = 16, normalized size = 1.23 \begin{align*} -\frac{4}{a x - 1} + \log \left (x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.78683, size = 46, normalized size = 3.54 \begin{align*} \frac{{\left (a x - 1\right )} \log \left (x\right ) - 4}{a x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.355664, size = 8, normalized size = 0.62 \begin{align*} \log{\left (x \right )} - \frac{4}{a x - 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.12011, size = 77, normalized size = 5.92 \begin{align*} -a{\left (\frac{\log \left (\frac{{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2}{\left | a \right |}}\right )}{a} - \frac{\log \left ({\left | -\frac{1}{a x - 1} - 1 \right |}\right )}{a} + \frac{4}{{\left (a x - 1\right )} a}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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