Optimal. Leaf size=13 \[ \frac{4}{1-a x}+\log (x) \]
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Rubi [A] time = 0.0251725, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {6126, 88} \[ \frac{4}{1-a x}+\log (x) \]
Antiderivative was successfully verified.
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Rule 6126
Rule 88
Rubi steps
\begin{align*} \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 = 0.0091648, size = 13, normalized size = 1. \[ \frac{4}{1-a x}+\log (x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 13, normalized size = 1. \begin{align*} \ln \left ( x \right ) -4\, \left ( ax-1 \right ) ^{-1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.983844, 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 = 1.98258, 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 = 0.47497, 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 [A] time = 1.24653, size = 18, normalized size = 1.38 \begin{align*} -\frac{4}{a x - 1} + \log \left ({\left | x \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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