Optimal. Leaf size=13 \[ \frac {4}{1-a x}+\log (x) \]
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Rubi [A] time = 0.04, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6167, 6126, 88} \[ \frac {4}{1-a x}+\log (x) \]
Antiderivative was successfully verified.
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Rule 88
Rule 6126
Rule 6167
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*}
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Mathematica [A] time = 0.01, size = 13, normalized size = 1.00 \[ \frac {4}{1-a x}+\log (x) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 18, normalized size = 1.38 \[ \frac {{\left (a x - 1\right )} \log \relax (x) - 4}{a x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 57, normalized size = 4.38 \[ -a {\left (\frac {\log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a} - \frac {\log \left ({\left | -\frac {1}{a x - 1} - 1 \right |}\right )}{a} + \frac {4}{{\left (a x - 1\right )} a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 13, normalized size = 1.00 \[ \ln \relax (x )-\frac {4}{a x -1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 12, normalized size = 0.92 \[ -\frac {4}{a x - 1} + \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.03, size = 12, normalized size = 0.92 \[ \ln \relax (x)-\frac {4}{a\,x-1} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.15, size = 8, normalized size = 0.62 \[ \log {\relax (x )} - \frac {4}{a x - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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