Integrand size = 12, antiderivative size = 13 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{x} \, dx=\frac {4}{1-a x}+\log (x) \]
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Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6302, 6261, 90} \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{x} \, dx=\frac {4}{1-a x}+\log (x) \]
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Rule 90
Rule 6261
Rule 6302
Rubi steps \begin{align*} \text {integral}& = \int \frac {e^{4 \text {arctanh}(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*}
Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{x} \, dx=\frac {4}{1-a x}+\log (x) \]
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Time = 0.46 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00
method | result | size |
default | \(\ln \left (x \right )-\frac {4}{a x -1}\) | \(13\) |
norman | \(-\frac {4 a x}{a x -1}+\ln \left (x \right )\) | \(15\) |
risch | \(-\frac {4}{a x -1}+\ln \left (-x \right )\) | \(15\) |
parallelrisch | \(\frac {a \ln \left (x \right ) x -4 a x -\ln \left (x \right )}{a x -1}\) | \(23\) |
meijerg | \(\frac {3 a x}{-a x +1}+\frac {2 a x}{-2 a x +2}+1+\ln \left (x \right )+\ln \left (-a \right )\) | \(33\) |
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none
Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.38 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{x} \, dx=\frac {{\left (a x - 1\right )} \log \left (x\right ) - 4}{a x - 1} \]
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Time = 0.09 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.62 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{x} \, dx=\log {\left (x \right )} - \frac {4}{a x - 1} \]
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none
Time = 0.18 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{x} \, dx=-\frac {4}{a x - 1} + \log \left (x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (12) = 24\).
Time = 0.27 (sec) , antiderivative size = 57, normalized size of antiderivative = 4.38 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{x} \, dx=-a {\left (\frac {\log \left (\frac {{\left | a x - 1 \right |}}{{\left (a x - 1\right )}^{2} {\left | a \right |}}\right )}{a} - \frac {\log \left ({\left | -\frac {1}{a x - 1} - 1 \right |}\right )}{a} + \frac {4}{{\left (a x - 1\right )} a}\right )} \]
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Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {e^{4 \coth ^{-1}(a x)}}{x} \, dx=\ln \left (x\right )-\frac {4}{a\,x-1} \]
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