Integrand size = 10, antiderivative size = 22 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x} \, dx=-\csc ^{-1}(a x)+\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right ) \]
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Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6304, 858, 222, 272, 65, 214} \[ \int \frac {e^{\coth ^{-1}(a x)}}{x} \, dx=\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )-\csc ^{-1}(a x) \]
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Rule 65
Rule 214
Rule 222
Rule 272
Rule 858
Rule 6304
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1+\frac {x}{a}}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a}-\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -\csc ^{-1}(a x)-\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right ) \\ & = -\csc ^{-1}(a x)+a^2 \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right ) \\ & = -\csc ^{-1}(a x)+\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x} \, dx=-\arcsin \left (\frac {1}{a x}\right )+\log \left (x \left (1+\sqrt {\frac {-1+a^2 x^2}{a^2 x^2}}\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(131\) vs. \(2(20)=40\).
Time = 0.12 (sec) , antiderivative size = 132, normalized size of antiderivative = 6.00
method | result | size |
default | \(-\frac {\left (a x -1\right ) \left (\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}+\arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {a^{2}}-a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )-\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right )}{\sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a^{2}}}\) | \(132\) |
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Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (20) = 40\).
Time = 0.25 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.59 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x} \, dx=2 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) + \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) - \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right ) \]
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\[ \int \frac {e^{\coth ^{-1}(a x)}}{x} \, dx=\int \frac {1}{x \sqrt {\frac {a x - 1}{a x + 1}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (20) = 40\).
Time = 0.33 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.14 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x} \, dx=a {\left (\frac {2 \, \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )}{a} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a}\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (20) = 40\).
Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.86 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x} \, dx=\frac {2 \, \arctan \left (-x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1}\right )}{\mathrm {sgn}\left (a x + 1\right )} - \frac {a \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right )}{{\left | a \right |} \mathrm {sgn}\left (a x + 1\right )} \]
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Time = 4.14 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \frac {e^{\coth ^{-1}(a x)}}{x} \, dx=2\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )+2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right ) \]
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