Integrand size = 8, antiderivative size = 37 \[ \int e^{-\coth ^{-1}(a x)} \, dx=\sqrt {1-\frac {1}{a^2 x^2}} x-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]
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Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6303, 821, 272, 65, 214} \[ \int e^{-\coth ^{-1}(a x)} \, dx=x \sqrt {1-\frac {1}{a^2 x^2}}-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \]
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Rule 65
Rule 214
Rule 272
Rule 821
Rule 6303
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1-\frac {x}{a}}{x^2 \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right )}{a} \\ & = \sqrt {1-\frac {1}{a^2 x^2}} x+\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {1-\frac {x}{a^2}}} \, dx,x,\frac {1}{x^2}\right )}{2 a} \\ & = \sqrt {1-\frac {1}{a^2 x^2}} x-a \text {Subst}\left (\int \frac {1}{a^2-a^2 x^2} \, dx,x,\sqrt {1-\frac {1}{a^2 x^2}}\right ) \\ & = \sqrt {1-\frac {1}{a^2 x^2}} x-\frac {\text {arctanh}\left (\sqrt {1-\frac {1}{a^2 x^2}}\right )}{a} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.14 \[ \int e^{-\coth ^{-1}(a x)} \, dx=\sqrt {1-\frac {1}{a^2 x^2}} x-\frac {\log \left (a \left (1+\sqrt {1-\frac {1}{a^2 x^2}}\right ) x\right )}{a} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(90\) vs. \(2(33)=66\).
Time = 0.11 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.46
method | result | size |
risch | \(\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{a}-\frac {\ln \left (\frac {a^{2} x}{\sqrt {a^{2}}}+\sqrt {a^{2} x^{2}-1}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}\, \left (a x -1\right )}\) | \(91\) |
default | \(-\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (a \ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right )-\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\right )}{\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a \sqrt {a^{2}}}\) | \(99\) |
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none
Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.73 \[ \int e^{-\coth ^{-1}(a x)} \, dx=\frac {{\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}} - \log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right ) + \log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a} \]
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\[ \int e^{-\coth ^{-1}(a x)} \, dx=\int \sqrt {\frac {a x - 1}{a x + 1}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (33) = 66\).
Time = 0.19 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.43 \[ \int e^{-\coth ^{-1}(a x)} \, dx=-a {\left (\frac {2 \, \sqrt {\frac {a x - 1}{a x + 1}}}{\frac {{\left (a x - 1\right )} a^{2}}{a x + 1} - a^{2}} + \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} + 1\right )}{a^{2}} - \frac {\log \left (\sqrt {\frac {a x - 1}{a x + 1}} - 1\right )}{a^{2}}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.41 \[ \int e^{-\coth ^{-1}(a x)} \, dx=\frac {\log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) \mathrm {sgn}\left (a x + 1\right )}{{\left | a \right |}} + \frac {\sqrt {a^{2} x^{2} - 1} \mathrm {sgn}\left (a x + 1\right )}{a} \]
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Time = 4.63 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.57 \[ \int e^{-\coth ^{-1}(a x)} \, dx=\frac {2\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{a-\frac {a\,\left (a\,x-1\right )}{a\,x+1}}-\frac {2\,\mathrm {atanh}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )}{a} \]
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