Integrand size = 12, antiderivative size = 25 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^2} \, dx=-a \sqrt {1-\frac {1}{a^2 x^2}}-a \csc ^{-1}(a x) \]
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Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6304, 655, 222} \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^2} \, dx=a \left (-\csc ^{-1}(a x)\right )-a \sqrt {1-\frac {1}{a^2 x^2}} \]
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Rule 222
Rule 655
Rule 6304
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1-\frac {x}{a}}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -a \sqrt {1-\frac {1}{a^2 x^2}}-\text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{a^2}}} \, dx,x,\frac {1}{x}\right ) \\ & = -a \sqrt {1-\frac {1}{a^2 x^2}}-a \csc ^{-1}(a x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.04 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^2} \, dx=-a \left (\sqrt {1-\frac {1}{a^2 x^2}}+\arcsin \left (\frac {1}{a x}\right )\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(76\) vs. \(2(23)=46\).
Time = 0.12 (sec) , antiderivative size = 77, normalized size of antiderivative = 3.08
method | result | size |
risch | \(-\frac {\left (a x +1\right ) \sqrt {\frac {a x -1}{a x +1}}}{x}-\frac {a \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right ) \sqrt {\frac {a x -1}{a x +1}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{a x -1}\) | \(77\) |
default | \(\frac {\sqrt {\frac {a x -1}{a x +1}}\, \left (a x +1\right ) \left (-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a^{2} x^{2}+\left (a^{2} x^{2}-1\right )^{\frac {3}{2}} \sqrt {a^{2}}-\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}\, a x +\ln \left (\frac {a^{2} x +\sqrt {a^{2} x^{2}-1}\, \sqrt {a^{2}}}{\sqrt {a^{2}}}\right ) a^{2} x -a x \sqrt {a^{2}}\, \arctan \left (\frac {1}{\sqrt {a^{2} x^{2}-1}}\right )+\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}\, a x -\ln \left (\frac {a^{2} x +\sqrt {a^{2}}\, \sqrt {\left (a x -1\right ) \left (a x +1\right )}}{\sqrt {a^{2}}}\right ) a^{2} x \right )}{\sqrt {\left (a x -1\right ) \left (a x +1\right )}\, x \sqrt {a^{2}}}\) | \(220\) |
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Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (23) = 46\).
Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.88 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^2} \, dx=\frac {2 \, a x \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right ) - {\left (a x + 1\right )} \sqrt {\frac {a x - 1}{a x + 1}}}{x} \]
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\[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^2} \, dx=\int \frac {\sqrt {\frac {a x - 1}{a x + 1}}}{x^{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (23) = 46\).
Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.20 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^2} \, dx=-2 \, a {\left (\frac {\sqrt {\frac {a x - 1}{a x + 1}}}{\frac {a x - 1}{a x + 1} + 1} - \arctan \left (\sqrt {\frac {a x - 1}{a x + 1}}\right )\right )} \]
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Exception generated. \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^2} \, dx=\text {Exception raised: TypeError} \]
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Time = 4.19 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.20 \[ \int \frac {e^{-\coth ^{-1}(a x)}}{x^2} \, dx=2\,a\,\mathrm {atan}\left (\sqrt {\frac {a\,x-1}{a\,x+1}}\right )-\frac {2\,a\,\sqrt {\frac {a\,x-1}{a\,x+1}}}{\frac {a\,x-1}{a\,x+1}+1} \]
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