Integrand size = 10, antiderivative size = 38 \[ \int \frac {e^{\text {csch}^{-1}(a x)}}{x} \, dx=-\sqrt {1+\frac {1}{a^2 x^2}}-\frac {1}{a x}+\text {arctanh}\left (\sqrt {1+\frac {1}{a^2 x^2}}\right ) \]
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Time = 0.02 (sec) , antiderivative size = 38, 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 = {6471, 30, 272, 52, 65, 214} \[ \int \frac {e^{\text {csch}^{-1}(a x)}}{x} \, dx=\text {arctanh}\left (\sqrt {\frac {1}{a^2 x^2}+1}\right )-\sqrt {\frac {1}{a^2 x^2}+1}-\frac {1}{a x} \]
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Rule 30
Rule 52
Rule 65
Rule 214
Rule 272
Rule 6471
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {1}{x^2} \, dx}{a}+\int \frac {\sqrt {1+\frac {1}{a^2 x^2}}}{x} \, dx \\ & = -\frac {1}{a x}-\frac {1}{2} \text {Subst}\left (\int \frac {\sqrt {1+\frac {x}{a^2}}}{x} \, dx,x,\frac {1}{x^2}\right ) \\ & = -\sqrt {1+\frac {1}{a^2 x^2}}-\frac {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 ) \\ & = -\sqrt {1+\frac {1}{a^2 x^2}}-\frac {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 ) \\ & = -\sqrt {1+\frac {1}{a^2 x^2}}-\frac {1}{a x}+\text {arctanh}\left (\sqrt {1+\frac {1}{a^2 x^2}}\right ) \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.11 \[ \int \frac {e^{\text {csch}^{-1}(a x)}}{x} \, dx=-\sqrt {1+\frac {1}{a^2 x^2}}-\frac {1}{a x}+\log \left (\left (1+\sqrt {1+\frac {1}{a^2 x^2}}\right ) x\right ) \]
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Leaf count of result is larger than twice the leaf count of optimal. \(108\) vs. \(2(34)=68\).
Time = 0.05 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.87
method | result | size |
default | \(-\frac {\sqrt {\frac {a^{2} x^{2}+1}{a^{2} x^{2}}}\, \left (a^{2} \left (\frac {a^{2} x^{2}+1}{a^{2}}\right )^{\frac {3}{2}}-\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\, a^{2} x^{2}-\ln \left (x +\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}\right ) x \right )}{\sqrt {\frac {a^{2} x^{2}+1}{a^{2}}}}-\frac {1}{a x}\) | \(109\) |
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Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.68 \[ \int \frac {e^{\text {csch}^{-1}(a x)}}{x} \, dx=-\frac {a x \log \left (a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} - a x\right ) + a x \sqrt {\frac {a^{2} x^{2} + 1}{a^{2} x^{2}}} + a x + 1}{a x} \]
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Time = 2.27 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.08 \[ \int \frac {e^{\text {csch}^{-1}(a x)}}{x} \, dx=- \frac {a x}{\sqrt {a^{2} x^{2} + 1}} + \operatorname {asinh}{\left (a x \right )} - \frac {1}{a x} - \frac {1}{a x \sqrt {a^{2} x^{2} + 1}} \]
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Time = 0.24 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.42 \[ \int \frac {e^{\text {csch}^{-1}(a x)}}{x} \, dx=-\sqrt {\frac {1}{a^{2} x^{2}} + 1} - \frac {1}{a x} + \frac {1}{2} \, \log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} + 1\right ) - \frac {1}{2} \, \log \left (\sqrt {\frac {1}{a^{2} x^{2}} + 1} - 1\right ) \]
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Exception generated. \[ \int \frac {e^{\text {csch}^{-1}(a x)}}{x} \, dx=\text {Exception raised: TypeError} \]
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Time = 5.16 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.89 \[ \int \frac {e^{\text {csch}^{-1}(a x)}}{x} \, dx=\mathrm {atanh}\left (\sqrt {\frac {1}{a^2\,x^2}+1}\right )-\sqrt {\frac {1}{a^2\,x^2}+1}-\frac {1}{a\,x} \]
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