Integrand size = 12, antiderivative size = 46 \[ \int \frac {e^{-\text {sech}^{-1}(a x)}}{x} \, dx=-\frac {2}{1+\sqrt {\frac {1-a x}{1+a x}}}-2 \arctan \left (\sqrt {\frac {1-a x}{1+a x}}\right ) \]
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Time = 0.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6472, 815, 209} \[ \int \frac {e^{-\text {sech}^{-1}(a x)}}{x} \, dx=-2 \arctan \left (\sqrt {\frac {1-a x}{a x+1}}\right )-\frac {2}{\sqrt {\frac {1-a x}{a x+1}}+1} \]
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Rule 209
Rule 815
Rule 6472
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x \left (\frac {1}{a x}+\sqrt {\frac {1-a x}{1+a x}}+\frac {\sqrt {\frac {1-a x}{1+a x}}}{a x}\right )} \, dx \\ & = -\left (4 \text {Subst}\left (\int \frac {x}{(1+x)^2 \left (1+x^2\right )} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\right ) \\ & = -\left (4 \text {Subst}\left (\int \left (-\frac {1}{2 (1+x)^2}+\frac {1}{2 \left (1+x^2\right )}\right ) \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right )\right ) \\ & = -\frac {2}{1+\sqrt {\frac {1-a x}{1+a x}}}-2 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {\frac {1-a x}{1+a x}}\right ) \\ & = -\frac {2}{1+\sqrt {\frac {1-a x}{1+a x}}}-2 \arctan \left (\sqrt {\frac {1-a x}{1+a x}}\right ) \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.04 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.61 \[ \int \frac {e^{-\text {sech}^{-1}(a x)}}{x} \, dx=-\frac {1}{a x}+\left (1+\frac {1}{a x}\right ) \sqrt {\frac {1-a x}{1+a x}}+i \log \left (-2 i a x+2 \sqrt {\frac {1-a x}{1+a x}} (1+a x)\right ) \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.89 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.09
method | result | size |
default | \(a \left (-\frac {1}{x \,a^{2}}+\frac {\sqrt {-\frac {a x -1}{a x}}\, \sqrt {\frac {a x +1}{a x}}\, \left (\arctan \left (\frac {\operatorname {csgn}\left (a \right ) a x}{\sqrt {-a^{2} x^{2}+1}}\right ) a x +\operatorname {csgn}\left (a \right ) \sqrt {-a^{2} x^{2}+1}\right ) \operatorname {csgn}\left (a \right )}{a \sqrt {-a^{2} x^{2}+1}}\right )\) | \(96\) |
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Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.65 \[ \int \frac {e^{-\text {sech}^{-1}(a x)}}{x} \, dx=\frac {a x \sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}} - a x \arctan \left (\sqrt {\frac {a x + 1}{a x}} \sqrt {-\frac {a x - 1}{a x}}\right ) - 1}{a x} \]
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\[ \int \frac {e^{-\text {sech}^{-1}(a x)}}{x} \, dx=a \int \frac {1}{a x \sqrt {-1 + \frac {1}{a x}} \sqrt {1 + \frac {1}{a x}} + 1}\, dx \]
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\[ \int \frac {e^{-\text {sech}^{-1}(a x)}}{x} \, dx=\int { \frac {1}{x {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}} \,d x } \]
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\[ \int \frac {e^{-\text {sech}^{-1}(a x)}}{x} \, dx=\int { \frac {1}{x {\left (\sqrt {\frac {1}{a x} + 1} \sqrt {\frac {1}{a x} - 1} + \frac {1}{a x}\right )}} \,d x } \]
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Time = 7.48 (sec) , antiderivative size = 184, normalized size of antiderivative = 4.00 \[ \int \frac {e^{-\text {sech}^{-1}(a x)}}{x} \, dx=\ln \left (\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}+1\right )\,1{}\mathrm {i}-\ln \left (\frac {\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}}{\sqrt {\frac {1}{a\,x}+1}-1}\right )\,1{}\mathrm {i}-\frac {1}{a\,x}-\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2\,8{}\mathrm {i}}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2\,\left (1+\frac {{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^4}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^4}-\frac {2\,{\left (\sqrt {\frac {1}{a\,x}-1}-\mathrm {i}\right )}^2}{{\left (\sqrt {\frac {1}{a\,x}+1}-1\right )}^2}\right )} \]
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