Integrand size = 4, antiderivative size = 21 \[ \int \text {sech}^{-1}\left (\frac {1}{x}\right ) \, dx=-\sqrt {-1+x} \sqrt {1+x}+x \text {arccosh}(x) \]
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Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6462, 5879, 75} \[ \int \text {sech}^{-1}\left (\frac {1}{x}\right ) \, dx=x \text {arccosh}(x)-\sqrt {x-1} \sqrt {x+1} \]
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Rule 75
Rule 5879
Rule 6462
Rubi steps \begin{align*} \text {integral}& = \int \text {arccosh}(x) \, dx \\ & = x \text {arccosh}(x)-\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx \\ & = -\sqrt {-1+x} \sqrt {1+x}+x \text {arccosh}(x) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \text {sech}^{-1}\left (\frac {1}{x}\right ) \, dx=-\sqrt {\frac {-1+x}{1+x}} (1+x)+x \text {sech}^{-1}\left (\frac {1}{x}\right ) \]
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Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95
method | result | size |
parts | \(x \,\operatorname {arcsech}\left (\frac {1}{x}\right )-\sqrt {x -1}\, \sqrt {1+x}\) | \(20\) |
derivativedivides | \(x \,\operatorname {arcsech}\left (\frac {1}{x}\right )-\sqrt {-\left (-1+\frac {1}{x}\right ) x}\, \sqrt {\left (1+\frac {1}{x}\right ) x}\) | \(29\) |
default | \(x \,\operatorname {arcsech}\left (\frac {1}{x}\right )-\sqrt {-\left (-1+\frac {1}{x}\right ) x}\, \sqrt {\left (1+\frac {1}{x}\right ) x}\) | \(29\) |
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none
Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \text {sech}^{-1}\left (\frac {1}{x}\right ) \, dx=x \log \left (x + \sqrt {x^{2} - 1}\right ) - \sqrt {x^{2} - 1} \]
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\[ \int \text {sech}^{-1}\left (\frac {1}{x}\right ) \, dx=\int \operatorname {asech}{\left (\frac {1}{x} \right )}\, dx \]
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none
Time = 0.22 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \text {sech}^{-1}\left (\frac {1}{x}\right ) \, dx=x \operatorname {arsech}\left (\frac {1}{x}\right ) - \sqrt {x^{2} - 1} \]
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\[ \int \text {sech}^{-1}\left (\frac {1}{x}\right ) \, dx=\int { \operatorname {arsech}\left (\frac {1}{x}\right ) \,d x } \]
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Time = 0.07 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int \text {sech}^{-1}\left (\frac {1}{x}\right ) \, dx=x\,\mathrm {acosh}\left (x\right )-\sqrt {x-1}\,\sqrt {x+1} \]
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