Integrand size = 10, antiderivative size = 46 \[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\text {sech}^{-1}\left (\sqrt {x}\right )^2-2 \text {sech}^{-1}\left (\sqrt {x}\right ) \log \left (1+e^{2 \text {sech}^{-1}\left (\sqrt {x}\right )}\right )-\operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}\left (\sqrt {x}\right )}\right ) \]
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Time = 0.07 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6416, 5882, 3799, 2221, 2317, 2438} \[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x} \, dx=-\operatorname {PolyLog}\left (2,-e^{2 \text {sech}^{-1}\left (\sqrt {x}\right )}\right )+\text {sech}^{-1}\left (\sqrt {x}\right )^2-2 \text {sech}^{-1}\left (\sqrt {x}\right ) \log \left (e^{2 \text {sech}^{-1}\left (\sqrt {x}\right )}+1\right ) \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5882
Rule 6416
Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {\text {sech}^{-1}(x)}{x} \, dx,x,\sqrt {x}\right ) \\ & = -\left (2 \text {Subst}\left (\int \frac {\text {arccosh}(x)}{x} \, dx,x,\frac {1}{\sqrt {x}}\right )\right ) \\ & = -\left (2 \text {Subst}\left (\int x \tanh (x) \, dx,x,\text {arccosh}\left (\frac {1}{\sqrt {x}}\right )\right )\right ) \\ & = \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )^2-4 \text {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\text {arccosh}\left (\frac {1}{\sqrt {x}}\right )\right ) \\ & = \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )^2-2 \text {arccosh}\left (\frac {1}{\sqrt {x}}\right ) \log \left (1+e^{2 \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )}\right )+2 \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {arccosh}\left (\frac {1}{\sqrt {x}}\right )\right ) \\ & = \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )^2-2 \text {arccosh}\left (\frac {1}{\sqrt {x}}\right ) \log \left (1+e^{2 \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )}\right )+\text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )}\right ) \\ & = \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )^2-2 \text {arccosh}\left (\frac {1}{\sqrt {x}}\right ) \log \left (1+e^{2 \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )}\right )-\operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}\left (\frac {1}{\sqrt {x}}\right )}\right ) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.98 \[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x} \, dx=-\text {sech}^{-1}\left (\sqrt {x}\right ) \left (\text {sech}^{-1}\left (\sqrt {x}\right )+2 \log \left (1+e^{-2 \text {sech}^{-1}\left (\sqrt {x}\right )}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}\left (\sqrt {x}\right )}\right ) \]
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Time = 0.47 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.41
method | result | size |
derivativedivides | \(\operatorname {arcsech}\left (\sqrt {x}\right )^{2}-2 \,\operatorname {arcsech}\left (\sqrt {x}\right ) \ln \left (1+\left (\frac {1}{\sqrt {x}}+\sqrt {-1+\frac {1}{\sqrt {x}}}\, \sqrt {1+\frac {1}{\sqrt {x}}}\right )^{2}\right )-\operatorname {polylog}\left (2, -\left (\frac {1}{\sqrt {x}}+\sqrt {-1+\frac {1}{\sqrt {x}}}\, \sqrt {1+\frac {1}{\sqrt {x}}}\right )^{2}\right )\) | \(65\) |
default | \(\operatorname {arcsech}\left (\sqrt {x}\right )^{2}-2 \,\operatorname {arcsech}\left (\sqrt {x}\right ) \ln \left (1+\left (\frac {1}{\sqrt {x}}+\sqrt {-1+\frac {1}{\sqrt {x}}}\, \sqrt {1+\frac {1}{\sqrt {x}}}\right )^{2}\right )-\operatorname {polylog}\left (2, -\left (\frac {1}{\sqrt {x}}+\sqrt {-1+\frac {1}{\sqrt {x}}}\, \sqrt {1+\frac {1}{\sqrt {x}}}\right )^{2}\right )\) | \(65\) |
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\[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\int { \frac {\operatorname {arsech}\left (\sqrt {x}\right )}{x} \,d x } \]
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\[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\int \frac {\operatorname {asech}{\left (\sqrt {x} \right )}}{x}\, dx \]
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\[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\int { \frac {\operatorname {arsech}\left (\sqrt {x}\right )}{x} \,d x } \]
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\[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\int { \frac {\operatorname {arsech}\left (\sqrt {x}\right )}{x} \,d x } \]
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Timed out. \[ \int \frac {\text {sech}^{-1}\left (\sqrt {x}\right )}{x} \, dx=\int \frac {\mathrm {acosh}\left (\frac {1}{\sqrt {x}}\right )}{x} \,d x \]
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