Integrand size = 10, antiderivative size = 40 \[ \int \frac {\text {arccosh}\left (\sqrt {x}\right )}{x^2} \, dx=\frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{\sqrt {x}}-\frac {\text {arccosh}\left (\sqrt {x}\right )}{x} \]
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Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6017, 12, 271} \[ \int \frac {\text {arccosh}\left (\sqrt {x}\right )}{x^2} \, dx=\frac {\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{\sqrt {x}}-\frac {\text {arccosh}\left (\sqrt {x}\right )}{x} \]
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Rule 12
Rule 271
Rule 6017
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arccosh}\left (\sqrt {x}\right )}{x}+\int \frac {1}{2 \sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}} \, dx \\ & = -\frac {\text {arccosh}\left (\sqrt {x}\right )}{x}+\frac {1}{2} \int \frac {1}{\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}} x^{3/2}} \, dx \\ & = \frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{\sqrt {x}}-\frac {\text {arccosh}\left (\sqrt {x}\right )}{x} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arccosh}\left (\sqrt {x}\right )}{x^2} \, dx=\frac {\sqrt {-1+\sqrt {x}} \sqrt {1+\sqrt {x}}}{\sqrt {x}}-\frac {\text {arccosh}\left (\sqrt {x}\right )}{x} \]
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Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(-\frac {\operatorname {arccosh}\left (\sqrt {x}\right )}{x}+\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}}{\sqrt {x}}\) | \(29\) |
default | \(-\frac {\operatorname {arccosh}\left (\sqrt {x}\right )}{x}+\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}}{\sqrt {x}}\) | \(29\) |
parts | \(-\frac {\operatorname {arccosh}\left (\sqrt {x}\right )}{x}+\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}}{\sqrt {x}}\) | \(29\) |
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Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.65 \[ \int \frac {\text {arccosh}\left (\sqrt {x}\right )}{x^2} \, dx=\frac {\sqrt {x - 1} \sqrt {x} - \log \left (\sqrt {x - 1} + \sqrt {x}\right )}{x} \]
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\[ \int \frac {\text {arccosh}\left (\sqrt {x}\right )}{x^2} \, dx=\int \frac {\operatorname {acosh}{\left (\sqrt {x} \right )}}{x^{2}}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.48 \[ \int \frac {\text {arccosh}\left (\sqrt {x}\right )}{x^2} \, dx=\frac {\sqrt {x - 1}}{\sqrt {x}} - \frac {\operatorname {arcosh}\left (\sqrt {x}\right )}{x} \]
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Time = 0.48 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.12 \[ \int \frac {\text {arccosh}\left (\sqrt {x}\right )}{x^2} \, dx=-\frac {\log \left (\sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \sqrt {x}\right )}{x} + \frac {2}{{\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} + 1} \]
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Timed out. \[ \int \frac {\text {arccosh}\left (\sqrt {x}\right )}{x^2} \, dx=\int \frac {\mathrm {acosh}\left (\sqrt {x}\right )}{x^2} \,d x \]
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