Optimal. Leaf size=40 \[ \frac{\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{\sqrt{x}}-\frac{\cosh ^{-1}\left (\sqrt{x}\right )}{x} \]
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Rubi [A] time = 0.0256731, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5903, 12, 265} \[ \frac{\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{\sqrt{x}}-\frac{\cosh ^{-1}\left (\sqrt{x}\right )}{x} \]
Antiderivative was successfully verified.
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Rule 5903
Rule 12
Rule 265
Rubi steps
\begin{align*} \int \frac{\cosh ^{-1}\left (\sqrt{x}\right )}{x^2} \, dx &=-\frac{\cosh ^{-1}\left (\sqrt{x}\right )}{x}+\int \frac{1}{2 \sqrt{-1+\sqrt{x}} \sqrt{1+\sqrt{x}} x^{3/2}} \, dx\\ &=-\frac{\cosh ^{-1}\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{\cosh ^{-1}\left (\sqrt{x}\right )}{x}\\ \end{align*}
Mathematica [A] time = 0.0135517, size = 40, normalized size = 1. \[ \frac{\sqrt{\sqrt{x}-1} \sqrt{\sqrt{x}+1}}{\sqrt{x}}-\frac{\cosh ^{-1}\left (\sqrt{x}\right )}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 29, normalized size = 0.7 \begin{align*} -{\frac{1}{x}{\rm arccosh} \left (\sqrt{x}\right )}+{\sqrt{-1+\sqrt{x}}\sqrt{1+\sqrt{x}}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54244, size = 26, normalized size = 0.65 \begin{align*} \frac{\sqrt{x - 1}}{\sqrt{x}} - \frac{\operatorname{arcosh}\left (\sqrt{x}\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17386, size = 73, normalized size = 1.82 \begin{align*} \frac{\sqrt{x - 1} \sqrt{x} - \log \left (\sqrt{x - 1} + \sqrt{x}\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acosh}{\left (\sqrt{x} \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12095, size = 61, normalized size = 1.52 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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