Optimal. Leaf size=26 \[ -\frac{\sqrt{x+1}}{\sqrt{x}}-\frac{\sinh ^{-1}\left (\sqrt{x}\right )}{x} \]
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Rubi [A] time = 0.0131465, antiderivative size = 26, 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 = {5902, 12, 37} \[ -\frac{\sqrt{x+1}}{\sqrt{x}}-\frac{\sinh ^{-1}\left (\sqrt{x}\right )}{x} \]
Antiderivative was successfully verified.
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Rule 5902
Rule 12
Rule 37
Rubi steps
\begin{align*} \int \frac{\sinh ^{-1}\left (\sqrt{x}\right )}{x^2} \, dx &=-\frac{\sinh ^{-1}\left (\sqrt{x}\right )}{x}+\int \frac{1}{2 x^{3/2} \sqrt{1+x}} \, dx\\ &=-\frac{\sinh ^{-1}\left (\sqrt{x}\right )}{x}+\frac{1}{2} \int \frac{1}{x^{3/2} \sqrt{1+x}} \, dx\\ &=-\frac{\sqrt{1+x}}{\sqrt{x}}-\frac{\sinh ^{-1}\left (\sqrt{x}\right )}{x}\\ \end{align*}
Mathematica [A] time = 0.0092362, size = 26, normalized size = 1. \[ -\frac{\sqrt{x+1}}{\sqrt{x}}-\frac{\sinh ^{-1}\left (\sqrt{x}\right )}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 21, normalized size = 0.8 \begin{align*} -{\frac{1}{x}{\it Arcsinh} \left ( \sqrt{x} \right ) }-{\sqrt{1+x}{\frac{1}{\sqrt{x}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.61248, size = 27, normalized size = 1.04 \begin{align*} -\frac{\sqrt{x + 1}}{\sqrt{x}} - \frac{\operatorname{arsinh}\left (\sqrt{x}\right )}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.66803, size = 74, normalized size = 2.85 \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{asinh}{\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 = 1.21315, size = 47, normalized size = 1.81 \begin{align*} -\frac{\log \left (\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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