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 = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 12
Rule 37
Rule 5902
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*}
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Mathematica [A] time = 0.01, size = 26, normalized size = 1.00 \[ -\frac {\sqrt {x+1}}{\sqrt {x}}-\frac {\sinh ^{-1}\left (\sqrt {x}\right )}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 25, normalized size = 0.96 \[ -\frac {\sqrt {x + 1} \sqrt {x} + \log \left (\sqrt {x + 1} + \sqrt {x}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 35, normalized size = 1.35 \[ -\frac {\log \left (\sqrt {x + 1} + \sqrt {x}\right )}{x} + \frac {2}{{\left (\sqrt {x + 1} - \sqrt {x}\right )}^{2} - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 21, normalized size = 0.81 \[ -\frac {\arcsinh \left (\sqrt {x}\right )}{x}-\frac {\sqrt {1+x}}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 20, normalized size = 0.77 \[ -\frac {\sqrt {x + 1}}{\sqrt {x}} - \frac {\operatorname {arsinh}\left (\sqrt {x}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {asinh}\left (\sqrt {x}\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asinh}{\left (\sqrt {x} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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