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 = 0.03, antiderivative size = 40, 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 = {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 12
Rule 265
Rule 5903
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*}
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Mathematica [A] time = 0.02, size = 40, normalized size = 1.00 \[ \frac {\sqrt {\sqrt {x}-1} \sqrt {\sqrt {x}+1}}{\sqrt {x}}-\frac {\cosh ^{-1}\left (\sqrt {x}\right )}{x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.58, size = 26, normalized size = 0.65 \[ \frac {\sqrt {x - 1} \sqrt {x} - \log \left (\sqrt {x - 1} + \sqrt {x}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.69, size = 45, normalized size = 1.12 \[ -\frac {\log \left (\sqrt {\sqrt {x} + 1} \sqrt {\sqrt {x} - 1} + \sqrt {x}\right )}{x} + \frac {2}{{\left (\sqrt {x - 1} - \sqrt {x}\right )}^{2} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 29, normalized size = 0.72 \[ -\frac {\mathrm {arccosh}\left (\sqrt {x}\right )}{x}+\frac {\sqrt {-1+\sqrt {x}}\, \sqrt {1+\sqrt {x}}}{\sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.30, size = 19, normalized size = 0.48 \[ \frac {\sqrt {x - 1}}{\sqrt {x}} - \frac {\operatorname {arcosh}\left (\sqrt {x}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {acosh}\left (\sqrt {x}\right )}{x^2} \,d x \]
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 \[ \int \frac {\operatorname {acosh}{\left (\sqrt {x} \right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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